Cell Theory of Nature

CELL THEORY OF NATURE

Work in process

David Ritz Finkelstein1

October 17, 2008

1Physics, Georgia Institute of Technology, Atlanta, Georgia. ﬁ[email protected] 2

Time is the number of motion with regard to before and after. Aristotle, Physics

I would indeed admit these inﬁnitely small spaces and times in geometry, for the sake of invention, even if they are imaginary. But I am not sure whether they can be admitted in nature. G. W. Leibniz [3]

It is, for example, true that the result of two successive acts is unaffected by the order in which they are performed; and there are at least two other laws which will be pointed out in the proper place. These will perhaps to some appear so obvious as to be ranked among necessary truths, and so little important as to be undeserving of special notice. And probably they are noticed for the first time in this Essay. Yet it may with confidence be asserted, that if they were other than they are, the entire mechanism of reasoning, nay the very laws and constitution of the human intellect, would be vitally changed. A Logic might indeed exist, but it would no longer be the Logic we possess. G. Boole [14]

To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is, to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day ﬁnd methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space. A. Einstein [29]

And so I suggested to myself that electrons cannot act on themselves; they can only act on other electrons. This means that there is no ﬁeld at all. R. P. Feynman [34] Contents

1 Strata of actuality 5 1.1 Atoms of atoms...... 5 1.1.1 Simplicity and stability ...... 7 1.1.2 Locality ...... 8 1.1.3 Swaps ...... 10 1.1.4 Beneath geometry ...... 11 1.1.5 The cosmic crystal film ...... 13 1.1.6 Praxics ...... 15 1.1.7 Indefinite probability forms ...... 17 1.1.8 Probability vector spaces and algebras ...... 17 1.1.9 The need for full quantization ...... 18 1.1.10 A cellular hypothesis ...... 19 1.1.11 Quantum time ...... 23 1.2 The idea of the queue ...... 26 1.2.1 Strata ...... 29 1.2.2 The Clifford algebras of Fermi and Dirac ...... 36 1.2.3 The origins of i ...... 37 1.2.4 The origins of H ...... 38 1.2.5 Notation ...... 40 1.2.6 Q terminology ...... 41 1.2.7 The origins of g ...... 42 1.2.8 Fields and queues ...... 44 1.2.9 Dynamical law of the queue ...... 45 1.2.10 The vacuum queue ...... 48 1.2.11 The cosmic crystal ...... 49 1.3 Quantization ...... 51 1.3.1 Canonical quantization ...... 53 1.3.2 Full quantization strategy ...... 55 1.3.3 Regularization ...... 58 1.3.4 Quantification ...... 59

3 4 CONTENTS

1.3.5 Cellularization ...... 59 1.3.6 The choice of statistics ...... 60 1.3.7 Internal structure of the photon and graviton ...... 62 1.3.8 Indefinite forms ...... 63 1.3.9 The representation of symmetry ...... 64 1.3.10 Finiteness ...... 64 1.4 The groups of nature ...... 66 1.4.1 Canonical strata ...... 68 1.4.2 Metric forms ...... 69 1.4.3 Fully quantum strata ...... 69 1.4.4 Fully quantum self-organization ...... 70 1.4.5 Full quantization tactics ...... 71 1.5 Fully quantum regularization ...... 74 1.5.1 Fully quantum dynamics ...... 74 1.5.2 Gauge ...... 74 1.5.3 Spin ...... 78 1.5.4 Real quantum theory ...... 79 1.5.5 Root vectors and quanta ...... 80 1.5.6 Full Fermi quantization ...... 81 1.5.7 Physical Lie algebras ...... 82 1.6 Gravity and other gauge fields ...... 83 1.6.1 History as quantum variable ...... 84 1.6.2 Fully quantum equivalence principle ...... 85 1.6.3 Weyl gauge strategy ...... 88 1.6.4 Kaluza gauge strategy ...... 89 1.6.5 Queue gauge strategy ...... 90 1.6.6 Fully quantum gauge group ...... 91 1.6.7 The space-time truss ...... 92 1.6.8 The gravitational and gauge potentials ...... 93 1.6.9 Vacuum ...... 94 1.7 Unifications ...... 95 1.7.1 Being and becoming ...... 95 1.7.2 Gravity and quantum theory ...... 96 1.7.3 Products ...... 96 1.7.4 Non-commutativity and granularity ...... 97 1.7.5 System and metasystem ...... 98 1.8 Outline ...... 99 CONTENTS 5

2 Linear praxics 101 2.1 Praxics in general ...... 101 2.2 Heisenberg and von Neumann praxics ...... 102 2.2.1 The system itself ...... 105 2.2.2 The equatorial bulge in Hilbert space ...... 106 2.2.3 Commutative reduction ...... 107 2.3 Standard semantics ...... 107 2.3.1 The orthogonal group ...... 108 2.3.2 Probability vectors ...... 108 2.3.3 The probability form ...... 110 2.3.4 The linear operators ...... 110 2.3.5 The projectors ...... 111 2.4 Change is a quantum eﬀect ...... 112 2.5 Simple systems ...... 112 2.6 Probabilities ...... 113 2.7 Mixtures ...... 113 2.8 Transformations ...... 114 2.9 States, proper and coordinate ...... 115 2.9.0.1 Proper state ...... 116 2.9.0.2 Coordinate state ...... 116 2.10 Praxiology and its singular limit ...... 116 2.10.1 Ritz combination rule...... 122 2.10.2 Probability Principle...... 122 2.10.3 System catenation ...... 123 2.10.4 Relation to probability ...... 123 2.10.5 The von Neumann ambiguity ...... 124 2.10.6 Schr¨odinger’s frozen cat ...... 132 2.10.7 Fundamental law ...... 134

3 Polynomial quantum logic 137 3.1 Set algebras ...... 137 3.1.1 The random set ...... 137 3.1.2 The queue ...... 138 3.1.3 Input-output processes ...... 139 3.2 Clifford algebra ...... 141 3.2.1 Clifford semantics ...... 142 3.2.2 Fermi Clifford algebra ...... 142 3.2.3 Fermi vectors ...... 143 3.2.4 Grade operator ...... 144 3.2.5 Mean-square form ...... 144 3.3 Quantification ...... 145 6 CONTENTS

3.3.1 Choosing a quantification ...... 145 3.3.2 The cumulator ...... 147 3.3.3 Algebra unification ...... 149 3.3.4 Spinor spaces ...... 149 3.4 Fermi algebra ...... 150 3.4.1 The duplex space ...... 152 3.4.2 A spin is a queue of semiquanta...... 153 3.4.3 Spin-statistics correlation ...... 155 3.5 Clifford statistics ...... 158 3.5.1 Spin-statistics anomaly ...... 159 3.6 Measurement ...... 160

4 Exponential quantum logics 163 4.1 Spinors ...... 163 4.1.1 Baugh numbers ...... 164 4.1.2 Random sets ...... 165 4.1.3 Quantum cells ...... 165 4.1.4 Bracing ...... 166 4.1.5 Critique of the brace operation ...... 168 4.1.6 Queues ...... 170 4.2 Simplifying quantization ...... 171 4.2.1 Choose a simple Lie algebra ...... 172 4.2.2 Choose a vacuum organization ...... 172 4.2.3 Choose a faithful irreducible representation ...... 172 4.3 Fermi full quantization ...... 174 4.4 Fully quantum vacuum organization ...... 175 4.4.1 Stratum assignments ...... 176

5 Quantum space-times 179 5.1 Problems of classical space-time ...... 182 5.1.1 Structural instability of time ...... 182 5.1.2 Dynamical instability of time ...... 182 5.2 Earlier quantum space-times ...... 184 5.2.1 Event energy ...... 184 5.2.2 Indeﬁnite probability form ...... 186 5.2.3 Feynman space ...... 186 5.2.4 Snyder space ...... 189 5.2.5 Segal space ...... 189 5.2.6 Penrose space ...... 191 5.2.7 Palev statistics ...... 191 5.2.8 Vilela-Mendes space ...... 193 CONTENTS 7

5.2.9 Baugh, Shiri-Garakani spaces ...... 194 5.3 Fully quantum event spaces ...... 194 5.3.1 Fully quantum spaces ...... 196 5.3.2 A fully quantum space-time ...... 197

6 Fully quantum kinematics 201 6.1 History space ...... 201 6.1.1 Canonical quantum histories ...... 202 6.1.2 Fully quantum histories ...... 203 6.1.3 Physics without functions ...... 203 6.1.4 Field variables ...... 204 6.1.5 The cosmic crystal ...... 206 6.1.6 The organization of the imaginary unit ...... 208 6.1.7 Statistical form ...... 208 6.2 Fully quantum scattering ...... 210 6.2.1 Experiment time ...... 212 6.2.2 Momentum vectors ...... 212 6.2.3 Quantum topology ...... 216 6.2.4 Origins of the coordinates ...... 217 6.2.5 Fully quantum fermions ...... 217 6.3 Fully quantum gravity ...... 219 6.3.1 Fully quantum events ...... 219 6.3.2 Time form ...... 220 6.3.3 Fully quantum gravitational potentials ...... 221 6.4 Construction of the vacuum ...... 223 6.4.1 Bosonization ...... 227 6.4.2 Antiparticles ...... 228 6.4.3 Flavor ...... 228 6.5 Fully quantum gauge theories ...... 228 6.5.1 Fully quantum gauging ...... 229 6.6 Fully quantum metrics ...... 231 6.6.1 Fully quantum probability forms ...... 232 6.6.2 Fully quantum causality form ...... 233 6.7 Covariant diﬀerentiator ...... 237 6.8 Reciprocity ...... 238 6.9 Fully quantum covariance ...... 239

7 Fully quantum dynamics 243 7.1 The history vector ...... 243 7.2 Higher-order time derivatives ...... 245 7.3 Spinorial dynamics ...... 245 8 CONTENTS

7.4 Gravity action ...... 246 7.4.1 Cosmological constant ...... 246

8 Output 249

9 ACKNOWLEDGMENT 251

BIBLIOGRAPHY 251

INDEX 258 CONTENTS 9

THINQ

Think quantum. 10 CONTENTS Chapter 1

Strata of actuality

1.1 Atoms of atoms.

The Atomic Theory of Matter and the Cell Theory of Life are now well established, Leib- niz’s monadology and Whitehead’s philosophy of organism are cellular, and Von Neumann formulated a cellular automaton model of reproduction, but an atomic or cellular theory of nature as a whole is still nascent. Here I [hopefully] develop such a theory to the point where it can accommodate the current gauge groups and predict particles and their interactions. Two deep but widespread philosophical reifications seem to have unduly blocked progress in physical understanding since the discoveries of general relativity and the quantum theory, preventing their synthesis within one more comprehensive theory. One is the postulation of absolute space-time, shared by Einstein and Heisenberg, which can only be corrected by a grand extension of relativity. The other is the inconsistent but common formulation of quantum theory expressed in the term “state vector”, which blocks the required relativization. Here the common formulation is replaced by one closer to Heisenberg’s and Bohr’s, which it is then natural to apply to space-time. The formulation is purely algebraic, as Einstein suggested, and so in principle it is simpler than the usual mix of classical differential geometry and quantum matrix algebra, but the algebra has an unfamiliar structural feature: Its elements are not only graded but ranked. The algebra is stratified into nested subalgebras. An initial chapter presents the main ideas in familiar language, in preparation for a more formal development in later chapters. So everything is said at least three times. An important defect in the current quantum theories is the apparently seamless continuity of time, which is representative of several continuities of special relativity general relativity, and the standard model. This leads to unphysical infinities in the predictions of quantum field theory and to an unphysical singularity at the core of black holes. Infinity in, infinity out.

11 12 CHAPTER 1. STRATA OF ACTUALITY

It is understood that time is what clocks meter out. This understanding then permits the question, What is time? What do clocks meter out? The ostensive definition gives meaning to the structure question. Naturally the continuity of time has no direct experimental support. The absence of evidence for time jumps is not evidence of their non-existence. The opposite possibility, discrete time, implies a breakdown of energy conservation for which there is also no evidence. This possibility is not further contemplated here. In the past, discarding continuous symmetries has not led to improved theories, while small variations in the structure of the symmetry groups has. Therefore we follow that avenue here. We synthesize the continuous and discrete concepts of time in the sense that quantum mechanics synthesizes wave and particle. We quantize time, rather than merely discretize it. This is one way to a finite physics of the domain presently described by quantum field theory and general relativity. A molecular theory of water resolves the singularity predicted at the core of vortices by continuum hydrodynamics of an ideal non-viscous fluid. A cellular theory of the event space does the same for the gravitational singularities. Dynamics operates on a higher logical stratum than time in present physical theory. Dynamical variables are functions of the time variable and not conversely. This made it possible to quantize dynamical variables without quantizing the time variable, which remained singular. To remedy this omission we make the stratification of physics more explicit, formulating a stratified algebra to express it. Canonical quantization removed an infinity that resulted from the continuity of energy in classical physics, while respecting the continuous symmetries of the Lorentz and Poincaré groups. Canonical quantization repaired only the dynamical stratum, however, and that only partially, quantizing some quantities and not others, and it preserved the singularities of the lower strata intact. Many concluded that a fuller quantization should eliminate the universal continuity of time as canonical quantization eliminated the universal continuity of energy. One impetus for the present effort is to eliminate infinities resulting from continuity while respecting all empirical continuous symmetries. But in fact the project took root when an encounter with the ideas of von Neumann made it clear that the logic that Euclid used for points was but a singular limiting approximation to the quantum mode of reasoning that Heisenberg and Bohr used for atoms. It seemed clear that in the long run the quantum mode of reasoning must be used for all strata of physics. For this purpose a stratified algebra and quantum theory are constructed. Graded algebras are familiar tools of quantum mechanics. A stratified algebra has both grade and rank. Von Neumann proposed to regard quantum theory as a reform of Boolean logics that replaced distributive by non-distributive lattices. Boole himself favored an algebraic rather than a lattice formulation, and Heisenberg reformed it by replacing a commutative algebra by a non-commutatice one. The algebraic road is the only one that has led to new physical 1.1. ATOMS OF ATOMS. 13 theories, and it is followed here. A cellular logical engine that operates according to a stratified finite-dimensional quantum logic would be finite on every stratum and still have continuous invariance groups like those seen in nature. This motivated the earliest phase of this project, though neither von Neumann’s logics nor his automaton theory are prominent in the product.

1.1.1 Simplicity and stability The quantum theory is often described in negative terms: It is non-deterministic and non- commutative. Here we turn to the positive side of the ledger. What did physics gain by this “painful renunciation”? (Bohr’s term.) Granted that physics is headed away from classical completeness and determinism, but towards what? Simplicity, it is supposed here. The history of physics in the 20th century has suggested to some that physics is on a slow climb from searching for a bcompletec and final theory to searching for a bsimplec and provisory one. This refers especially to Lie algebraic structure of our theories. A bsimplec (or birreduciblec) structure in general is one with no proper invariant substructures; that is, none but the trivial two, itself and the empty structure. A simple Lie algebra, however, is required to be non-abelian as well as having no proper invariant Lie subalgebras; this excludes the 1-dimensional case. A bsemisimplec, or decomposable, structure is a a union of simple structures. A structure that is not semisimple is bcompoundc.A bsimplification strategyc is at least implied by bSegalc [66]. His principle is stretched here to accommodate the Pauli exclusion principle: Physical systems have simple graded Lie algebras. This suggests that a compound graded Lie algebra that works is only an approximation to a simple one that works better. The strategy that follows is to make all the graded Lie algebras that enter in the formulation of a physical theory simple by a variation of their constants. An arbitrarily small variation will always suffice, provided that constraints, attributable to organization, freeze out some variables. We adopt this in the following but it does not go far enough; the problem it poses it is still seriously undetermined. It is sharpened in two stages. First §1.4.5 gives a fully quantum (= Q; q = canonical quantization, in all its variant forms) kinematics. To determine dynamics, Q correspondents of the familiar principles of bminimal couplingc and bminimal differential orderc — which can be understood as maximal locality — are formulated and applied to the existing theories of gravity and the standard model. Group simplicity has immediate consequences for structural stability of the theory [66]. This is defined as follows in the present context. The various Lie algebras that can be defined on a fixed vector space have structure tensors that form a certain curved algebraic bstructure manifoldc in an associated tensor space. An algebra with a neighborhood in the structure manifold composed of isomorphic algebras is said to be bstructurally stablec, bregularc, bgenericc, brobustc, or brigidc. If on 14 CHAPTER 1. STRATA OF ACTUALITY the contrary every neighborhood of a given algebra contains a non-isomorphic algebra, the given algebra is bstructurally unstablec, bspecialc, bsingularc, or bfragilec. Semisimple Lie algebras are structurally stable and many bcompoundc Lie algebras are not. Any unstable Lie algebra has structurally stable ones in every neighborhood [66]. Here we suppose that one of these is more physical than the unstable one.

1.1.2 Locality Locality has become an indispensable guide for theoretical physics since Newton used it to criticize his own law of gravity. The standard model freezes the gravitational ﬁeld at the zero value it has in special relativity, but draws nevertheless on the insights of New- ton, Faraday, and Einstein concerning dynamical locality of gravity and electromagnetism. extending them to hypercharge, electroweak, and strong interactions:

Assumption 1 (Locality) All action is by contact.

In the cellular quantum theory developed here, two cells are defined to be in contact when they share vertices, elements of a lower stratum. Then blocalityc restricts them to interaction by bcontactc. Locality is imposed here by requiring the action operator to be a polynomial of low degree in swap operations. Einstein’s law of gravity is the simplest localization of Newton’s. To take locality as seriously as quantum theory permits, one should attribute all global symmetries to organizations of local structures by local interactions. It is possible, to be sure, to have maximal information about a quantum particle and not know where it is localized. Canonical quantum theory does not modify dynamical locality when it introduces this kinematic non-locality. Faraday incorporated this locality in his pre-quantum concept of field, Maxwell and Einstein developed the field concept into their pre-quantum theories, and Dirac developed it into his quantum theories. The notion of local gauge invariance, the basis of the theories of gravity and the standard model, is the peak of locality. Here a quantum reform of canonical gauge theories is carried out while preserving the locality principle. The gauge group arises here from the invariance group of a cell of a stratum beneath that of space-time. Quantum field theory was derived from a classical theory by bcanonical quantizationc. This was a reasonable first step, but there are signs that the journey has just begun.

Canonical quantization is a partial quantization.

It is lacking in both horizontal and vertical dimensions, in the following senses. In the first place, as was pointed out, canonical quantization leaves deeper strata unquantized. Classical physics is stratified, in that it can be built up in strata of increasing complexity, also called levels, stages, generations, phases, types, and orders. The medium usually used to represent such stratification is set theory. Sets of each stratum are made 1.1. ATOMS OF ATOMS. 15 up of elements of lower strata and are the elements of sets of higher strata, forming a great ladder of beings. The number of rungs below a set on this ladder is called its brankc here. Stratum L, or S[L], consists of the sets of rank ≤ L. Classical field theories, for example, represent a field as a continuous bfunctionc on space-time to field values, and represent space-time as a set of events. Therefore a field belongs to a higher bstratumc than the space-time event and the field value, which are elements of its elements, or its second elements. It is assumed here that nature too is stratified in this way. The constructs of bstratumc and brankc are therefore basic for this discussion. They are developed further in §1.2.1. A set of classical elements is classical. Therefore one cannot assemble a quantum system out of classical elements. Since canonical quantization partially quantizes the dynamical stratum, but leaves the deeper space-time stratum unquantized, it breaks the ladder of strata. To maintain this ladder the system must be quantum, quantum, quantum all the way down. In the present work, each stratum is a quantum structure assembled from those of lower strata by a regular quantification process like Fermi statistics. Near the bottom of the ladder is the empty set, which is trivially both classical and quantum. What separates the classical and quantum ladders is not their bottom but the stratum-raising operation by which they are assembled. Quantization converts atomic beings into atomic processes of creation and annihilation, while preserving the stratification. In the second place, a canonical quantization is but a first step even on its own stratum. The canonical relations qp − pq = i, iq − qi = ip − pi = 0 (1.1) say that p and q are not quantized. (They have continuous unbounded spectra.) This canonical non-quantization is the main source of infinities in present-day quantum theories. Canonical quantization quantizes some quantities, like the harmonic oscillator energy, eliminating some infinities, but its quantization is unfinished, and infinities remain. The problem with canonical quantization is that its Lie algebra (1.1) is singular and has only infinite-dimensional representations, aside from a trivial one. The Lie algebra of space-time coordinates and derivations has the same singular nature. Quantization can be total. The prototype of complete quantization is the so(3) Lie algebra of commutation relations qp − pq = i, iq − qi = p, pi − ip = q. No matter how small > 0 is, these relations allow p, q, and i to have discrete spectra. In an irreducible representation of these reformed relations, all physical quantities can have bounded discrete spectra with a finite number of values. By a bfull quantizationc is meant one that replaces every singular Lie algebra of every stratum by a nearby simple Lie algebra. In what follows, q means quantum, including the canonical quantum theory, and bQc (or bqueuec) means fully quantum, simple on every stratum. A full quantization replaces the infinitesimal translations of a canonical quantum theory by infinitesimal generators of a simple group. 16 CHAPTER 1. STRATA OF ACTUALITY

1.1.3 Swaps In the theory most fully developed here, all commutation relations arise from those which define statistics. The elementary quantum process is a creation or annihilation, and these pair into bswapcs, or pair exchanges, when they enter dynamics. We lose nothing of interest by building all physical processes out of swaps. Indeed, the earliest double-valued representations to be studied were of swaps, not rotations, the algebras being isomorphic. Full quantization as thus defined is an excessively underdetermined problem. One plausibly reduces the possibilities by limiting the Lie algebras to those that result from iterating one quantification process. These then belong to one of the classical Killing- Cartan sequences of Lie algebras, determined by the quantification process. Here Fermi quantification is adopted and the so(n+, n−) Lie algebras result. Several questions confront the explorer at once: The standard model uses a global causality form g, probability (amplitude) form H, and imaginary unit i. As global constants they violate Einstein locality. What is their origin? Common assumptions of batomismc, blocalityc, and bsimplicityc suggest that they arise from local elements of a lower stratum by organization. Whether this is correct remains for experiment to decide. Atomism analyzes apparent continua into more elemental individuals, atoms, or monads. The atomism presented here resolves space-time processes into atomic (first-grade) processes of a lower stratum.

Notation: NR is the vector space N z }| { NR = R ⊕ ... ⊕ R . and similarly for NC. V1 V2 is the direct sum V1 ⊕ V2 furnished with the diﬀerence norm

k ... k = k ... k1 − k ... k2. SO(V ) is the special orthogonal group of the quadratic space V ; so(V ) is its Lie algebra. A homogeneous Grassmann polynomial of Grassmann grade g is called a g-adic, and the system it represents is called a g-ad (bcenadc, bmonadc, bdyadc, ..., bpolyadc).

The theory most studied here, bϑoc, is based on the line of orthogonal Lie algebras so(n+R n−R). A theory bϑlcbased on the linear Lie algebras sl(n) is also considered. They replace the global Minkowski space-causality form g, Hilbert-space metric H, and global imaginary number i of the canonical quantum theory by local elements of structure. Simplicity is meant here in the sense of Lie algebra theory. Einstein introduced two classical non-commutativities, that of boosts in special relativity, and that of translations 1.1. ATOMS OF ATOMS. 17 in general relativity. Heisenberg later introduced the deeper quantum non-commutativity of coordinates like position and momentum. The theories ϑo,l deduce both Einstein’s and Heisenberg’s non-commutativities as singular limits of simple quantum non-commutatvities. The superlative of “non-commutative” is “simple”. The bsimplicity principlec requires the groups of all strata to be simple groups. This eliminates commutative subgroups and maximizes non-commutativity. Infinitesimal generators of translation groups are called momenta in quantum theory. Those of rotation groups are called spins or angular momenta. These Lie algebras arise here from the graded Lie algebras of quantum statistics. All physical operations are reduced to permutations, which in turn reduce to dyadic permutations, swaps. The prototype of all swaps is the move of a man in a game of checkers, which swaps adjacent squares. Spins and unitary charges are other forms of swaps. All momenta are regarded here as singular approximations to high-stratum representations of underlying swaps. A continuum of pre-quantum physical theory is usually assembled by iterating set- S exponentiation S → 2 and is represented by a bclassical setc or bc setc, briefly a bseac. Analogously, a fully quantum system may be assembled by iterated Fermi quantification. It is then called a bquantum setc or bq setc; or briefly a bqueuec when it is supposed to be not only a mathematical system but a physical one. The constituents of a queue are queues of a lower stratum. The queue is fully quantum. The hypothesis is that a queue hides beneath every sea, and the project is then to find it. In the theory ϑ the vector spaces of all strata are Clifford algebras of real o b c √ orthogonal groups SO(N; σN ) with signatures σN = N. Fully quantum theories based on the sequences SU(NC) and SL(NR) are also considered. It is most likely, apriori, that the quantification process varies from stratum to stratum, but a fixed process is explored for the present. An infinitesimal generator of an orthogonal space-time group or its covering group can be called a bspinc for short. An infinitesimal generator of one of the unitary groups of the standard model is called a bchargec. These two concepts meld here into one, the more combinatorial construct of a bswapc. The prototype swap, the move of a checkerpiece, annihilates a piece on one square and creates one on a touching square. Queue theory permits the swap of swaps, constructed by bracing. It is accepted here that continuous space-time coordinates and field variables are not basic but emerge from multitudes of organized swaps. This builds on quantum space proposals of Segal [66], Weizsäcker [81], Bohm [12], Roger Penrose [60], and Vilela Mendes [75] among others.

1.1.4 Beneath geometry Events of the sea are supposedly completely defined by their space-time coordinates. Some Lorentz-invariant proposals for quantum space-time, however, adjoin momentum-energy coordinates to the space-time coordinates [69]. Some adjoin internal coordinates related 18 CHAPTER 1. STRATA OF ACTUALITY to charges as well. Events of the queue may have more than twice as many coordinates as those of the sea. Galileo’s conceptual reformation adjoined a time variable to the space variables of Euclid. This can be regarded as refining rather than enlarging the pre-Galilean space, by resolving its points into events, atoms of eternal being into atoms of transient process. It deepens the description of the same domain. Quantizing space-time further analyzes the event into parts. It deepens the Galilean reformation. When we look about us we see objects with color, temperature, weight, and many other properties besides space-time coordinates. Indeed, nothing with only space-time coordinates has ever been seen. From a contemporary empirical viewpoint the concept of pure space-time seems like an archaic myth with a remarkable longevity. How did this myth arise and persist? The question is asked not for its historical interest but to help us recognize and reform similar myths that control us today. For Euclid, a point is that which has no parts, an atomos, and all in his universe was made up of these atoms. Euclid was therefore an atomist. His atom was likely an abstraction from a speck of the medium underlying the diagrams of his geometry, such as a writing slate or a box of sand, and ultimately the flooded fields along the Nile. Visible specks are composed of millions of chemical atoms. The timelessness of Euclid’s point seems to result from its alleged immobility, which ultimately expresses the geocentric cosmology of Plato and older sources, in which the Earth, being the center of the universe, is at absolute rest. Evidently velocity and momentum-energy were abstracted away at the same time as time (§5.2.1). This omission is equally unphysical. Again, we never see such an impoverished object. Therefore geometry will not be regarded as the substratum of physics but as a high- stratum statistical approximation to a solid. Field theory restores some of the omitted event variables by attaching them to imaginary space-time events, still supposed to have independent existence on a lower stratum. It is a further patch on the myth of Euclid, not a reformation. These underlying events are still unphysical, unobservable. Theories of the Kaluza-Klein variety enlarge the concept of event to include further physical variables, but they still lack the dynamical dimensions of momentum(-energy), including the canonical conjugates of the extra dimensions. Kaluza introduced an extra electromagnetic coordinate axis so that he could identify electric charge with a momentum along that axis. In this post-quantum era, however, these continua are no longer necessary for this end. We have two kinds of angular momenta to build with, spin and orbital, one essentially quantum, the other originally classical. Generalizing these constructs to other groups, we may speak of quantum and classical generators respectively, based on finite quantum systems and infinite classical systems respectively. In the time of Kaluza the bquantum generatorsc — spin and charge — were not widely accepted as basic. Spins were still being modeled as rotating objects as late as [37], for example. Here atomicity and simplicity are taken seriously, however. Spins and charges need 1.1. ATOMS OF ATOMS. 19 not be modeled as objects moving in a classical continuum. On the contrary, a classical continuum may today be modeled as an assembly of quantum spins. Roger bPenrosec assembled a 2-sphere in Euclidean space from su(2C) spins 1/2 [60], a construction later extended to Minkowski space-time [38]. Likewise, Connes attached the Lie algebra of quantum charges to a classical space-time, instead of attaching a Lie group [21]. What we require of the Kaluza-Klein dimension is an electric charge 0, ±1 in units of the quark charge. To represent this quantum variable no classical continuum is required. One quantum swap in its defining representation is sufficient. In a quantum theory there is no reason to attach an entire manifold with infinite multiplicity to space-time to account for the gauge group generators, as Kaluza did. It suffices to attach the finite-dimensional quantum system of the observed gauge charges. Then the gauge Lie algebra enters at a quantum stratum below the classical stratum, as a theory of quantum swaps. And the underlying event has only the charge values 0, ±1, higher charges resulting from catenation of several events. Kaluza-Klein theory created the compactification problem: What energies curve the gauge dimensions into circles? This problem never arises for spins or swaps. Then the physicist’s predeliction for parsimony of concepts suggests that the classical space-time coordinates also be resolved into quantum swaps. The quantum physical event has many more coordinates the the usual space and time variables of special or general relativity, including dimensions usually considered dynamical. Calling its quantum space a “space-time” might then mislead. It will be called “event space” here. This in turn is just stratum E of a stratified queue, designated by Q. The daughter stratum of E is that of queues composed of events. In honor of Faraday this will still be called the field stratum and designated by F. The adjunction of dynamical variables to positional is also suggested by the Trautman ladder, though in a classical context.

1.1.5 The cosmic crystal film To save the phenomena, the ambient queue field of stratum F must be somewhat like a crystal film, a graphene, a fullerene ball, a truss dome, a cortex, or a bReggec skeleton with four long dimensions and small extra dimensions [?]. Now the fundamental unit of the field is a quantum bsimplexc or bcellc, a fermionic queue of fermionic events. Its variables are swaps with only three values, those of the defining representation, which may always be normalized to 0, ±1 by choosing units. A theoretical surface of zero thickness is sometimes called a membrane or brane. A bfilmc, on the contrary, is allowed here to be several cells thick, like a diamond film. The cosmic film does not occupy some space but rather constitutes one. The space-time manifold is [hopefully] a singular limit of the queue film. The usual external and internal variables of the quanta are now the longitudinal and transverse coordinates relatove to the film.h The longitudinal directions in the film run along the crystal hypersurface for cosmologically many cells. The transverse directions, normal to the cellular hypersurface, 20 CHAPTER 1. STRATA OF ACTUALITY are only one or several cells in extent. They correspond to the stiffeners of the dome. The difference between spins and charges is the difference between swaps along longitudinal and transverse axes of a quantum cell. The stable particles are then [hopefully] excitations of the crystal film to which the film is transparent along its external dimensions. Less stable excitations are less stable particles. When the canonical commutation relations are regularized, as in the above illustration, the resulting relations establish a symmetry between q, p, and i. In the ϑo,l theories, therefore, two extra quantum edges are assumed for each cell, to provide a swap that organizes over many cells into i and a central algebra of complex numbers. It would be confusing to call these additional two dimensions the “complex” dimensions, since they are real; they will be called bcomplexualc. If each cell is a hypercube (for example) with one vertex as origin, then adjacent to the origin are four external vertices corresponding to Dirac spin operators γ1,2,3,4 that organize across cells into global space-time in a singular limit; two complexual vertices γ5,6 contributing to the global complex plane; and 10 internal vertices providing standard model charge operators γ7,...,16. Classical space-time is to be the singular limit of the external extension of the crystal film. The fields and sources in the standard model involve the internal extension as well. The imaginary constant i of quantum probability amplitudes corresponds to the generator i of an so(2) that rotates each of the two complexual dimensions γ5,6 into the other. The swap i becomes central only in the canonical limit. The canonical quantum field variables are singular canonical limits of collective properties of the individual cells of the crystal film involving their internal dimensions. In the film phase, cells of the queue can be addressed approximately by four numerical space-time coordinates, and the field construct emerges. The queue has strata corresponding to the field stratum F of the canonical quantum theory, the deeper space-time event stratum E, and a still deeper differential stratum D. Charges arise from a still deeper charge stratum C. The relation between these strata is no longer functional but is still a relation of set theory, expressed by the bbracec operation I described briefly in §3.1.2 and more fully in Chapter 4. It is widely supposed today, and in this study too, that physical events have more than the classical four space-time coordinates, that the extra coordinates provide field variables, and that they are small compared to the four classical coordinates. This smallness here suggests that field variables arise in a canonical limit from queue variables of a lower stratum than the space-time coordinate variables of the events in the queue. In the quantum space-time of ϑo the basic Fermi operators generate events; the usual scalar coordinates xµ of Minkowski and Einstein are singular limits of swaps in an orthogonal Lie algebra so(N+R N−R) of the underlying quadratic space; the fermionic quanta of the standard model are quantum event themselves; and the gauge bosons of the standard model and gravity are bound catenations of such events. Replacing the field by the queue as basic variable solves a major conceptual problem 1.1. ATOMS OF ATOMS. 21 that has beset such quantum studies: how to formulate the general concept of a single- valued field function when the space-time argument is a quantum variable rather than a random variable (§6.1.3). The field passes for a single-valued function of the event in the canonical quantum theory because in Q theory, the ambient queue has an empirical structure analogous to that of a long, thin bcrystal filmc. The cells are (say) 16-dimensional, but four coordinates suffice to locate a small patch of cells; these are the space-time coordinates. Field coordinates are surrogates for the other coordinates of the cell patch. Space-time coordinates run along the long film-dimensions and field variables along its short ones. Each basic vector of one stratum is composed of vectors from the next lower stratum somewhat as a simplicial bcellc is composed of vertices. The long space-time coordinates commute in the canonical limit, and their values specify a unique patch of cells in the canonical field, whose short quantum dimensions define the field variables. Such a crystalline structure does not fit well into a field theory over a manifold, like Kaluza-Klein or string theories, which require one to go far beyond experimental data. A fully quantum theory accommodates quantum experimental data with less strain.

1.1.6 Praxics

The physical distinction between the classical and the quantum is basic to this study. Both classical and quantum physics can be expressed in the terms of hit-and-miss relations between ideal metered sharp input processes and output processes usually represented by the directions or rays of vectors and dual vectors. Heisenberg called the components of such vectors “probability functions”, so the vectors themselves may be called bprobability vectorsc, or more fully, bprobability amplitude vectorsc. If an input process p is associated with a probability vector hp] then an output process p0 is associated with a dual probability vector [p0h so that the transition probability is [p0hp][p < p0] P (p0 ←p) = . (1.2) [p0hp0][php] [p0hp] is the evaluation of the dual vector on the vector. hp][p0h = ip0][ph is the linear operator or dyadic that maps hw] 7→ hp][p0hw].

Superpositions of input and output processes for a system are called bterminal processesc. Terminal processes correspond to the atoms or points of a Boolean logic. The duality between input and output does not occur in Boolean logics, which purport to deal with atemporal being rather than temporal doing. The ancient insistence on absoluteness and timelessness still pervades the study of logic as it did natural philosophy through the Renaissance. Some natural philosophers escaped the dead hand of Plato by changing their job title to physicist and starting new departments. Perhaps logic must go a similar route. The temporal, relativistic, stratiﬁed physical theory of quantum epistemic processes that has logic for its classical limit is called 22 CHAPTER 1. STRATA OF ACTUALITY

bpraxicc here. It does not claim the absolute authority that logic did, but denies that authority and declares itself to be empirical and revisable. One stratum of praxic can be formulated in terms of input and output spaces V,V D, an involutory mapping H : V ↔ V D and a relation ø on V D × V . When iv0] = iHhv], the transition v → v0 is assured, occurs on every trial; the input “hits” the output channel. The symbol H honors Hermite and Hilbert. In the usual vector β α α representation, it becomes a symmetric tensor of the form H = Hβαe e in the basis e . The form H and any a real multiple rH (r 6= 0) have the same physical meaning. When the relation v0 ø v holds, the transition v → v0 never occurs; input v “misses” the outp channel v0. In the usual projective representation, v0 ø v becomes the duality 0 α relation vαv = 0. To distinguish the two metrical forms that are central in this work, we call H the bprobability formc (more properly, the probability-amplitude form) and g the bcausality formc. In the theory ϑo they coincide on a stratum C and diﬀer on higher strata. In both quantum and pre-quantum versions of process praxics, the probability form H deﬁnes an involutory symmetry between input and output processes: H2 = 1. Symmetries in the usual sense are permutations of input processes that respect some structure of the system, such as H and ø ; or the dynamical development as well. In contrast:

The probability form is a symmetry of the metasystem.

Namely, it interchanges inputs and outputs. H is not a time reversal in an ordinary sense, since both input and output processes are generally thermodynamically irreversible and feasible, while the time reversal of an irreversible process is unfeasible. A bHermitianc operator h is one with the symmetry property

[w iH h H−1 iv] = [v ihhw] =: [w ihThv]. (1.3)

The quantum principle is usually stated in a vector representation: The relation p0 ø p holds if and only if an associated vector hp] and dual vector [p0h obey [p0hp] = 0. More generally put,

Assumption 2 (Quantum Principle) Every isolable physical system has a 1-1 correspondence from its input and output processes p, p0 to Hermitian probability operators hph, hp0h on an associated vector space such that the transition probability for p → p0 is Tr hp0hph.

In most current practice H is positive deﬁnite and physical probability operators are positive deﬁnite and of unit trace. A process p is sharp or irreducible if the associated probability operator hph is a projector on some vector hp]: hP h = hp][ph. 1.1. ATOMS OF ATOMS. 23

Vectors used in this way are called probability vectors (more properly, probability- amplitude vectors). They belong to the stratum of beams or collections of systems, not to the stratum of the individual system they guide. One does not measure a probability vector on one system, as one does a force or momentum vector; a probability vector represents a measurement itself. This use of a vector to specify an input or output process for a system is different from all uses of vectors in pre-quantum physics. The change is at the level of the laws of probability (§2.9). The form H is more descriptively written as iHh, showing that it accepts two vectors. With respect to a basis hα], H is defined by the coefficient matrix [βhHhα] =: Hβα.

1.1.7 Indeﬁnite probability forms

The relations ø and H are usually assumed to be contradictory; then H is bdefinitec. To be Lorentz-invariant, however, a fully quantum theory, having a finite-dimensional probability vector space, must use an indefinite probability form (§5.2.2). There are two well-known ways to give physical meaning to indefinite probability forms, one associated with bGuptac and bBleulerc, the other with bDiracc. If H is indefinite, then there are probability vectors hp] for which the probability form kpk = [php] > 0, = 0, < 0. They are said to be positive, negative or null vectors. [zhz] = 0 says that on every trial, hz] misses the output that it hits on every trial. The inference seems unavoidable: There are no trials of hz]; null probability vectors represent unfeasible processes. They are still necessary in order to express the basis of one frame in terms of those of another. Moreover if Hp ◦ p > 0 > Hn ◦ n then the transition probability P (p←n) is negative: of N > 0 trials, transitions occur in P N < 0 trials. Given that the number of trials of any kind is non-negative, this implies N = 0; inputs of negative signature are unfeasible. In the Gupta-Bleuler interpretations the null cone formed by null vectors is an impassable frontier for transitions. Dirac, however, proposes that Nature sometimes extends us credit, allows us to with- draw systems even before we input any. This results in a negative number of systems. Such an output against future inputs can be regarded as the input of a dual system. This seems especially natural for fermions, where there is symmetry between the full and the empty probability vector of the many-fermion system, and there is an algebraic isomorphism between increasing the fermion number from 0 and decreasing the fermion number from its maximum value.

1.1.8 Probability vector spaces and algebras

For a discrete bclassical systemc the relations ø and H deﬁne a discrete state space and their symmetry group is the discrete permutation group of the state space. 24 CHAPTER 1. STRATA OF ACTUALITY

For classical mechanics the state space is a symplectic manifold, and its group is the contact transformation group. This group is singular. For a bcanonical quantum theoryc the group of the ø -relation is commonly an infinite- dimensional unitary group. That is the trouble. In a fully quantum theory the groups of ø and H are classical groups and the probability vector space is finite-dimensional but indefinite. Automorphisms of ø are projectively represented by elements of SL(V ), the regular linear mappings V → V of unit determinant. Endomorphisms of the quantum system are projectively represented by linear operators V → V , which make up the algebra Alg(V ). Infinitesimal automorphisms of the quantum system are bijectively represented by operators in Alg(V ). Canonical probability vector spaces are provided with a symmetric bprobability formc H, usually definite. Then the bMalus-Bornc relation holds:

The transition probability is the squared value of a unit dual vector on a unit vector.

Infinitesimal transformations V → V respecting H form the bisometryc Lie algebra so(V ) of the probability vector space V . The Lie algebras so(3R R, sl(NR), and sl(NC) have no finite-dimensional unitary representations with definite probability form H.

1.1.9 The need for full quantization For Faraday and Einstein, both field values and events were classical and their pairing was managed using informal classical logical intuition. For Dirac, the electromagnetic field values were operators and handled by formal algebraic methods, but space-time was classical, as though field-meters and clocks were classical objects. In canonical quantum theories in general, the metasystem is ordinarily treated classically and the system quantally. The space-time variables of field theory are metasystem variables. The split of nature into classical and quantum strata is unphysical and creates problems: It seems to be impossible to represent the action of a quantum system on a classical one without quantizing the classical system. Approximating a sufficiently small classical field-meter or target with a quantum system likely creates a black hole, and neither of the two major principle theories of the day, general relativity and canonical quantum theory, covers this phenomenon. These considerations suggest that quantum field theories need further quantization, if only to work near the Planck energy.

Any classical system is but a crudely observed quantum system. 1.1. ATOMS OF ATOMS. 25

Canonical quantization modiﬁes and partially regularizes only one stratum of a canonical classical theory and leaves deeper strata such as that of space-time still classical and singular.

1.1.10 A cellular hypothesis

The present bfull quantizationc quantizes and regularizes every stratum by means of a radical

Assumption 3 (Cellular Hypothesis) Physical processes on every stratum are quantum cells composed of a ﬁnite number of ﬁnite quantum processes of lower strata with Fermi statistics.

Cellularity is intended here to express stratification. The cell overlies a deeper strata of organelles and underlies a higher strata of organs. Classically this stratification is expressed by iterated quantification, usually within set theory. Full quantization carries this classical stratification over into the quantum realm. The bqueuec, a quantum structure assembled by iterating Fermi(-Dirac) statistics (§3.1.2), is finite and its groups on every stratum are simple Lie groups. The bfull quantizationc strategy is group-guided:

Reduce a canonical theory to a singular limit of a queue theory keeping exactly the same groups where possible, and approximately the same groups where not.

Dirac’s Lagrangian quantum dynamics approximates a propagator u for infinitesimal time as [q(t + dt)huhq(t)] ∼ exp i[q(t + dthLhq(t)], the exponential of a matrix element of the action. This has no unitary-invariant meaning. Operator exponentials are frame- independent, but exponentiating matrix elements singles out a small family of frames. In this case, the frames singled out are those in which each basis element is associated with a definite space-time event at which the Lagrangian coordinates are evaluated. This principle refers on the dynamical stratum F to an underlying event stratum E. Q respects that stratification in its way. Since there are no classical systems, a sum over classical histories is not a part of a quantum theory but only a heuristic precursor. It is also too singular to be well-defined. In Q theory it survives as a finite trace over a Q history taken to be a queue of events, with a Q dynamics taken to be a dual probability amplitude vector [Dh assigning probability amplitudes to every experimental history vector hE]. It is easier to take the q limit if q theory too is expressed in terms of histories ??. D Fermi quantification maps vectors v ∈ V, u ∈ V to operators v,b ub on the Grassmann algebra of V , subject to {u,b vb} = u(v); (1.4) 26 CHAPTER 1. STRATA OF ACTUALITY

This is a full quantum theory as far as it goes, with discrete bounded spectra only, but it covers only one stratum. One may cover all strata by iterating it. This results in the queue. Eliminating the field in favor of the queue eliminates the classical puzzle of infinite self- interaction [34]. Present physical theories of gravity and of the standard model [hopefully!] become singular large-scale limits of one fully quantum queue theory that is also valid on and near the Planck scale. Fully quantum events have coordinates complementary to their positional coordinates, resembling momentum coordinates. Such events are no longer atomic elements of space- time position but of a quantum dynamical process. Classical space-time emerges when these event momenta can be neglected because they are small, perhaps 0. In a fully quantum theory, variables on every stratum have bounded discrete spectra. In particular in the theories ϑo,l the spectrum of the time coordinate is bounded and uniformly spaced in multiples of a btime quantumc bXc, which with c also defines a space- quantum. Due to the non-canonical commutation relations, exact Lorentz invariance is nevertheless preserved. Spectra on lower strata are much sparser than those on higher strata. The Heisenberg uncertainty principle between space-time and momentum-energy variables breaks down badly on lower strata and reappears on a higher stratum as a singular limit. This is large effect, and may be able to explain the large discrepancy between the empirical cosmological constant action density and that expected from field theory, which is on the order of one Planck unit. General relativity and quantum theory are dominated by two forms: H]/, a hermitian form on vectors with associated imaginary i, such that Hi + iH = 0, defining transition probabilities, called the bprobability formc. g, a symmetric bilinear form on space-time differentials dx defining bcausal structurec of the event stratum, called the bcausality formc. It is measured with chronometers in the classical limit, and by graviton emission and absorption in the canonical quantum theory. µ µ The form g = (gνµ) defines the proper time dτ for a differential displacement dx near x so that dτ 2 = −g dx dx. (1.5)

R 2 It also defines the action S = − m0c dτ for a projectile of rest mass m0. Fermi quantification is usually carried out within a unitary quantum theory. It has, however, a larger invariance group than the unitary. It makes use of neither a probability form H nor a central imaginary i, nor does it produce any. If the quantum kinematics has a group SU(NC), the invariance group of the Fermi algebra is at least GL(NC). This raises the question of how the canonical g, H and i manifest themselves in a queue. Since they are global in dominion, the strong blocalityc principle of Einstein suggests that they are not intrinsic but bemergentc, arising from more local constructs by organization and spontaneous symmetry breaking. The bfully quantumc theory ϑo assumes that a lower stratum C has intrinsic constant g, H, i and that these organize into variables of higher 1.1. ATOMS OF ATOMS. 27 strata like F. The operators of a unitary representation of the Poincarégroup respect a definite probability form H on one stratum, while the Poincarétransformations being represented respect a time form g on a deeper stratum. Both the unitary group and the Poincaré group can be imbedded within SO(N+,N−) groups appropriate to their strata: In ϑo, the unitary representations of the Poincar´regroup used in physics are singular limits and organizations of higher-stratum SO(N+,N−) representations of lower-stratum SO(N+,N−) groups, which arise naturally for a queue. The Riemannian curvature of space-time is then expressible as a near-classical vestige of the quantum non-commutativity of the momentum-energy variables of the quantum event. Even in the classical theory of gravity the probability form and the causality form are not independent, making it plausible that the have a common origin at a deeper stratum. General covariance and all other singular gauge invariances are approximations to a fully quantum gauge invariance whose gauge group is regular, and the approximation breaks down at high energy. Space-time [hopefully!] becomes an organized quantum system with a melting point analogous to a Curie temperature. There are no events in the classical sense:

What has space-time position has momentum-energy, angular momentum, spin, and the standard charges too.

In the canonical limit, the momentum-energy of quantum events may still contribute to the cosmological constant action density. To bfully quantizec a physical theory it suffices to assemble every stratum of the theory from quantum elements of a lower stratum with Fermi statistics. All quantum commutation relations for a fully quantum system derive from the quantum statistics of its quantum elements. The usual canonical quantization is expressible as a first approximation — but a singular one — to such a full quantization, which necessarily quantizes time. It is easy to see the basic ingredient of queue models. The basic generating operators all represent operations of creation and annihilation of quanta of Fermi statistics. These act before the organization of space-time as well as after, so it would be misleading to say that these building bricks are spins, although they have Clifford commutation relations. In Fermi statistics, annihilation of a quantum is isomorphic to creation of a dual quantum, so we may say briefly that queue models are made of creation operations, or bcreationsc, represented by creation operators, or bcreatorsc. Full quantization works on the Lie algebras that occur on every stratum of the canonical theory in various roles:

1. Abrelativityc Lie algebra deﬁnes the frame transformations for the system.

2. An bexperimentalc Lie algebra deﬁnes feasible reversible experimental operations on the adiabatically isolated system. 28 CHAPTER 1. STRATA OF ACTUALITY

3. A bsymmetryc Lie algebra defines the transformations that fix some specified feature of the system such as the dynamics.

Some of these Lie algebras are made up of operators that act on probability vectors. In the canonical theory those of lower strata act on classical objects, for example, on space-time points, or tangent vectors to space-time. In principle, full quantization preserves roles as it regularizes Lie algebras, and its Lie algebras of every stratum act on vectors.

A bfully quantumc system is quantum on all strata.

In the fully quantum theories ϑo,l the probability vector spaces of all strata are real Grassmann algebras. In the theories of the orthogonal group line like ϑo they are Clifford algebras as well. The canonical Lie algebras and unitary Lie algebras that occur at various strata of the standard model and gravity theory are singular limits or reductions of finite- dimensional invariance Lie algebras so(n) of these Clifford algebras. Fluctuations about the classical predictions of the electromagnetic field theory helped to convince Einstein of the existence of photons. Analogous fluctuations about the canonical quantum predictions are expected to signal the existence of quantum space-time events. A discrete space-time theory must have difficulty accounting for the continuous groups that have been the main navigational aids in the quantum exploration. A quantum space- time theory does not; its probability vector spaces are born with these groups or close approximations. Queue theories like ϑo,l replace the classical logical construct of set by the praxic construct of a queue. Einstein pointed out that his general relativity did not geometrize physics, it physicalized geometry, presumably the geometry already used in physics. In a similar sense, queue theories do not set-theorize physics but physicalize the set theory already used in physics, while following in the footprints of the existing canonical quantum theory as closely as possible. This goes deeper than geometry, which can after all be regarded as an exercise in set theory. The result is no longer a theory of sets as objects but a theory of queues as reflexive processes of creation (and annihilation), an example of a bpraxiologyc. In a bqueuec theory a Grassmann product of g single-quantum probability vectors can be regarded as the probability vector of a cell with g quanta as vertices or elements. The Cellular Hypothesis of Assumption 3 can be sharpened:

Every physical system is a queue.

A (many-system) balgebrac, unless otherwise speciﬁed, is a probability vector space that is also a Grassmann algebra whose product represents the composition or catenation of probability vector processes. The most familiar example is the Cliﬀord algebra that serves as vector space of a collection of fermions. 1.1. ATOMS OF ATOMS. 29

1.1.11 Quantum time

The quantization of time leads to several conceptual changes in quantum theory taken up in this section..

Recall that a system is bquantumc, to put it in terms of laboratory experience, when the filtration operations for it do not all commute with one another. Only the simple or irreducible case is considered here, with no proper central filtrations, 0 and 1 being considered improper. This is the case of unlimited quantum superposition. The axioms for a projective geometry provided with a polarity characterize such systems in a coordinate-free way. Filtrations are then isomorphically represented by the projectors of the probability vector space. The canonical relations originate in the differential calculus in the form

∂x − x∂x = 1, ∂tt − t∂t = 1. (1.6) and migrate to the Lie algebra of Poisson Brackets of classical canonical mechanics, from which they enter canonical quantum theory. To be sure, t is not a canonical observable of the system stratum, and ∂t is not a canonical observable of any classical stratum, but the instability of their Lie algebra propagates to that of space coordinates x in a relativistic theory, and integrals over events enter into observables and produce infinities. On the other hand, finite dimensional Fermi algebras, with their graded-canonical commutation relations, are structurally stable, represented in finite-dimensional probability vector spaces, and require no regularization. They are preserved intact in the present full quantizations. Gauge-fixing fields and ghosts are introduced into canonical theories to make them renormalizable. They are not needed in a Q gauge theory that is finite from the start. The gauge is fixed in actuality by the experimenter, who breaks all symmetries [84]. The gauge-fixer can be imagined, for example, as a Mandelstam loop system [55], each loop including a quantum interferometer, or a generalized voltmeter in the classical limit. A gauge transformation replaces one such loop system by another. A gauge-fixing field is then a surrogate for a loop system in the metasystem. Quantizing it therefore violates Bohr’s early principle of the classical metasystem, but it works. This provides some precedent for the deeper violations perpetrated here. In order to eliminate the unphysical ultraviolet and infrared infinities of canonical quantum theories, many considered slightly varying the space-time-momentum-energy commutation relations to quantize space, time, or both, including bAmbarzumianc and bIvanenkoc [1], bEinsteinc, Heisenberg, bSnyderc, bSegalc, bPenrosec, bConnesc, and bVilela-Mendezc (§5.2). This strategy is not at all iconoclastic, merely biconoelasticc. It is extended to all strata of assembly here. Admittedly, Einstein mentions the approach only to declare it daunting [29]. 30 CHAPTER 1. STRATA OF ACTUALITY

It takes a fine mill to grind time. Time passes untouched through the canonical quantization machine. One rationalized the persistence of classical space-time when all else is quantum by allocating space-time to a different stratum of physical theory. Time t is not an observable variable of the system stratum in canonical quantum theory; not because it is unobservable, since it is observed with clocks, but because it is an observable of another stratum. The clock on the laboratory wall is not part of the system under study. This concept of stratum merits further discussion. Physical theories are usually structured, at least tacitly, into mathematical bstratac recursively generated by bracing and catenation. Making one variable a variable function of another, for example making position a function of time, allocates the dependent variable to a higher mathematical stratum than the independent. Canonical quantization accepts the stratum structure of the classical theory, modifies the algebra on the stratum of dynamical variables, and leaves the algebras of the deeper strata unmodified, including the stratum of time in mechanics and space-time in field theory. This agrees well-enough with low-energy experiments except that it builds in infinities from the start. But we never measure an infinity on any stratum, having neither divided any physical line infinitely often nor extended any line to infinity. Euclid built these infinities into our theory of space with no experimental evidence for either, only an absence of clear evidence for finiteness. And since it takes a theory to interpret the evidence, this may simply be an absence of adequate theory. The ill-defined formulations from which present-day quantum field theory sets out, with its non-existent integrals over infinite-dimensional spaces and its unbounded operators that cannot be multiplied, reflect limits to our imagination and mathematical powers as much as inferences from experiment. All these unphysical infinities trace back to our choices of groups. Euclid incorporated infinities into plane geometry in order to have translational and rotational invariance under the Euclidean group. A Lie algebra is called bsingularc or bregularc according to its bKilling formc; a Lie group, according to its Lie algebra. The Euclidean and canonical groups are singular, the classical groups are regular. Singular groups generally require infinite- dimensional unitary representations, but classical groups have enough finite-dimensional matrix representations to make a finite quantum theory possible in which the groups act on finite-dimensional probability vectors, and expectation values are finite traces. And there are classical groups near most singular groups, including the canonical groups, though the approximation is not uniform, and may require freezing some degrees of freedom of the Lie algebra. Full quantization is not a well-defined algorithm but a heuristic method, even more so than canonical quantization. Astronomical evidence for the Big Bang already suggests that one might not be able to extend the time translation operator beyond about 10∼21 s into the past, so that it is not completely absurd to renounce the time translation group as is done here. A canonical quantization reduces the infinities of classical physics by replacing some commutative groups by canonical ones. A full quantization, for example ϑo, eliminates the remaining infinities by replacing every bsingular Lie algebrac — which includes all canonical 1.1. ATOMS OF ATOMS. 31

Lie algebras — with a nearby regular Lie algebra — for example, an so(n+, n−) —- on every stratum, with organizations as needed. The theory ϑo assumes that the constituents of each stratum have Fermi statistics. The reasoning behind this choice is given in §1.3.6. It unifies spins and charges with a quantum space-time somewhat as MankoˇcBorˇstnikunifies spins and charges with a classical space- time [58]. Its simple Lie algebras groups are unitary ones su(n+, n−) of various signatures, in the A series of Cartan. A quantum space-time theory based on the D series has also been studied [68]. Segal proposed to quantize the space-time stratum with a simple Lie group instead of a canonical group on the grounds of structural stability [66]. Since there are always errors in data, physical Lie algebras should be insensitive to small errors in the commutation relations; should, in other words, be structurally stable. A Lie algebra is bsingularc or bregularc as its Killing form is. Thus the canonical Lie algebras h(n) are singular, while the linear Lie algebras sl(n) is regular, as are the orthogonal Lie algebras so(n) and the unitary Lie algebras su(n) of whatever signatures. Singular Lie algebras are not structurally stable, and regular Lie algebras are, which includes the simple ones (§1.1.1). But finiteness is an even stronger motivation than structural stability for regarding all the singular groups of physics as approximations to nearby classical groups. The deepest conceptual problem in quantizing space-time arises from the fact that the time variable of mechanics and the space-time variables of field theory are not observables of the system but of the metasystem. Canonical quantization is limited to observables of the system, and this limitation has been elevated to a decree: Do not quantize the metasystem. Two observables do not commute if they are complementary; that is, if measuring either invalidates a prior measurement of the other. It would be hard to understand how two quantities can fail to commute if they are not supposed to be measured. Time is usually defined today as what clocks read, and as such is measured; but within the metasystem, not the system. But if the time that enters into the dynamics of the system were to be identified with the time reading of the laboratory clock, it would create a mystery of how the clock on the wall is so rigidly connected to the particle in the experimental chamber. Here the two times are kept conceptually separate, further relativizing time. It is no longer posited that there are ideal clocks that read the ideal time. Each clock actually reads its own time. The space-time structure connects clocks, keeping them in approximate synchrony. The limit of ideal time is a singular one, and here underlying regular theories are discussed. Special and general relativity introduced new acceptable time variables. A fully quantum relativity introduces many more. It allows different times on different strata, and introduces a complex organization of many quantum variables below the perceptions of canonical physics to account for the correspondence between these times, a form of self-organization that degenerates to a monolithic classical time in the singular canonical limit. 32 CHAPTER 1. STRATA OF ACTUALITY

As a Lorentz-invariant model of quantum space-time point that can approach a point of Minkowski space-time in a singular limit, Segal suggested an so(6R; σ) angular momentum of large quantum number in a 6-dimensional quadratic space of unspecified signature σ. Thus Segal chose the defining representation of an extension of the Lorentz group as the seed of space-time-energy-momentum space. This has a substrate of Minkowski space-time vectors, with Lorentz spinors playing a secondary role. Segal does not consider a stratified ∼ quantum theory. The conformal Lie algebra so(4, 2) and the Lie algebra so(3, 3) = sl(4R) are also possibilities. In an independent approach, Penrose combined many so(3) spins 1/2 to make an SO(3)-invariant quantum space that can approximate a Euclidean 2-sphere in a classical limit. A Segal model can be constructed by similarly combining many so(6; σ) elementary spins. The queue theory ϑo is constructed by iterating a Fermi quantification. Fermi statistics is the natural choice, on the grounds of its regularity. This assigns odd exchange parity to even rotational parity, but can be reconciled with the empirical spin-statistics correlation [hopefully] if the vacuum moves a half unit of spin to remote regions S:SPINSTATISTICS. Dirac spinors transform as probability amplitude vectors of Fermi aggregates of lower- stratum entities [16]. Therefore the substrate of queue theory too will be spinorial rather than vectorial. Since a Fermi algebra is also a Clifford algebra, its substrate is appropriately spinorial. Such a full quantization expands every canonical Lie algebra, whatever its stratum, to a linear Lie algebra sl(NR) on the same stratum, decentralizing and conditioning its i.

1.2 The idea of the queue

A bqueuec is a hypothetical Fermi-statistical quantum system whose basic probability vectors are generated from 0 by (1) the operations of Grassmann polynomial formation, with addition representing quantum superposition and Grassmann multiplication representing physical combination, and by (2) bracing, I(ψ) = {ψ}, representing physical unitization. The stratum of a queue is the maximal number of its nested braces. This leads to the stratum scheme defined by (1.7). To define a quantum kinematics one must further specify the brace I, a probability form H, and their physical interpretations (Chapter 4). The Lie-algebraic commutation relations defining any classical group can be interpreted as a statistics [59] called bPalev statisticsc. Bose statistics is a singular limit of Palev statistics. The dyads or pairs in a queue obey a Palev statistics. They can be considered as generalized bosons, or bpalevonsc. Canonical quantum theory uses an infinity of fermions to make a boson. In fully quantum theory a fermion pair is a palevon, as close to a boson as desired. This makes the Cellular Hypothesis of Assumption 3 more tenable. A probability form H on a probability vector space V is not needed to define the Fermi 1.2. THE IDEA OF THE QUEUE 33 algebra over V , but it is needed to represent physical probabilities. Its presence gives the Grassmann algebra over V a second structure, that of a Clifford algebra over V . The two products are written as v ∨ v0 and v t v0. The generations or strata have probability vector spaces S[L], L = −2, −1, 0, 1, 2,... defined recursively:

S[−1] = ∅ D S[L + 1] = Poly I ’ S[L], D where I : S[L] → Grade1 S[L + 1]. (1.7)

The dimension of S[L] is dL. The Whitehead-Russell bapostrophec after an operation means that the operation is to be executed on the following set element by element. f’ X can be read as (the set of) “the f’s of the X’s”. S[−1] is the empty set regarded as probability vector space. The sum of no vectors is the vector 0 that belongs to every grade and rank but has no operational meaning. This is the sole element of the vector space 2S[0]. The product of no vectors is the vector 1, representing the empty input. This is a basis vector for S[1]. The resulting probability vectors support grade and rank operators, designated by bGradec and bRankc. The grade is the Grassmann polynomial grade; the rank is the number of nested bracings. Stratum L consists of all probability vectors with rank r ≤ L. The Fermi system algebra S has a fractal, self-similar structure: A tree isomorphic to the entire structure grows from each element in the structure. S = 2S is its own Fermi probability vector algebra. As a result, groups of any stratum are represented naturally on higher strata. In physical application the recursion is cut off at some stratum; stratum 7 suffices for the dynamics of ϑo. The concept of probability assumes an unlimited supply of systems and an unlimited number of trials. In a fully quantum Fermi praxic, the probability vector space of a queue is a finite-dimensonal subspace of the self-Grassmann algebra S generated by the sequence of generators begun in Table 1.1. One probability amplitude vector, the bdynamics vectorc of stratum F, is supposed to improve on the standard model and canonical quantum gravity, at least in the small. Probability vectors are statistical, and properly belong to a higher stratum than the quantum system they pertain to. This technicality is ignored here to simplify discussions. A system and its variables are assigned to the same stratum as its probability vectors. If this practice leads to confusion it should be corrected. The exploding sequence of stratum multiplicities of S corresponds to that of classical set theory. It describes the materiel made available for theory construction by this algebraic language. There is no compulsion for the theorist to use it all, but then it is incumbent on the theorist to delimit the subalgebra that is actually physical. The lowest stratum L able to describe a system of given multiplicity D must have multiplicity dL ≥ D.A practical upper bound is set on L by the economy principle, of not using more materiel 34 CHAPTER 1. STRATA OF ACTUALITY than necessary. The theory ϑo uses the 4-dimensional stratum 2 for its Lorentz group, the 16-dimensional stratum 3 for the Lorentz group and the other groups of the standard model, and stratum 5 for the generic event of a quantum history. Full quantization thus converts infinite fine structure on one stratum into finite structure on several strata. Structure that has been attributed to the very small is now allocated to a deeper stratum. Extra classical space-time dimensions of the Kaluza kind are represented by probability vector space dimensions several strata below space-time, and so prior in order of construction. Space-time itself is replaced by a huge-dimensional probability vector space, with one dimension per independent event possibility. This elimination of all classical continua also reduces the number of probability vector dimensions, both ordinary and extra, from the infinity of field theory to a finite number for a queue. A fully quantum theory must cover the canonical theory; that is, agree with it where it works. One therefore constructs one by slightly revising the canonical (or classical) theory on each stratum, not by starting from scratch; by replacing canonical commutation relations with classical Lie algebra commutation relations. For cross-section computations one requires queue correspondents of input and output momentum eigenvectors, although queue momentum components do not commute with each other. To start the inquiry, radical simplifying assumptions seem practically necessary in order wring experimental statements out of the theory. In the theory ϑo one provisionally assumes that that the various canonical strata correspond to fully quantum strata whose probability vectors lie in corresponding subspaces of one bgrandc probability vector space S; and that every stratum is assembled from the previous with quantum statistics in the same Cartan class. This is less restrictive than it may seem, since it turns out to be easier to make bosons out of fermions with regular statistics than singular. The model ϑo then quantizes even time. The main work is then to fit the complexities of canonical quantum physics, stratum by stratum, into the simpler fully quantum framework. Recall that the process of canonical quantization can be divided into three parts: 1. Construct a family of canonical quantum theories, defined by canonical Lie algebras with an adjustable parameter ~ in their commutation relations. 2. Express the classical theory as a singular limit of these canonical quantum theories as ~ → 0. 3. Choose one of the family to succeed the classical theory, by fixing the optimal value of ~ experimentally Correspondingly, one fully quantizes (all strata of ) (say) the standard model in three stages:

1. Deﬁne the general fully quantum theory, using one sequence of classical Lie algebras for its commutation relations. 1.2. THE IDEA OF THE QUEUE 35

2. Express the standard model and gravity as a singular limit X → 0 of such fully quantum theories.

3. Fix the optimum physical value of X experimentally.

Since time is to be quantized, it is convenient that the quantum constant X be a fundamental natural unit of time or bchronc forming a complete set of units with the natural units of speed c (the rømer?) and action ~ (the planck?). Graded Lie algebras are used to deﬁne the statistics of the queue as they are used to deﬁne the statistics of fermions in canonical quantum theories.

1.2.1 Strata Quantum theory inherits a concept of assembly by stages from classical set theory. For both classical and quantum set theory the strata form a hereditary family in which any collection of individuals of one stratum deﬁnes an individual of the next by a stratum-raising process represented by a bbracec operator

D D I : w 7→ {w} ≡ I w ≡ w (1.8)

The brankc of a set is the number of nested braces required to produce it from the empty set, or from whatever proper elements serve as foundation for the set theory. To build up a quantum correspondent, one may ﬁrst translate the pre-quantum procedure into a matrix language resembling that of quantum theory, but kept pre-quantum by commutativity restrictions. To arrive at the quantum theory one then drops the commutativity conditions. Classical sets can be described by rays in a preferred orthonormal basis for a real D Euclidean Grassmann algebra Sc, with R, +, I , ∨, H, and · (§3.2). Superpositions of the basis vectors are given no physical meaning, or forbidden, in the classical theory. Random sets are described by (positive) statistical operators on Sc that are diagonal in a natural basis. This leads to nested strata S[L] of random sets, S[L] ⊂ S[L+1], related by bracing:

D [L] L+1 I : S → S . (1.9)

This sets the stage for quantum set theory. One need only turn on superposition. Canonical Fermi quantification goes from a one-quantum Hilbert space to a Grassmann algebra over a Hilbert space. The Fermi (operator) algebra over that Hilbert space is the Clifford algebra over the bduplex spacec with the bduplex formc. The queue follows the pattern of the random classical set and the canonical Fermi assembly in using a Grassmann algebra S for its vectors, now assumed to be real. Its vector catenation ∨ is that of Fermi statistics. To iterate Fermi statistics requires converting a high-grade polyadic x of one stratum into a first-grade monadic {x} of the next by bracing. In this way one constructs 36 CHAPTER 1. STRATA OF ACTUALITY the hereditary family (1.7) of finite-dimensional vector spaces S[L], one for each stratum L = −1, 0, 1, 2,... of bracing. The set theory presently used in classical and canonical quantum physics is the most S familiar hereditary family. It is generated by bexponentiationc S → 2 , which catenates monads of one stratum into polyadics of that stratum, and bracing, which forms polyadics of one stratum into monadics of the next stratum. S The exponential set 2 is often called the bpower setc of S. This can lead to confusion here; if S is a numerical variable, its powers are Sn, not nS. Here 2S is called the (binary) exponential of S. Correspondingly, a space V of queue probability vectors is mapped into a superspace by an analogous functor V → 2V , where 2V is the Clifford algebra over V . This process too is called (vector space) bexponentiationc. A vector of brankc r is one with exactly r nested bracings. Its symbol in the bar notation of Table 1.1 is r bars high. A vector of stratum L is one of rank ≤ L. Its symbol in the bar notation of Table 1.1 is ≤ L bars high. Some trivial cases: The probability vector space for stratum L = −1 is the empty set, of dimension −1. The probability vector space for stratum L = 0 is the one-point set {0}, of dimension 0. The probability vector space for stratum L = 1 is R, of dimension 1. This is a customary representation of a vacuum vector in Fock space. Starting from stratum 0, L catenated bracings still remains within the vector space S[L] of stratum L. Its dimension is 2L. In canonical theories, space-time events and their coordinates, like time, are allocated to an bevent stratumc, or stratum E. The probability vector is assigned to a higher bfield stratumc F. These stratum designations are retained for the queue. Presumably the present construction of a quantum bracing operator I (§4.1.4) is not the last word on the subject. This operator completely ignores the grade of its operand, embracing a million monads as readily as two, like a mathematical black hole. But tinkering with the brace would take us far from current physical theories and open too many possibilities. While it seems ab- surdly optimistic to assume that the stratified structure of quantum nature is close to that of classical sets, accepting classical set theory uncritically, as the standard model does, is even more optimistic, and yet the standard model already works rather well. Assuming that classical logic works exactly on all strata but one is betting on a case of measure 0 in a large family. Assuming a small change that leads to regularity is following the successful precedent of canonical quantization, with no new concepts, and no new degrees of freedom. Rather, full quantization further prunes vector dimensions already pruned by canonical quantization. The main hazard of this exploration is getting lost in the jungle of possibilities. The strategy of uniform statistics greatly reduces that risk; but it is another speculative unification. 1.2. THE IDEA OF THE QUEUE 37

A field of coefficients, R or C, is regarded as given from the start. The exponential D space of a vector space is defined using familiar operations of +, I = {...}, ∨, and † on vectors. These have the following physical interpretation. The operational characterization of quantum addition or superposition is suggested by projective geometry. If the transition probability between an input process and an output process is 0, they are said to bmissc or boccludec each other. One input process w is a bsuperpositionc of others u and v if and only if every output process that occludes u and v also occludes w. In classical theories the vectors are restricted to a bsis and become states and a superposition of u and v is either u or v. D The bbracec operator I = {...} of set theory is indispensable in physics today, un- D derlying both quantization and gauging, including general relativization. In physics bI c represents physical unitization or association. When we say in classical mechanics (for example) that the position of a particle is a function x(t) of time, we express that function D D as a set of bracings {x, t} := I x ∨ I t of position values and time values. To respect quantum superposition, the quantum brace is defined as a linear operator. It converts a probability vector of any grade and rank into a first-grade vector of the next rank. Catenation, the multiplication ∨, represents performing input operations in sequence, and composes a sequence of g vectors into a tensor of bgradec g. Every element can be decomposed into homogeneous parts by bgradec; Gradeg V designates the g-grade subspace of a Grassmann algebra V . V The bexponentialc 2 of a vector space V is defined here as the vector space of formal polynomials in the braced vectors of V , reduced modulo the commutation relations of the statistics, usually Fermi. 2V is the vector space of a collection of a variable number of V -systems, and the vector space of one 2V system. The brace is convenient when V itself is a space of polynomials, permitting one to use the same multiplication sign, usually none, for both algebras V and 2V without confusion. The queues of all strata taken together form the minimal family of quantum systems closed under exponentiation. To define the exponential space, it remains to specify formally the bracing operator I and a probability form H. Iterated Fermi statistics without the probability form leads to a nested family of linear groups. The resulting vectors are polynomials in the bmonadicsc (elements of grade 1) of Table 1.1, which are anticommuting vectors of norms kψk = Hψ ψ = ±1. All dynamical variables of a queue are polynomials in its generators and all their commutation relations derive from statistics. The generators in turn are expressed in D terms of +, I , and ∨. If any things are fundamental in this theory, at least for the present, they are the D meta-processes +, I , ∨. Quantum theory today has a mixed salad of groups: unitary groups of quantum origin, and canonical, orthogonal, and permutation groups of classical origin. Some canonical 38 CHAPTER 1. STRATA OF ACTUALITY

......

16 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ F 6 222 ...... ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ E 5 2216 ...... ˜ ˜ ˜ ˜ ˜ D 4 216 C 3 16 ˜ B 2 4 A 1 2 0 1 −1 0

L L dL MONADICS OF STRATUM L

Table 1.1: Monadics resulting from ≤ L iterations of the classical exponential-set functor P, the Grassmann functor P0, or the Clifford functor P, for L ≤ 4, with small samples of strata L = 5, 6. The known universe is too small to print the monadics of stratumL = 6 in a Planck-size font. Those of stratum5 would fit onto 64 pages of 12-point roman. KEY: L is the tentative stratum assignment of (4.33). L is the absolute stratum number. S[L] is the space of all vectors of rank ≤ L. dL := Dim S[L] = 2L.A bar represents bracing D I . A stroke represents 1, the empty. A tilde ˜labels a monadic of negative square. 1.2. THE IDEA OF THE QUEUE 39 groups come classical ones by way of canonical quantization, others come from the differential calculus of space-time coordinates. The main orthogonal group is the Lorentz group. The main unitary groups are the U(1), SU(2), SU(3) of the standard model and the SU(∞) of quantum field theories. Full quantization as practiced here extracts all these groups from the invariance groups of quantum statistical assemblies or queues. The Fermi operator algebra alone is not an adequate grammar for quantum physics. It expresses the misses ( ø ) but not the hits (H). It must be augmented with a probability form H for the many-quantum assembly. This is generally induced by a one-quantum form also designated by H. The theory ϑo uses an indefinite mean-square form for this purpose, defined in §1.3.8. The resulting H-algebra has su(N+C N−C) symmetry for complex coefficients C, so(N+R N−R) for real coefficients R. The Fermi operator algebra is trivially stable against small changes in H because H does not enter into its structure. The H-algebra is stable against small changes in H provided that H is regular; which is therefore assumed. The algebraic grammar used in this work is not dramatically different from some considered earlier [39, 40]. Every bsharpc — that is, maximally determinative — input process is still represented by a ray in a recursively generated Grassmann-algebraic vector space. What is added is the strategy of full quantization and its application to the standard model and general relativity. The identification of the organization of i, the quantized i, as the organization of the metrical structure of space-time and gravity is a corollary. Houses are built from the bottom up. Some try to construct the next theory from the bottom up – that is, from intuitively appealing axioms; I admit that I have. This is a trap. Our intuition develops in our least critical, least experienced years. It is rarely quantum or relativistic enough. One might consider constructing a theory from the top down, from raw experimental data, instead. This too is impractical. Only theories tell us what the experiment is about. The next theory develops from the most advanced form of the last, according to Schwinger; from the bottom downward, according to Laughlin. Trees grow upward, downward, and sideways all at once, outward from their center. Here we imitate trees, not builders. D In the theory ϑo, some basic constructs +, ∨, I , R, g, H, i are supplied at or near the bottom. The causality form g at the field stratum F is variable because the cellular content of the cosmos is variable. It is likely that the same holds for H and i. The bdynamicsc is specified by a bdynamics vectorc Dh, usually exponential in an action operator, that assigns a probability amplitude EhD to the experiment specified by an experiment vector Eh. The theory ϑo draws its dynamics vector from the existing theories of gravity and quanta. It is likely that the global quantum i is emergent like g, because of how it composes when systems are composed. Infinitesimal symmetry generators of separate systems add; 40 CHAPTER 1. STRATA OF ACTUALITY and finite symmetry transformations multiply; but i’s equate. This signals that i comes from outside the systems being composed, from a place that is not changed by composition, as from stratum G, the metasystem. The centrality of i presumably results from the law of large numbers, so it must emerge on a high stratum; stratum G will do. A similar argument applies to g, which is shared by all systems at a common event in the canonical theory. Einstein’s general relativization can be seen as a singular limit of a quantum gauging of g (§1.2.7). The emergence of H is discussed in §1.2.4. S The model ϑo is expressed within a general-purpose self-Grassmann algebra S = 2 large enough to represent a queue of any finite stratum number. By a bbilinear spacec is meant a vector space provided with a bilinear form. Ev- ery probability vector space is provided with a bilinear form H that defines its bassured transitionsc or hits ψ → Hψ and the relative transition-probability amplitudes Hφ ψ. A bilinear form with this physical interpretation is called here a bprobability formc. It vanishes for forbidden transitions. If V is the vector space of a queue of stratum L then a bClifford algebrac over V designated by 2V is the vector space of the queue of stratum L + 1. 2V consists of polynomials in the vectors v ∈ V identified modulo the Clifford Clause v2 = kvk = H v v. This raises the question of how nature determines the probability form H. The initial 2 space S[0] = R has a natural norm krk = r , the numerical square. If a norm is given on V it propagates to 2V = Poly(V ) in a well-known natural way. One may then propagate the norm from one stratum to the next. Unfortunately this convenient form is positive definite, and therefore unsuitable for the probability form of a fully quantum theory. Queues of any finite stratum are free of the infinities mentioned above. All their variables have discrete bounded spectra. Every canonical quantum theory is as close as desired in any given domain of experiment to a queue theory.

A canonical quantum theory is an approximation to a better queue theory.

In this scheme, a queue of stratum 0 has only one vector, the number 0. The queue of stratum 1 has the 1-dimensional vector space R. The first non-trivial queue is the quantum bit, or bq bitc, of stratum 1, with two-dimensional vector space. As several quanta combine into one, polynomials in monadics form polyadics, which form an associative algebra, graded by algebraic grade and generated by individual vectors. This many-system algebra of a system is not to be confused with the operator algebra of the system. In the case of odd statistics, adducing the multiplication operation enlarges the vector space from V to a Grassmann algebra 2V , an exponential growth in dimension. In the exponentiation process every polyadic of the current stratum serves as a monadic D of the next stratum. This process is represented here by the bracing operator I , a linearized version of the bracing operation {...} of set theory. To construct an algebra that is its own exponential, S = 2S, it suffices to close S under finite superposition and multi- 1.2. THE IDEA OF THE QUEUE 41 plication, and bracing. Bracing is used in theoretical physics to express the idea of “at”; for example, to link the variables of classical mechanics to the time at which they are measured, and field operators with the point of space-time at which they act. But it is usually left implicit and therefore classical, part of the infrastructure of the theory, without operational physical meaning. To complete the formulation of the physical theory one should define bracing itself by laboratory operations, as one does superposition (in (2.14)) and multiplication. That cannot yet be done, Instead, such operational meaning is given to many structures formed by bracing. It is supposed, however, that under sufficiently drastic condition a set can separate into its factor event elements. Bracing cannot be identified with physical binding, as of a proton and an electron bound into a hydrogen atom. But physical examples suggest that bracing underlies physical binding. Electric binding of atomic electrons to the nucleus rests on the bracing of electric bfieldc variables to their space-time points. A passage from a theory of yes-or-no questions — predicates — about an individual to a theory of how-many questions — occupation numbers — about a collection can been called bquantificationc; William Hamilton used the word in 1846. In quantum physics it was somewhat inappropriately named “bsecond quantizationc”. Here the older word is retained and extended, for the sake of interdisciplinary consistency. In classical theory a quantification defines a functor from state spaces of individuals to state semigroups of collections generated by the individual state spaces, with the semigroup product operation representing the symmetric union or xor of collections or classes. In both classical and quantum usage, the product operation that is introduced by quantification melds two systems into one. In the full quantification of ϑo a many-quantum vector space is a Grassmann algebra over the one-quantum vector space. Its Grassmann product ∨ is analogous to the classical partial operation por, for which

v por v = v ∨ v = 0, (1.10) defining no ray. In such quantum theories, the classical semigroup becomes a quantum Grassmann algebra, thanks to quantum superposition. Iterated Fermi quantification provides a nested sequence of linear groups SL(N), one each for each iteration. The variables of any stratum are finite matrices, becoming arbitrarily large as the stratum climbs. The spectra of these observables are finite and bounded. Space-time and quantum can then be fitted together into one space-time-quantum construct of several nested strata; six non-trivial strata suffice here. The lower-stratum algebras are used here for the internal Lie algebras of the standard model; the higher-stratum ones for space-time and Fermi-Dirac quantification (graded) Lie algebras. The classical analogue of a queue is a brandom setc, a classical finite pure set as a statistical object; the bpure setsc being those generated from nothing by unions and 42 CHAPTER 1. STRATA OF ACTUALITY braces. (§3.1.1). This analogy is useful as a guide to setting up the algebraic language for queues. The classical structure has multiple strata of fan-out and fan-in, and is random in a classical sense. The classical finite pure sets are elements of a group S, the state space of the random set, with the xor operation as group product. It is convenient to represent random sets in the real Grassmann algebra S generated by finite addition, bracing, and catenation, subject to the Grassmann Relation that the square of a unit set is 0. The only elements in S interpreted as descriptions of random sets, however, are the monomials in the basic unit sets, which represent sets of every cardinality, without their superpositions. Their rays form a ray basis for S isomorphic to S, whose elements correspond to the classical sets. Statistical descriptions of a random set are probability operators ρ : S → S having the rays of S as eigenrays and probabilities as eigenvalues. S is a bself-Grassmann algebrac in that S = 2S. (1.11)

Stratum −2 has the empty vector space. Stratum −1 has the one-point vector space {0}. Stratum 1 is the bit, with two-dimensional vector space. These are seeds for the whole tree of classical recursive, hereditarily finite random sets. The canonical quantum algebra in use today replaced classical probabilities by quantum probability amplitudes, and modified the algebra of addition and multiplication. It D has only one quantum stratum, and no quantum unitizer I . This poverty blocks the formulation of stratified quantum theories, leaving quantum theory in what seems to be a stage of arrested development. The quantum analogue of the classical random set is a hypothetical quantum element, the quantum set, or queue, whose vectors are the elements of S, now with unlimited superposition, without the restriction to S. S is used here to replace, regularize, and extend the canonical algebra, replacing both Fermi-Dirac and Bose-Einstein canonical algebras.

The bcanonical quantizationc strategy in a nutshell is: Express the classical canonical system and its algebras at the dynamical stratum as singular limit of a canonical quantum system and its associated algebra. This tacitly freezes the deeper-stratum algebras in their classical forms and perpetuates a singularity at the dynamical stratum.

The bfull quantizationc strategy is analogous but extends to all strata: Express the canonical quantum theory and its algebras at all strata as singular limits of a fully quantum theory and its stratiﬁed algebra. For example the theory ϑo replaces all canonical Lie algebra relations [p, q] = i of the singular theory by fully quantum Lie algebra relations

[p, q] = r, [r, p] = q, [q, r] = p (1.12) in an orthogonal Lie algebra, on all strata. 1.2. THE IDEA OF THE QUEUE 43

1.2.2 The Cliﬀord algebras of Fermi and Dirac

Two major Clifford algebras of the standard model must be reconstructed within the queue algebra S⊗SD and so can be used to guide its design: the Dirac Clifford algebra of electron µ spin with four anti-commuting generators γ ; and the bFermi operator algebrac of (say) an electron assembly with generators ψξ(x) distributed over a classical space-time variable {x} and a spin-charge variable {ξ}. These are not strictly comparable. The Fermi Clifford algebra is constructed over the direct sum of a vector space and its dual. Namely, it is a Clifford algebra over a neutral duplex space NR NR, reviewed in §3.4.1. As the operator algebra over a Grassmann algebra, it fits naturally into S ⊗ SD. The Dirac Clifford algebra is constructed over a Minkowski space-time tangent space. It is also presented as a Fermi algebra over semivectors by Brauer and Weyl [§3.3.4]. In µ either case it has anti-commuting tensors γ of integer spin. This violates the usual bspin- statisticsc correlation for particles, but that applies to quantum particles, which belong to stratum E, while spin γµ seems to reside on stratum C. The Fermi algebras used in physics impose anti-commutation relations on spin 1/2 vectors in keeping with the bspin-statisticsc correlation. The Dirac algebra used in physics imposes anti-commutation relations on operators γµ that transform as Minkowski vectors and as operators on spinors. This is no violation of the usual spin-statistics correlation, which is asserted only for the particle stratum E. The γµ, being so few in number, can be assigned to the operator algebra of a lower stratum, here designated as stratum C while its absolute stratum is being decided. Evidently 4 = C < E = 6 as conservative trial estimates.

1.2.3 The origins of i

In a real-number formulation of this fully quantum theory, the central i of the quantum equation of motion, the canonical Lie algebra, and quantum superposition, is replaced by one real antihermitian dyadic re not intrinsically different from pe and qe. The operator re that becomes i in the singular limit is supposedly fixed in magnitude at its maximum possible absolute value by a physical organization process like magnetization, and can be considered to be a quantized i except for its magnitude. Suitably renormalized re becomes an antisymmetric i = −iH such that the eigenvalues of −i2 are ≤ 1. In the ambient queue and in the canonical limit, i → i, a constant. Such an i-organization is the opposite of complexification, a mathematical process which adjoins an i from outside the algebra. Complexification enlarges a real algebra to a complex over-algebra of fourfold greater real dimension; while i organization reduces a real algebra to a subalgebra of half the dimension, isomorphic to a complex algebra. The i of complexification is absolute and central. The i of ϑo is contingent on organization and non-central except in a singular limit, and is expected to be a frozen form of a gauge 44 CHAPTER 1. STRATA OF ACTUALITY variable. Organizations of i and g on stratum F out of elements on stratum C can be compared. Indeed i and g have similar basic tasks, to convert generators of infinitesimal transformations to real variables, the conversions being bilinear and linear respectively:

µ ν µ g : v 7→ gνµv v , i : qe 7→ q = iq.e (1.13) Einstein replaced the constant chronometric form of special relativity by the variable one of general relativity on the basis of the Gravitational Equivalence Principle. The existence in present quantum theory of a constant global gauge generator i flouts the Einstein locality principle underlying general relativity, by establishing a remote comparison of vector phase angles. The basic relations of a local theory relate variables at a point to those in its infinitesimal neighborhood according to Einstein. The fully quantum version of this principle is that the basic relations relate a bcellc to adjacent cells only. Then the fully quantum theory must be chosen to be real not only for the sake of structural stability but also for the sake of upholding the strong locality requirements underlying the theory of gravity, localizing i and g at the same time. The imaginary i should be the limit value of a dyadic gauge variable bi associated with a gauge connection and a physical interaction. With the dyadic 8i, the familiar exponent of quantum gravity, ix R (dx)R, formerly bilinear in the dyadics of events, is now trilinear i in the dyadics rather than dyadic. This trilinearity [hopefully] unifies the gauge fields with their sources and approximates the usual Y-shaped Feynman diagrams of the standard model. (§7.4). It is not difficult to surmise which interaction this might be, if any. In an early study i was proposed for a Higgs variable connected with the electroweak interactions. Nowadays this is obviously wrong. The coupling to i is as universal as quantum theory. It cannot be limited to the electroweak interactions. The only known interaction as universal as i is gravitation, but this is a tensor force. It seems that the gauge theory of i is the gauge theory of some totally new and equally universal scalar interaction.

1.2.4 The origins of H A Fermi statistics has the commutation relations

D ∀v ∈ V, u ∈ V : ubvb + vbub = u ◦ v (1.14) where the hats are worn by creation-annihilation operators and the circ operation evaluates the dual vector u on the vector v. The Fermi commutation relation (1.14) neither assumes nor provides a probability form, nor an imaginary unit, not a causality form for its input vectors v. They must have another origin. They could be provided from the bottom up, from the top down, or from both top and bottom at once. An operation is local if it can be carried out point by space-time point. The probability form H deﬁnes the transition amplitude Hv u between two input vectors. Is H local? If not it is likely to be emergent., provided by a higher stratum. 1.2. THE IDEA OF THE QUEUE 45

The probability form seems to be local in the canonical theory; no integral kernel is needed to form the H-adjoint of an operator. Therefore H is induced from the bottom up. To maintain Lorentz invariance of the fully quantum theory, the induced H must be indeﬁnite, on every stratum where it exists. And to reconstruct the standard model H must become deﬁnite in the singular canonical limit. The fully quantum theory ϑo, which constructs its variables within the line of Lie algebras so(NR) appropriate to Fermi statistics, sets out to fully quantize a loop theory of gravity, for the reasons indicated in §1.2.3, with i emergent from global organization on a higher stratum and H induced by one on a lower stratum. Quantum variants of the brace and de-brace operations are linear operators on S assumed to obey the ladder conditions

D D D D −1 T −1 T I I − II = 1, I 1 = 0, I = H I H = (HIH ) , (1.15) in which T indicates the transpose. These commutation relations between I and its adjoint D D I define a separate Lie algebra with three generators I, I , 1 that is structurally unstable and therefore suspect ([66, 32]); but this instability seems harmless, since only a finite number of I strata are used, stopping here at stratum G=8. Classical physics carries finite groups, Lie groups, and some functional groups with arbitrary functions as parameters; including permutation groups, the Poincarégroup, the canonical group of classical mechanics, the diffeomorphism group of gravity, and the gauge groups of the standard model. Many of these groups are singular. Nevertheless canonical quantization seeks unitary representations of them all. In a full quantization they are first approximated by simple Lie groups, here finite- dimensional linear groups acting on strata §[L] in one space S, by assuming ad hoc organizations as necessary. The so(16R) of stratum-4 vectors is large enough to include the local bgauge groupsc of gravity and the standard model. It is used in this way in the theory ϑo. Each stratum of a physical theory has its own Lie algebra to be simplified. A canonical quantum field theory has a quantum field stratum F and a classical event stratum E; in a fully quantum theory all strata are quantum. The Kaluza-Einstein-Mayer theory [30, 31] has extra dimensions on the stratum of space-time differentials (stratum D) and tangent vectors but not in the space-time event stratum (stratum E). Somewhat similarly, ϑo has extra dimensions on a lower quantum stratum C. Its stratum E has the four observed macroscopic dimensions, and inherits the microscopic dimensions of stratum C and D as well. Quite varied catenation processes connect these strata in a canonical quantum theory. One passes from stratum D to E by integration, and from E to F by quantification with an appropriate statistics. In classical set theory, nevertheless, one mode of catenation, set formation, generates the self-exponential set S = 2S of all finite sets, which forms a common syntactical framework for all other modes of catenation. 46 CHAPTER 1. STRATA OF ACTUALITY

The strata of ϑo are treated more uniformly. They all fit into a fully quantum theory based on one statistics with simple Lie algebras of one Cartan class. Stratum L has a real Grassmann algebra S[L] ⊂ S within the embracing self-Grassmann algebra S. This is analogous to classical random set theory, which is its projectively commutative restriction of S to rays of one basis. Classically, one integration x = R dx carries us from infinitesimal differentials to cosmological distances; from stratum D to stratum E. The quantum correspondent of such an integration is exponential space formation P. There seems to be little sign of a distinct physical space-time unit in the range of sizes between subnuclear and cosmic. One possible indication of such a unit, to be sure, might be the cosmological constant, a certain zero-point energy per unit space volume, or zero-point action per unit space-time hypervolume. The space-time cell can be chosen in the form of a light diamond or double cone. A hypercube about 0.1 mm across contains one unit ~ of bzero-point actionc. There seems to be no effective cell structure of that scale. Therefore a single application of the exponential functor P must make the bcosmological leapc from subnuclear to cosmological strata. Roughly speaking, each monadic of a queue stratum corresponds to a quantum vertex of that stratum, made up of vertices of the parent stratum beneath. The cosmological leap must go from a merely large dL to a cosmologically large dL+1:

The electron and the cosmos are one stratum apart.

A glance at the values of dL in Table 1.1 shows that the first candidate for the cosmological leap is that from stratum 5 to stratum 6. Moreover, stratum 6 is already so large that there seems to be no physical motive at present to place the bcosmological leapc at a higher stratum. ϑo therefore pegs the charge, differential, event, field, and meta-strata at the values shown in (4.33).

1.2.5 Notation Rank and stratum are related by r ≤ L; neither determines the other. A bracketed argument as in ω[L] and S[L] generally speciﬁes a stratum. The bhyperexponentialc 2L is deﬁned by

(2L) 20 = 1, 2L+1 = 2 . (1.16)

Then 65536 2L = 1, 2, 4, 16, 65536, 2 , 26,..., for L = 0, 1, 2, 3, 4, 5, 6,.... A basis for the vectors of the strata A, B, ..., G will be indexed by lower-case forms a, b, . . . , g of the stratum designations. For example se, se0 ,... designate basis elements for S[E]. 1.2. THE IDEA OF THE QUEUE 47

A basis for the Lie algebra of each stratum is indexed by corresponding upper-case Latin letters as in ωA, ωB, . . . , ωG. For example the ωE are operators on S[E], and span the Lie algebra so[E]. The index A (say) can be regarded as a skewsymmetric pair aa0 with a traceless condition. By a bcanonicalc quantum theory is meant one with at least some operators in the canonical relations α α [q , pβ] = iδβ , α, β = 1,... (1.17) deﬁning the canonical Lie algebra h(n). An banti-canonicalc quantum theory is one with α α anti-commutation relations [q , pβ]− = iδβ among its basic variables. A bcanonical quantizationc respects canonical Lie algebras on every stratum where they occur, representing them in a Hilbert space for the dynamical stratum and and preserving them intact for the lower strata. A bfull quantizationc replaces all canonical Lie algebras on all strata by Lie algebras of classical groups, which are simple groups of isometries of vector spaces. Quantum systems are considered with real vector spaces as well as complex. If V is a real vector space then C ⊗ V is a complex vector space; if U is a complex vector space ∼ then RU designates — improperly — any real vector space V such that U = C ⊗ V . The concepts of the real and complex theories are structured so that an operation R extracts a real theory from the complex. For either a real or complex vector space, an borthonormalc frame of reference F consists of vectors of norm Hψ.ψ = ±1 Hψ.ψ = ±1 that are mutually orthogonal: Hψ.ψ0 = 0 for F 3 ψ 6= ψ0 ∈ F . A vector of positive norm represents a feasible input-output operation. The interpretation of a vector of negative norm is discusssed in §1.3.8.

1.2.6 Q terminology A queue has a many-system algebra with the usual stratum-preserving Grassmann operations of vector addition + and composition b∨c, with the customary physical interpretations. It also has a brace operator I on its vectors that raises stratum by 1. This is a quantum version of the brace operator of set theory, which presumably developed from considering collections of entities like sheep, stones, or taxpayers, but I applies to collections of quanta. It can also be written with braces or a bar:

Is = {s} = s. (1.18)

I metaphorically boxes its argument, converting many into one, and a polyadic into a monadic. A many-quantum vector is a tensor with a string of indices like Tzy...ba, one for each quantum. The operator I converts this tensor to one with a single compound index, written as T{zy...ba}. The compound index {zy . . . ba} may then enter as a single index in the string of indices on a tensor of the next stratum. 48 CHAPTER 1. STRATA OF ACTUALITY

Since classical set theory has worked reasonably well in quantum theory, only changes in it necessary to conform to the quantum principle are made here. Since the vector space of the standard model has a probability form, one seeks a probability form for the queue. A probability form H on a vector space V induces several diﬀerent forms H on the vector space 2V of a Fermi assembly. A H is assumed to be an anti-automorphism of the Grassmann algebra:

H(u ∨ v) = Hv ∨ Hu. (1.19)

Stratum L = 1 has vector space R, and it is natural to assume that the hermitian adjoint H of a real number is itself. This H propagates up the strata if one makes a suitable assumption about the eﬀect of bracing and multiplication on the norm. The theory ϑo uses the bmean-square normc (E:MEANSQUARENORM) for H. In addition organization provides a quantum probability form H and imaginary i that provide the canonical H and i in a singular limit. The composition of no systems is the trivial queue whose vector space is the empty set; this starts the recursive construction at stratum −2 The quantum bbracec operator Is is a linearized version of the usual brace operation {s} of classical set theory. Left-multiplication by a vector in a many-system algebra is a linear operator interpreted physically as a quantum bgenerationc — creation — operation, and usually called a creator. A set is usually a mathematical object, not a physical one. A queue is supposed to be a physical process, not to be confused with any of its vectors. Unless otherwise declared, queues have Fermi statistics. The theory bϑoc [hopefully!] is a fully quantum variant of the standard model and gravity and their space-time. It has operators that generate events themselves, and the events have Fermi statistics. If vectors in the space V represent input processes, those in the dual space V D represent output processes, so that w(v) =: w.v, the value of w on v, is the relative transition probability amplitude. These vector processes are system–metasystem interactions. Operators represent processes that go on entirely within the system, when no one is watching. Nevertheless, since a dyadic product of vectors is a linear operator, vectors and generators are assigned to the same stratum here.

1.2.7 The origins of g In the canonical theory a field g(x) of causality forms guides classical point projectiles like planets along their trajectories. Quanta with spin like electrons require more guidance, provided by a spin form γµ(x). In a quantum theory of gravity these forms become operators on the graviton field as well as on its material sources. In the weak field limit they create and annihilate gravitons. 1.2. THE IDEA OF THE QUEUE 49

If a full quantization enlarges the event group from SO(3, 1) to (say) SO(10, 6), presumably these forms grow correspondingly. The spinors of SO(10, 6) SPINOR Briefly put, Dirac defined how spins move in special relativity by mapping momentum µ D vectors p = e pµ in the cotangent space (dM) of Minkowski space-time into Clifford- µ algebra elements γp = γ pµ so that Clifford’s law holds in the form

(γp)2 = g p p, (1.20) and imposing the equation of motion

[γp + m, q] = 0 (1.21) on dynamical variables q not explicitly depending on time, in which m is the quantum rest mass. Here γ is called the (Dirac) bspin vectorc. In classical general relativity g and γ become functions g(x) and γµ(x). In a canonical quantum gravity g(x) and γ(x) generate gravitons. The Dirac spin algebra is both a Clifford algebra over MInkowski vectors and an operator algebra over spinors. Its operators are observables in a generalized sense, of a quantum system that may be called a bspinc, whose probability vectors are spinors. For a quantum theory with probability vector space V one also requires a probability form H : V → V D and (especially to form an action principle) an imaginary unit i : V → V . The 16 basic polyadics sc ∈ S[C] for C = 3 give rise to an operator algebra Alg S[C] that accommodates the spins and charges of the standard model and gravity. The four monadics among them are used for the spin vector γ of special relativity. In turn the four basic monadics arise from a lower stratum B. This transition B → C is represented by the transition from stratum 2 to stratum 3 of Table 1.1. The sb are already quantum, not classical, and require no regularization. In queue theory all monads are creation (and annihilation) processes, and all monadics are creators. µ Stratum D ≈ 4 is too small for the canonical Lie algebra L(x , ∂µ, 1) of coordinates µ x and differentiations ∂µ of classical special relativity; but the canonical Lie algebra µ h(x , ∂µ, 1) is a singular limit of the orthogonal Lie algebra so(6) ⊂ so[C]. General relativization too is an analysis of a system into atomic elements, though classical: namely, the analysis of space-time (stratum E) into tangent spaces (stratum D), one at every event. This classical analysis in turn is regarded here as a singular classical limit of a quantum analysis of a field queue of a dynamical stratum F into cells of stratum E which are also events, each endowed with a fixed causality form g. It can therefore be D expressed using the brace operation I on S. The variability of the gravitational space-time manifold then derives from the variability of this collection of cells. µ When the variables x , ∂µ of general relativity are fully quantized they become operators representing generators of so[C] within so[E]. Provisionally stratum E is two levels 50 CHAPTER 1. STRATA OF ACTUALITY higher than stratumC. The fully quantized coordinates and momenta and i now generate a regular Lie algebra so[C] instead of a singular canonical limit h(4). In the classical theory a change in frame in a space-time manifold M is represented by an element of Diff[M]. Since all is quantum, physical changes in frame might be automorphisms in SO[E] of an event vector space S[E] relative to a probability form H but they cannot actually be point mappings of some classical state space. Diff[M] does not represent the physical group of quantum gravity any more than the canonical group of classical dynamics represents the unitary group of the quantum theory. This conclusion is entered into line E of the tabulation in (4.33). Canonical quantization is nevertheless a useful guide to a full quantization, which may be required to have a canonical quantum theory as singular limit. Canonical quantization of gravity in the Einstein form introduces the bprobability formc H of the quantum theory and converts the classical causality form g to a canonical quantum hermitian operator g(x) that generates gravitons. Canonical quantization unifies both the commutator and the Dirac-Poisson Bracket Lie algebra into one commutator Lie algebra in which the elements of gνµ(x) of g(x) and νµ the canonically conjugate variable πb (x) no longer commute but obey canonical relations R 2 ν µ subject to gauge constraints. The integral dτ, where dτ = −dxgνµdx dx , is now a graviton generator as well as a proper time operator. The canonical quantization leaves stratum D and E classical, and quantizes only stratum F. A fully quantum theory like the theory ϑo still has quantum strata D, E, and F corre- µ µ sponding to the canonical D, E, and F. The sea variables dx , x , gνµ correspond to queue operators in associated Lie algebras so[D], so[E], so[F ], respectively. These Lie algebras are represented within Clifford algebras of successively higher strata D, E, and F, in a way that respects the stratum structure of present canonical quantum field theories.

1.2.8 Fields and queues

Once coordinates fail to commute, the usual classical bfieldc concept is not available. It requires the construct of functional relation; a fully quantum theory has the concept of relations between quanta but not that of functional relations. But canonical quantum field theory already suggests an alternative pathway: A canonical quantum field is also a catenation of quanta; a field operator represents the input or output of one quantum. Catenation can take over many duties of functional relation. A field operator ψ(x) inputs or outputs one quantum at x. In bFermi statisticsc, ψ(x) is an operator on the vectors of a quantum catenation; the field as a q system can be reconstructed from a queue whose elements are quantum events. Here a vector of each stratum is a catenation of associations of elements of a lower stratum obeying a certain statistics, in the way that a simplicial cell is a catenation of its vertices. Physics without functions is discussed further in §6.1.3. A queue is not a field in the familiar sense of a functional relation between a field- variable and a space-time. Einstein and others advocated a bunificationc of bfieldc and 1.2. THE IDEA OF THE QUEUE 51 space-time into one construct; Feynman proposed to eliminate the field. The queue does both. The quantum spaces of ϑo have vector spaces in one stratified bself-Grassmann algebrac S. Some dimensions in one stratum C of the algebra of such a quantum space will be assigned to space-time, other dimensions of the same bstratumc will be allocated to internal degrees of freedom of a bGrand Unified Theoryc the bstandard modelc, or to the complex plane. Dimensions of stratum F in the same algebra are to be used for dynamics and bgaugec transformations, including general covariance as a singular limit. All these vector spaces and their isometry groups and Lie algebras are represented in the stratified bself- Grassmann algebrac S.

1.2.9 Dynamical law of the queue In canonical theories, classical or quantum, a kinematical algebra gives some relations among dynamical variables at one time, and a dynamical law gives additional relations between dynamical variables at different times, reducing the kinematical algebra to a quotient algebra called the dynamical algebra. Here it is necessary to fully quantize the classical construct of natural law. The hypothesis of a fixed absolute fundamental law of nature seems to have no place in a fully quantum theory. Such a law controls the system only between input and output processes. Quantum measurement always violates the dynamical law of the system measured. The space-time event at which a field is measured or a quantum is created is not determined by the system but by the experimenter in the metasystem, so canonical field theory couples system field variables to metasystem space-time variables. Its most serious infinities therefore arise from its theory of the metasystem, not of the system. To regularize the theory by quantization requires quantizing at least the spatiotemporal variables of the metasystem. This contradicts early strictures of Bohr requiring the metasystem to be described classically; but it accords with Bohr’s later view on this matter [13]. To avoid possible conceptual inconsistency, it must be remembered that the experimenter and most instruments cannot function or even survive as such if they are observed with maximal quantum resolution, as a vector for the metasystem implies. For example, measuring the positions of the electrons in a structure like an organic molecule or a crystal probably breaks the bonds of the structure. A maximally observed experimenter can no longer experiment. Bohr avoided this inconsistency when he restricted the description of the experimenter to a classical stratum of resolution, which can be non-invasive. A sufficiently low-resolution or b gentle determinationc of the metasystem is described by a bprobability operatorc ρ, not a vector. Generally an experiment is described by two probability operators, an input one ρ0 and an output one ρ2, for the two ends of the experiment [45, 40]. 52 CHAPTER 1. STRATA OF ACTUALITY

A probability operator ρ representing a gentle determination of the metasystem can be expressed in terms of maximal determinations by a spectral resolution

X n ρ = pnP (1.22) n

n with probabilities pn and singlet projectors P . But this does not mean that performing the gentle measurement described by ρ entails performing one of the lethal measurements described by the P n. A fully quantum theory modiﬁes the canonical construct of dynamical law. Classically the law is treated as an extra-cosmic principle that governs the cosmos; an irresistible action of the unknown on the known. Quantum physics, however, recognizes that most of the cosmos by far is necessarily unknown, and that every experiment begins and ends with violations of the system dynamics. People and instruments cannot function as such when they are sharply determined at the quantum stratum of resolution. Therefore the bmetasystemc — the part of the world that studies the system — must remain almost unknown during an experiment. Furthermore the metasystem is necessarily enormous compared to the system. Then the unknown source of dynamical law need no longer be extra-cosmic, and the violation of dynamical law is not mysterious. Quantum theory permits one to replace the absolute concepts of the known cosmos and the unknown extra-cosmos by the relative concepts of a queue and its metasystem, neither of which admits complete symbolic representation. The dynamical law now appears as a surrogate for the metasystem. It is no longer extra- cosmic, merely extra-systemic. The law of the system might be a partial description of the metasystem. It is therefore useful to provide a standard representation for the metasystem. The main function of the metasystem during an experiment is to emit and absorb systems selectively. It is common in thermodynamics to represent a heat bath for a system, whatever its actual composition, as an ensemble of many replicas of the system. There is a reasonable extension of this practice to quantum theory. One may represent a general source for a system as an ensemble of many replicas. An input vector then represents a mode of selection from this assembly, and dually for an output vector. If the system has Fermi statistics, the metasystem algebra can be represented by the Fermi algebra over the probability vector space:

The metasystem is one stratum higher than the system.

The queue system being on stratum F, the queue metasystem is assigned to stratum G = F + 1. A special or general relativistic theory admits no invariant decomposition of history into instants. This makes history vectors more convenient than instant vectors, in the way introduced by Dirac and developed by Feynman, Schwinger and others. History vectors 1.2. THE IDEA OF THE QUEUE 53 support the representation of actions on the system between the first input and the last output process, rather than actions at one reference instant in that history, propagated to other instants by a given dynamics, Each term in a trace over histories represents an amplitude for transmission of the system through a dense network of filters between input and output. These filters represent drastic but adiabatic interventions in the system development. They change the dynamics without performing determinations. The trace over such histories expresses the adiabatically isolated dynamical development in terms of many different adiabatic ones. A dynamical law is mathematically associated with a history vector that assigns probability amplitudes to every other history vector. In the singular canonical quantum theory time is a continuous real variable and in a suitable frame the quantum phase of the history vector is the classical action divided by ~. Quantum phase, however, is relative to a choice of basis and has no absolute meaning. The Lagrangian function is regarded as the matrix element x0hLhx of a Lagrangian operator between coordinate values at two infinitesimally separated instants. Up to divergent normalization factors, Dirac showed, the exponential of the Lagrangian matrix element for an infinitesimal time can be replaced in the integral by the matrix element of the exponential of a Hamiltonian operator, a unitary transformation for an infinitesimal time: 0 ei x h Lh x dt/~ ∼ x0he−iH dt/~hx (1.23)

It is this unitary transformation, not the exponent Lagrangian, that has invariant physical meaning in the quantum theory. The usual concepts of Hamiltonian and Lagrangian are useful in the classical limit but even in the canonical quantum theory the invariant way to describe quantum dynamics has been by a history vector, a Dirac-Feynman path amplitude.

A history vector used to express a dynamics in this way is called a bdynamics vectorc. It represents what can go on in the system between input and output operations. Such global constructs are often used in gravity theory and the standard model. Using a history vector to describe a dynamics does not mean that the dynamics is a quantum entity like the system. Every vector has one foot in the metasystem and one in the system. The dynamics vector puts its weight on the foot in the metasystem. One might use a vector in 2R to describe the orientation of a macroscopic linear polarizing ﬁlter, although the orientation is regarded as a classical variable in the metasystem. The dynamics vector is supposed to be determined as precisely as one likes by many experiments under the same conditions, and therefore to be a classical object. Classical objects described by quantum vectors are also familiar from superconductivity, as descriptions of quantum organization of many electron pairs. It seems likely that the dynamics vector for stratum F is always an order parameter of a quantum organization of the metasystem on stratum G. In one extremely singular classical limit, the system is supposed to approach the entire cosmos, the metasystem disappears to inﬁnity, and the contingent quantum law becomes the absolute classical law. Quantum theory relativizes absolute law rather in accord with century-old expressions of C. S. bPeircec, Ernst bMachc, and other antinomians. 54 CHAPTER 1. STRATA OF ACTUALITY

In traditional terms, the algebra of the system is a necessary feature of the theory and each element of the algebra represents a contingent feature. One assumes that the system has a ﬁxed kinematical algebra of operators, giving the actions that can be carried out on the system and relating them. Here this algebra is a representation algebra for a Lie algebra deﬁned by a structure tensor C, in a quadratic vector space. This assumes good cosmic weather and large human resources, permitting experiments to be repeated as often as desired. Within that idealized framework we assume a free choice of experimental actions, represented by elements in that algebra. Features of experience that we take to be necessary elements of physical theory we incorporate in the structure of the algebra representing what can happen; those that we recognize as contingent we express in the structure of an element of the algebra representing what actually happens.

The event variables that enter into the fully quantum dynamics of stratum F are usually operators of the previous stratum E. Although (as has been pointed out) space- time variables that ﬁx the location of an input/output action are clearly metasystemic, of stratum G, those that enter into the ﬁeld operators of stratum F are presumably of the event stratum E and so have smaller spectra. In the theory ϑo, E = 5 and G = 7.

If t[E] is a time operator on stratum E and has N values in its spectrum, with a spectral interval of X, then one expects t[G], the time variable of the experimenter on stratum G, to have ∼ 22N values in its spectrum. From the fact that t[G] is quasi-continuous one cannot infer that tE is.

It was originally supposed that X is of the order of magnitude of the Planck time just on dimensional grounds, involving just the gravitational constant G, ~, and c. However we still know gravity only at the classical stratum, and our quantum theories have only much longer times, like the Compton periods of the particles. For current quantum experiments, gravity can be relegated to the metasystem.

The ratio of the Compton time of (say) the electron to the Planck time is a bLarge Numberc of physics. This Large Number is currently accounted for by quark confinement and asymptotic freedom [?]. In a fully quantum theory, low strata have a natural microscopically small limit to their space-time dimensions that makes a certain kind of confinement unavoidable. Naturally one hopes that this confinement is the origin of quark confinement. It is tentatively conjectured here that the time quantum X of the queue has an order of magnitude suggested by the particle masses and coupling constants of the standard model, like a Compton time for the most massive quantum, not the much smaller Planck time, which emerges in higher strata. The Large Number can then be related to the large multiplicities of strata ≥ 6. In the most satisfactory outcome, this quantum account of the bLarge Numberc would converge to the Wilczek theory of the Large Number in a canonical limit. 1.2. THE IDEA OF THE QUEUE 55

1.2.10 The vacuum queue

Probability amplitudes for experiments have the generalized Malus-Born vector form A = DhE, evaluating an experiment vector with a dynamics (vector). To use the theory requires a lexicon of experiment vectors expressing basic experimental input-output operations. For many experiments the experimental process is best described relative to an am- biant bvacuumc vector, which represents no input or output of quanta, and serves as the background for more interesting experiments. The condition of minimum energy seems indispensable for a useful concept of the vacuum, so we restrict the concept to the singular limit of a time-independent action. We divide the time into early, middle and late eras, in which the radius of the system is growing, near maximum, and shrinking. Then In practice there is is a good approximation to a vacuum in the middle era, which can be arbitrarily long in a singular limit. Of course, no experiment is done in a vacuum; there is usually an experimenter around. What is called the vacuum is a singular limit of the metasystem. In actuality there are always perturbations of the system by the metasystem not taken into account in the dynamical theory. In canonical ﬁeld theories a vacuum vector is an instant vector of minimum energy, and therefore of minimum Hamiltonian. The dynamics vector determines a Hamiltonian, and the Hamiltonian determines a vacuum projector. Usually the minimum energy is set to zero, and then the vacuum instant vector hΩ0 is independent of time. A canonical vacuum history vector might have a discretized form that begins and ends in the vacuum, connected by a chain of do-nothing identities. This raises the general problem of relating fully quantum history vectors to instant ones (§6.1.1). Instant vectors are presumably eigenvectors of a time operator; a typical time operator is given in (5.17). In a fully quantum theory, time on stratum E has a uniform discrete bounded spectrum between extreme times ±T . The eigenspaces of time have dimensions that depend on time, and they are not unitarily equivalent. Here it is supposed that we live in middle times, and the vicinity of the eigenvalue t = 0 is taken as typical for present physics. In practice one draws on experimental data as well to guess at the vacuum. These procedures of the canonical quantum theory must have fully quantum correspondents (§6.4), involving the experimenter and metasystem more explicitly than the canonical limit does.

1.2.11 The cosmic crystal

On a macroscopic scale and at low energies, the cosmos near each event today resembles Minkowski space-time. It seems to have Poincarégroup symmetry and to support but break a canoncal group. The translation subgroup and the canonical group, not the Lorentz group, force the continuity and homogeneity of event space. From a more microscopic 56 CHAPTER 1. STRATA OF ACTUALITY viewpoint, they may be illusions of low resolution. From the time of Kaluza to today it has become increasingly plausible that we are astronomically long in only four of our dimensions, but submicroscopically short in most of them, like a membrane. Sorkin and Samuel developed this membraneous perception of the field into a qualitative theory of the cosmological constant, by an analogy between quantum fluctuations of space-time and thermal fluctuations in fluid membranes [71, 65]. The cosmos is even more anisotropic than a DNA molecule or a soap film, with a larger aspect ratio between long and short dimensions. The model that is most appropriate in a quantum framework is not a membrane but a crystalline film. Quantum models are not constrained by the classical assumption that the extra dimensions are those of classical continua, as in Kaluza-Klein theories and string theories. All that is needed of the extra dimensions is the gauge Lie algebra they provide at each event. Classically, to be sure, it takes an infinite continuum of events to support even a finite-dimensional Lie algebra; but in the quantum theory, N independent quantum events suffice to support an so(N) or su(N) Lie algebra. These events may constitute just one queue cell of a cosmic crystal film. The problem is no longer what makes the short dimensions of the cosmic crystal so short. In a fully quantum theory the cell is a given architectonic unit and a crystal film thickness is about 1 atomic unit, requiring no explanation. Now the problem is how the four long dimensions of the cosmic film grew so large. But this is at least a familiar kind of phenomenon, calling for no unusual sources of curvature. Presumably the long space-time dimensions grew like the long dimensions of a snowflake or a graphene molecule: by organizing small queue cells into a long thin queue. Any mathematical formulation of this structure should be tested by comparing its Brillouin zones with the observed particle spectrum. A fit would acknowledge the insight of Newton and Fresnel, who argued for an adamantine ether in the face of much skepticism. The evidence for their crystal has not diminished with the years, if one replaces their assumption of broken Galilean invariance by one of near-Poincaréinvariance. This crystal has no rest frame and its vector has continuous symmetry as well as discrete structure. The extra dimensions mean that each quantum cell of a fully quantum system of the world can have a high-dimensional kinematic Lie algebra, for example, the 255-dimensional sl(16), while residing in a multicellular system with a lower-dimensional local Lorentz symmetry so(3, 1). Iron atoms in a magnet have a reduced symmetry because of the magnetic field they themselves create, and exhibit a Zeeman splitting in consequence. Presumably the quantum cells of the cosmic crystal film too are distorted by the anistropy of the ambient film. The crystal film vector can be regarded as composed of struts connected by pins. Each strut in the truss symbolizes a spin of a classical group; and may itself be a sub-truss assembled from spins of lower-dimensional groups, by addition, bracing, and composition Two struts are pinned by the elements they share. If the cells of the cosmic film are N dimensional, they may organize into four cosmological dimensions if they abut their 1.3. QUANTIZATION 57 neighbors on faces of N − 4 dimensions. As a result of how displacements of the structure are distributed among its cells, a truss dome can be much stiffer against local longitudinal displacements, which couple its long dimensions to each other, than against transverse displacements, which couple longitudinal to transverse dimensions, by a factor on the order of the aspect ratio. In gauge theory, gravitational gauge transformations couple longitudinal dimensions, and electroweak and color gauge transformations couple longitudinal with transverse, with couplings in the ratio of the Large Number. This suggests that the Large Number is essentially the aspect ratio of the cosmic dome. In the fully quantum theory the possibility arises of a self-organization like a spin- alignment that forms a macroscopic coordinate, the total spin, but also freezes it at the peak of its spectrum, so that in the canonical limit it does not appear as a coordinate at all. The theory ϑo uses this possibility to model i. It sets aside 4+2 = 6 vector dimensions of the cell for macroscopic organization: 4 for space-time and 2 for the complex plane. Since i is a scalar, the possibility must be examined that its organization is responsible for a cosmic inflation. In the theory ϑo, the quantum variables proper of the standard model, those with no classical correspondents, like isospin and color, are creators for the short or transverse dimensions of the cosmic crystal dome. These are conveniently represented by operators on stratum 3 of the queue, which has an sl(16) Lie algebra. A truss that is a quasi-continuum can be assembled by stratum 6, within a kinematic Lie algebra sl(264K) acting on fields of events. This suggests that the Large Number can conveniently be represented by the multiplicity of stratum 6 compared with stratum 5 or lower. By stratum 6, exponentiation approximates Fermi-Dirac quantification. The ingredi- ents of such models are intrinsic in the self-Grassmann algebra S. The non-commutative transport of classical gravity appears as a classical vestige of the non-commutativity of momenta in a quantum space-time. While the ordinary infinitesimal concept of locality breaks down when the spectra of space-time coordinates are discrete, a quantum correspondent can survive as a principle of bcontiguous actionc. This asserts that dynamical interactions and basic operators couple cells in the cosmic dome that share some elements.

1.3 Quantization

The canonical quantum theory of bBohrc and bHeisenbergc taught physicists to think about nature at the atomic stratum in a new way. It represented each adiabatic physical operation on an isolated atom — and ultimately on any isolated system under study — by a matrix of relative transition probability amplitudes. Classical transition probability matrices are composed of binary truth values 0 and 1 for deterministic processes, and of positive real 58 CHAPTER 1. STRATA OF ACTUALITY transition probabilities for Markov processes. The matrix for a canonical quantum process is composed of complex numbers instead. The canonical quantum theory thus revises the theory of probability for quantum processes: Catenation of two operations is still represented by the product of their matrices, classical or quantum. This product sums probabilities or amplitudes for all paths in the index space of the basis. Where the classical physicist adds probabilities for all paths, however, the quantum physicist adds probability amplitudes for all paths and squares the absolute value to form a probability. The canonical quantum theory also relativizes the basic construct of state. In the classical theory all experimenters share the same absolute states, differing only in the coordinates labeling the states. In the quantum theory each experimenter has a reference frame of proper vectors, different from all other experimenter frames. Relativity transformations between reference frames are also complex matrices. Filtra- tions, represented by projection operators P obeying P 2 = P = P †, that do not commute for quantum systems. When two filtrations do not commute, determining one property invalidates a prior determination of the other. That class-defining filtrations commute was first explictly postulated by Boole, a logician and psychologist of the 19th century, who spoke of “mental acts of election” [14], quoted on page 2, and recognized that this commutativity was empirical and subject to corrections that would be important. Each variable is defined by filtration operations that sort according to the value of the variable. The non-commutativity of filtrations suspended the commutative laws for the multiplication of variables that had been tacitly assumed for centuries. Classical filtrations were assumed to have no effect on the system filtered but able to provide complete determinations. Quantum filtrations are incomplete but omnipotent: any transition in a system can be effected by appropriately chosen successive filtrations. Bohr described the quantum way of thought as a “painful renunciation” of classical thought in the atomic realm. Quantum non-commutativity interferes with the Cartesian procedure of completely representing the physical system under study by a mathematical object, and with the Laplacian ideal of a fundamental all-determining law. It reconstructs classical bontologyc, a metaphysical theory of absolute beings, as a limiting approximation of a quantum bpraxiologyc, a more operational metaphysics, of operations on operands known only through those operations, and then not completely but only statistically. The vectors of quantum theory are not ontological in the usual sense but bpraxiologicalc; they are verbs describing actions rather than nouns describing absolute objects (§2.10). Bohr’s “renunciation” is a relativization, and was quickly compensated by deeper physical understandings of atoms, molecules, crystals, and nuclei. It also formed conceptual unifications unimaginable in a classical ontology. According to the Bohr correspondence principle classical and quantum physics converge for experiments on such large scales of action that ~ → 0 in comparison. The difference between classical and quantum theories is important on the scale of ~ and never entirely 1.3. QUANTIZATION 59 disappears. 1900 physicists naively treated a solar system as a set of masses, an atomic electron cloud as a set of electrons, a line in space or time as a set of points, and a random variable as a set of its possible states, a state space. One catenation process, set formation, served on every stratum. They tacitly supposed that the constitutive relations found in nature could be faithfully represented by those that hold among sets, and that useful physical probabilities could be found by counting sets of mutually exclusive mathematical possibilities. It was also taken for granted that an ideal measurement need affect the system no more than counting a mathematical set affects the set. Briefly, physics was assumed to be identifiable with a part of mathematics, the physical process with a symbolic one. These assumptions are part of an bontologyc that reduces the main task of physics to finding a part of mathematics that corresponds to reality. It uses a bstate strategyc to do this: Represent a physical system by a mathematical set of states, changes of the reference frame by permutations, our physical actions on the system by mappings of states, and physical variables by real functions of states.

The bcanonical classical strategyc is the state strategy supplemented by a canonical codicil: Represent dynamics by a continuous one-parameter group of canonical transformations, with time as the parameter and a Hamiltonian function as generator. The state strategy works reasonably well for celestial mechanics and much of elec- tromagnetics. Set theory thus serves classical physics as a “philosophical language” [88], although one that is still seriously context-dependent, in that one cannot read the physical meaning of a set from its set-theoretic formula, and highly redundant, in that most sets have no physical meaning. Quantum theory replaced the state strategy by a bvector strategyc: Represent input-output processes by vectors. Represent physical actions, including filtrations, by the matrices of transition probability amplitudes that they define between vectors. Represent physical variables by weighted sums of orthogonal filtrations.

The matrix product represents sequential action. A bprojectorc P (a matrix with 2 † P = P = P ) represents a bﬁltrationc. A weighted sum of orthogonal projectors, called an bobservablec or bvariablec, encodes a multi-channel sortation, each term corresponding to one channel, and each weight being a value accepted into that channel.

1.3.1 Canonical quantization In addition to deﬁning a new kind of theory, the quantum theories, Bohr and Heisenberg provided a heuristic strategy, canonical quantization, for reconstructing a quantum original from a classical approximation. The canonical quantum strategy supplements the quantum strategy with a bcanonical codicilc: 60 CHAPTER 1. STRATA OF ACTUALITY

The commutators of certain chosen basic variables are their bPoisson Bracketsc times −i~, and time is central. To extend the process of canonical quantization, three stages implicit in the usual formulation are ﬁrst made explicit in general terms:

1. Subquantization: Choose basic state variables q, p of the classical system whose linear combinations will serve as subquantum vector space V .

2. Quantiﬁcation: Generate the many-subquantum vector space W = Poly(V ) of polynomials in the subquantum vectors, deﬁning a subquantum statistics by commutation relations.

3. Correspondence: Express the original classical algebra as singular limit of the many- subquantum operator algebra.

For example, consider the quantum harmonic oscillator in units with mass m = 1, elas- tance l = 1, natural frequency ω = 1, and ~ = 1. If x is its coordinate and p its momentum, generating the operator algebra A(q, p) modulo the commutation relation [x, p] = i. The Hamiltonian is 1 H = (p2 + x2), (1.24) 2 Stage 1, subquantization: The system has a subquantum with one-dimensional vector space spanned by the basic vector a = 2−1/2x + ip. Stage 2, quantiﬁcation: The subquantum has bosonic statistics , and every subquantum has the same energy E = ~ω = 1. In this case the oscillator energy is the sum of the subquantum energies: H = Nω = a∗a. (1.25)

Stage 3, correspondence: The classical commutative algebra A0(q, p) of the oscillator consists of the elements of A(q, p) operators of the form : f(q, p) :; here f is an arbitrary polynomial expression in q and p, and the colons : ... : indicate that every product written between them is the commutative product on the non-commutative operator algebra defined by the bWickc bnormal orderingc, in which creators a∗ stand on the left and annihilators a on the right. Commonly the subquantum vector space of stage 1 is not brought in explicitly during the canonical quantization, the canonical relation [q, p] = i~ of step 2 is not interpreted as a statistics from the start, the quantification is tacit, the only quantum algebra mentioned is the many-quantum algebra of step 3, and the subquantum and its statistics are first discovered after quantization has been carried out. Full quantization, on the contrary, begins with a quantification and a statistics. Canonical quantization cannot be relied on to produce the physical constituents of a dynamical system. It can produce modes of collective oscillation instead, like phonons. One 1.3. QUANTIZATION 61 can canonically quantize the electromagnetic field, the elastic vibrations of a carbon crystal, or the transverse vibrations of a steel E string. The resulting subquantum is the photon, the crystal phonon, or the string phonon. It is known that this quantization does not produce the atomic constituents of crystals or strings, and we are not required to suppose that it does for the electromagnetic field. Here it is supposed that all these subquanta are like phonons; that photons too are excitations of a collective crystal-like system, namely the quantum space-time complex itself. Heisenberg suggested that since photons are at least approximately massless, they are, like phonons, Nambu-Goldstone bosons associated with a symmetry-reducing organization of the system, even though they have spin 1. Space-time is presumably not composed of things that move in space-time, and therefore its constituents cannot be found by a canonical quantization, but a full quantization puts forward candidates for the quantum elements of quantum space-time.

1.3.2 Full quantization strategy Full quantization is full relative to canonical quantization. Canonical quantization quantizes one or two strata, and not completely, leaving a radical. A fully quantum strategy quantizes all lower strata, and its quantizations are completed, no radical survives. Graded canonical Lie algebras, Fermi or Bose, are the standard statistics for step 2. Neither classical space-time nor Bose statistics is completely satisfactory for field theories, due to their singular natures, which call for infinite-dimensional representations. In the landscape of Lie algebras, however, any canonical Lie algebra is surrounded by orthogonal Lie algebras of various signatures, as well as by other algebras, both singular and regular, like a singular peak surrounded by several watersheds separated by divides. Any of these watersheds corresponds to a simple Lie algebra having the canonical Lie algebra as a singular limit, but having greater structural stability and fewer infinities. One of these is hopefully more physical as well. By suitably adjusting some parameters in the theory, any of these neighbors can be made experimentally indistinguishable from the present theory by present-day experiments, but inevitably marked differences between the predictions appear under extreme conditions. Canonical quantization effects only the dynamical stratum, while full quantization works on all strata of physical theory, including that of the time variable. The deeper stratum of space-time, in particular, lacks the canonical classical structure, but still has a Lie algebra structure, based on Lie Brackets of space-time vector fields instead of Poisson Brackets, which are Lie Brackets of vector fields over phase space. This Lie algebra too will be regarded here as a vestige of quantum non-commutativity. The canonical commutation relations have an essentially unique irreducible representation, a remarkable and convenient property. Segal [66], Palev [59], and Vilela Mendez [75], for varying reasons, replace the singular canonical commutation relations by regular commutation relations, which, unfortunately, lack this uniqueness of representation, and raising the question of which of the infinity of representations of these relations is physi- 62 CHAPTER 1. STRATA OF ACTUALITY

cal. Fermi algebras are regular and yet have unique irreducible representations, so bFermi statisticsc reduces the problem of choosing representations to the problem of choosing the Fermi algebra. Full quantization regularizes the statistics on all strata of theory, using anticommutation relations rather than commutation relations, resulting in a finite and structurally stable theory. It specifies a unique representation for each stratum. Fermi statistics has unitary Lie algebras su(p, n) instead of canonical Lie algebras. The Fermi algebra model pursued here approximates the the graded Lie algebras of the various strata of the system under study by those of various strata within one bFermi algebrac Alg S, a quantum analogue of random set theory. It reconstructs the canonical Lie algebras, including that of Bose statistics, as singular limits of simple unitary Lie algebras su(p, n) of Cartan class A, as Palev did with the linear Lie algebras sl(n). It is not yet clear which commutation relations come closest to experiment, but likely it is not the canonical, which are bstructurally unstablec (5.1.1). The matrix language of the canonical quantum theory is still the most practical one, and it is used here too, with additional deep structure: the vector space supporting the matrices is also a Fermi algebra and a matrix algebra.. Full quantization sacrifices the concepts of blocalityc, bunitarityc, and bfieldc, but a singular limit of classical space-time resurrects them. The fully quantum construct that replaces the canonical quantum field is the bqueuec of stratum F, typically of enormous grade. A vector of a queue has the formal structure of a chain in a classical simplicial complex. These simplicial chains are subject both to quantum superposition and orthogonal relativity transformations. In the classical interpretation, however, a product of three states like abc is associated with the solid triangle of convex linear combinations of its vertices a, b, c, representing statistical mixtures of the three; while in the present interpretation a product abc of three vectors is associated with the space of coherent linear superpositions of a, b, c with arbitrary probability amplitudes. The sacrifices mentioned are compensated at once by promising unifications. Some qualitative experimental predictions are clear from the start: 1. For a system of a given extent, a large violation of the Heisenberg uncertainty relation is predicted at very high energy, when the organization producing i melts down, as if i~ → 0 with increasing energy. 2. Space-time meltdown is accompanied by a bdegravitizationc’ analogous to demag- netization above the Curie temperature. A canonical field is a single-valued function of the event coordinates. The extension of quantum praxics to space-time events makes this field construct impossible, since the quantized space-time coordinates do not commute and cannot all be specified at once, and single-valuedness loses invariant meaning Field theory in the usual sense must therefore be relativized to a basis when the space-time is quantum (§6.1.3). What survives full quantization is the more general concept of the stratified system, now fully quantum on 1.3. QUANTIZATION 63 many strata. Single-valuedness of the field as a function of the event, a difficult problem when field value and event are separate quantum entities, becomes trivial when the field value and the event are identified, and the field becomes a quantum set of events. The theories of the standard model and gravity indicate that the gross structure of the ambient fully quantum space-time is highly anisotropic in the sense some of its dimensions are very long, others are very short. The ratio of the extensions of the long and short dimensions can be called the baspect ratioc of the system. In queue theory this anisotropy is likened to those of a snowflake, truss dome, and DNA molecule: The long dimensions result from the binding of many cells, the short dimensions are composed of one or several cells. The cosmic dome is about one cell thick and cosmologically-many cells long. In this simile, the two long dimensions of the dome stand for the four space-time dimensions of astronomy, usually called external. The short dimension symbolizes ∼ 10 real dimensions supporting an SO(10) usually called internal, of a theory like the Grand Unified Theory of Georgi and Glashow [43]. The long and short dimensions of the fully quantum are not classical continua as in the inspiring construction of Kaluza, or discrete as in an abstract simplicial complex, but are composed of spins, as suggested by Roger Penrose, quantum variables proper, spinning in various numbers of dimensions, depending on stratum. In addition, in the theory ϑo, one lower-stratum spin component is aligned among higher-stratum cells, as electron spins in a magnet are aligned among atoms, to account for the quantum imaginary, and another for the Higgs phenomenon. These spins assemble themselves not relative to some space-time infrastructure but intrinsically, by shared elements.

Queue cells do not occupy space-time, they constitute it.

In a fully quantum theory, coordinate variables are sums of spin variables of some orthogonal group of enormous dimensionality; the central coordinates, the longitudinal ones in the truss metaphor, are composed of enough spins so that the law of large numbers makes their sum approximately central; and the non-central, transverse to the truss, are composed of so few spins that their non-commutativity is salient. Basic coordinates along all these dimensions, being sums of quantum spins, have discrete, finite, evenly spaced spectral values of various spectral multiplicities, like the magnetic quantum number of an atom. Under ordinary macroscopic observation, the classical limit, the long coordinates blur into classical continuous unbounded variables, and the short ones, being properly quantum, disappear from view, except for several spins in each cell that enter into long-range correlations with those in other cells and manifest themselves as the classical electrical and gravitational fields. The class of functions X → Y , usually designated by an exponential Y X , is well defined when X and Y are classical spaces, defined by commutative coordinate algebras, or when X is classical and Y is quantum, but not when they are both quantum spaces. The statement that the field variables are single-valued functions of the space-time 64 CHAPTER 1. STRATA OF ACTUALITY variables is not strictly meaningful when the space-time variables do not commute but can become meaningful in some conditions in the singular limit of canonical quantum theory, and then may hold or not. The physical topology of a line (say) in a classical space can be built unconditionally into the kinematics, as for a world line, or it can be a contingent consequence of dynamical binding, as for an iron wire. The continuum topology of a line R in a fully quantum space-time has no invariant kinematic expression and so is likely a singular limit of a queue structure, which is expressible in terms of the kinematics of S. Gauge theory can be regarded as an enlargement of the gauge group of gravity beyond the space-time translations. The present study truncates the gauge group of gravity to fit within the orthogonal group SO(S). The diffeomorphism group is presented as a singular approximation to an orthogonal group of high but finite dimensionality. Einstein introduced dynamical interactions that account for the approximate local flatness and stiffness of space-time, but postulated its continuity as part of the kinematics, so one must consider whether new interactions are needed to account for the apparent continuity of low-energy space-time within a fully quantum theory. It is assumed here that self-organization and large numbers suffices. The vector space of the fully quantum theory bϑoc is formed by iterating Fermi statistics. Its enveloping vector space is a finite-dimensional subspace SF ⊂ S of the bself- Grassmann algebrac S.

1.3.3 Regularization A quantum system with finite-dimensional matrices, like a spin, still supports continuous groups, but its quantum variables, being finite-dimensional matrices, have discrete bounded spectra free of infinities. Predicates of a quantum system are represented not by sets of states but by flats of a projective geometry, or equivalently by projection operators in a vector space affiliated with the system. Points of the geometry represent bsharpc determinations, lines, planes, ..., represent bcrispc determinations, and the generic probability operators represent bdiffusec determinations. bCompletec determinations assign definite truth-valus 0 or 1 to every possible determination of the system. They are represented by points of state space in classical physics, and do not exist in quantum physics. Since the matrix elements are transition probability amplitudes, the quantum theory renounces determinism; since its predicates do not commute, it renounces complete description. In exchange it enjoys purely finite observables and continuous symmetry groups. The canonical quantum strategy did not fulfill all the historic hopes of the quantum reconstruction. Quantum theories still contain physical continua, such as space-time, and still give infinite answers to some reasonable questions, presumably because they still contain continua. Moreover they contain bquantum variables properc, quantum variables that disappear in the classical limit, such as electron spin and the isospin, hypercharge, and chromospin of the standard model. The canonical quantum strategy gave precious little 1.3. QUANTIZATION 65 guidance about how to discover and represent these. After the fact, the canonical quantum strategy was extended to admit quantum variables proper by allowing graded Lie algebras defined by anticommutation relations, as well as ungraded Lie algebras defined by commutation relations. Thus extended, the canonical quantum strategy is less divergent in some respects but still retains continua and their attendant infinities. Full quantization is a more powerful physical regularization process than canonical quantization.

1.3.4 Quantiﬁcation

Quantization began as a formal regularization process for radiation thermodynamics, and evolved into a procedure that produced a powerful theory of atomic spectra and dynamics. From the classical viewpoint it appears as a mysterious formal prescription, often described as meaningless. Conversely, one who describes it as a formal prescription adopts a classical viewpoint. From the quantum viewpoint, however, the process of quantization has a clear physical meaning, whose understanding impels significant further developments in the process. Ev- ery quantization is also a dissection, a transformation of one into many. The commutation relations defining the quantization are the commutation relations defining the statistics of the many. The process of quantization set out from a pre-quantum theory of an unbounded and unresolved continuum, a 2N-dimensional state space S. The canonical quantum theory regards S as a surrogate for a quantum space S with non-commutative coordinates. Quan- tization replaces the commutative Lie algebra of certain basic continuum coordinates of the phase space S by a non-commutative Lie algebra defining S that is closer to being simple. Here this quantization Lie algebra, whatever its structure, is interpreted as a statistics, describing how quantum units combine into the aggregate quantized system. This univer- salizes an interpretation already used in many special cases. The quantization Lie algebra, as vector space with Lie product forgotten, becomes the vector space of a hypothetical quantum unit of which the quantum system is actually composed.

Thus every bquantizationc process can be understood to infer the existence of a quantum unit or monad, and to resolve the given continuous system into a polyad of such monads. A quantification is then implicit in every quantization. The commutation relations of the quantization are the statistics of the implied quantification. The linear space spanned by the operators that enter into these commutation relations serves as the vector space of the quantum monad of the implied quantification. 66 CHAPTER 1. STRATA OF ACTUALITY

1.3.5 Cellularization

Full quantization produces theories with regular groups on every stratum, either orthogonal (as assumed here) or unitary. When the continuum is quantized, a fully quantum dynamics must then account for the emergence of a connected continuum in a singular limit. This continuum is simply assumed in the classical theory. The example of general relativity is not much use for this. There Einstein had to add new gravitational variables to a theory without them. Here instead one must reduce an infinity of local gravitational field variables to a finite number of non-local queue variables. Einstein had to account dynamically for the stiffness and approximate flatness of space-time with a new gravitational action. No new forces are needed in a fully quantum theory to account for the apparent continuity of space-time, [presumably!] merely a bcanonical limitc and fermionic statistics (§4.4). In a fully quantum theory classical space-time curvature arises as a classical vestige of the quantum non-commutativity of gauge invariant (kinetic) momentum-energy. The program proposed is to deduce classical gravity as a macroscopic quantum phenomenon, a vestige of quantum theory surviving in the near-classical limit as a result of an off-diagonal long-range cell organization analogous to magnetization and superconductivity: a “bgravitizationc”.

1.3.6 The choice of statistics

Every quantification process, and therefore every quantization process, requires us to choose a graded Lie algebra for its commutation relations, which define its statistics. The choice of statistics is one of the most vexed questions raised by this project since its inception, because the choice has affects far up the trail, but must be made at the outset. Moreover, groups of any Cartan family A, B, C, or D have groups of other families both as subgroups and as approximants in singular limits, so the some effects of the choice of Lie algebra can be subtle. Ordinarily quantum physics is pragmatic on this issue, using both Bose and Fermi statistics as experiment suggests. This has been practical for the the field stratum F. In a fully quantum theory, a quantum statistics must be chosen for lower strata B – E as well. To narrow the daunting choice, it has been optimistically assumed here that all the statistics in nature are of one kind, leaving corrections to this uniformity assumption until its predictions are in. A queue theory requires tensor products and vectors of very many dimensions to represent queues of many elements. The symplectic or exceptional Lie algebras are useless for this purpose and need not be considered for the line of statistics. They may still enter as reductions of higher stratum classical groups resulting from organization. Then the main choices to replace the canonical Lie algebras are the Lie algebras of real orthogonal matrices like the Lorentz group, or groups of complex unitary matrices like the 1.3. QUANTIZATION 67 unitary group of a Hilbert space, with whatever signatures work best. Correspondingly one must choose between (graded) Lie algebras generating real or complex commutation relations defining a Grassmann algebra or a nearby Clifford algebra. The real Fermi statistics is explored here before the complex because the absolute central imaginary is unstable and gauging the operator i might provide a theory of gravity convenient for full quantization. The usual Fermi statistics builds a many-quantum vector space 2V as the Grassmann algebra over the one-quantum vector space V . This seems at first like an unstable construction. If one made the least generic change in the right-hand side of Grassmann’s Relation v2 = 0 it would become v2 = kvk, a quadratic form in v. The Grassmann algebra over V would become a Clifford algebra over V . However the Fermi algebra makes no use of any form on V ; it is the Clifford algebra of the duplex form, which is determined by the vector space structure of V . The Fermi operator algebra does not vary, and its Grassmann Relation is an identity. This then fixes the initial line of groups as the groups of the Grassmann algebras S[L], the linear groups SL(2LR). These Lie algebras provides fully quantized p, q, r satisfying the fully quantum commutation relations (1.12). The vector space V of the dynamical stratum F uses a probability form H in the canonical theory. This is stable if H is regular, which is therefore assumed. It reduces the linear group to an orthogonal group on its stratum. If there is a central imaginary i, it reduces the orthogonal group to the unitary group of the canonical theory; with the difference that the unitary group has a non-compact form SU(N+C N−C) so that special relativistic invariance is possible. Then the canonical SU(∞) has to appear in the canonical limit, in which N → ∞. Canonical commutation relations are found on deeper strata as well, for example be- µ tween space-time position x and differentiation with respect to position peµ = ∂µ. These 2 4 can be represented as differential operators on the function space L (R ). This canonical Lie algebra, to be sure, is not a symmetry algebra of the canonical dynamics, which looks very different from its Fourier transform. But it may be assumed to be a symmetry of the canonical kinematics of the event stratum. It is singular and produces infinities, and must be fully quantized here. The most economical Lorentz-invariant simple variations of the µ canonical Lie algebra h(4) of x and ∂µ for µ = 1, 2, 3, 4 would be the Lie algebras so(6; σ) of Segal and Vilela-Mendes. For the reason already indicated, ϑo uses the so(10, 6) theory of quantum space-time. This has an so(6; σ) subtheory of the Segal-Vilela-Mendes kind. Fermi algebra is useful for representing skew-symmetric operator-valued tensors or forms but it contains no operator-valued symmetric tensors that can be used for a variable chronometric form g. In particular, the duplex norm k ... kDup on the duplex space W = V ⊕ V D vanishes identically on both subspace V and V D. The mean-square form provides a Minkowski metric form on the first-grade vectors of stratum 3, used in the model ϑo as the seed of the chronometrics of special and general relativity. 68 CHAPTER 1. STRATA OF ACTUALITY

Special relativity favors a real orthogonal group, the Lorentz group, but this originally represented experience at a macroscopic classical stratum, where quantum superposition is impossible and the quantum i is therefore unnecessary in any case. Every quantum system in use today has a unitary group, not an orthogonal one. Smaller internal Lie algebras are both unitary and orthogonal, as su(2) = so(3). The youngest group in particle physics, color su(3), is not an orthogonal group. The argument for the complex Fermi statistics is strong. If the real Fermi statistics is adopted, then the central imaginary i of the canonical quantum transition amplitudes has to be reconstructed as a singular limit of a non-central fully quantum imaginary operator i of a real quantum theory of the Stückelberg kind [72]. Both real and complex Fermi statistics are finite dimensional. The real is more stable, with smaller center, but the instability associated with a central imaginary i causes no infinities. The real quantum theory is pursued in ϑo mainly because Einstein locality suggests it, and because the quantum imaginary i might be the order parameter of gravitization.

1.3.7 Internal structure of the photon and graviton

One crevasse that ϑo must cross is that between the regular fermion and singular boson ﬁelds of the standard model and gravity. The vectors of ϑo form a Grassmann algebra incorporating the exclusion principle. Therefore ϑo faces the old problem of how to assemble bosons — the photon, graviton, and the others of their ilk — from more basic fermions, but brings to bear on the problem a new tower of quantum strata. Perhaps the way that bosonic alpha particles are assembled from fermionic nucleons can be used as a guide, but the great rest-mass and binding energy incurred by such a small assembly must be avoided. This might be possible because the association between small size and high momentum is a canonical one:

The Heisenberg uncertainly relation fails in a fully quantum theory.

Moreover the Bose-like statistics for a pair of fully quantum fermions is the sl(n) statistics of Palev. If ω is the column of basic coordinates and momenta of the quantum, the canonical quantum relations of Bose statistics have the schematic form

Bose :[ω, ω] = ih (1.26) where is a canonical skew-symmetric symplectic form. Scale factors X, E have been set equal to 1, so h is a dimensionless version of ~. The fully quantum commutation relations of Palev, which follow from Fermi statistics, have instead the form

Palev :[ω, ω] = Cω (1.27) 1.3. QUANTIZATION 69 with structure tensor C for the relevant Lie algebra sl[L] of stratum L. Bose is inhomogeneous in ω, Palev is homogeneous. For an extreme example, ω = 0 violates Bose grossly but satisfies Palev exactly. That is, canonical quantum coordinates and momenta cannot both vanish; but fully quantum coordinates and momenta can, at ranks below the organization of i → i. Comparing Bose and Palev makes it plain that h and therefore the effective action quantum ~ must include a factor l giving the maximum eigenvalue of the spin component ωYX in the symplectic XY plane for the stratum. The quantum number l varies hyperexponentially as 2L with the rank L. Assume for example that the organization of i occurs on the field stratum F ≈ 6. Then according to Table 1.1, Planck’s constant is effectively smaller on stratum E ≈ 5 than on 20 000 stratum F by a factor that can be as much as 26/25 > 10 , kin to the Large Number of the bcosmological leapc. This is in accord with the experience that the event stratum is successfully treated classically, as if ~ were 0. TO DO Complete this estimate 080910

1.3.8 Indeﬁnite forms

ν µ The real indefinite Minkowski form g = e ⊗ e gνµ on space-time requires a real indefinite probability form H on S in order to permit finite-dimensional representations of the Lorentz group by isometries. The probability form is not Minkowskian in general but may have arbitrarily many positive and negative elements in its diagonal form, depending on spin. Each orthonormal frame in S then defines a decomposition S = S+ S− into two subspaces with definite forms, positive and negative. The elements of S+ = in that frame represent feasible input processes. Plausible ways to interpret vectors of bnegative normc are provided by Dirac and by Bleuler and Gupta. In the Dirac interpretation nature has credit and can incur a system debt. A deposit of a negative number of fermions is merely withdrawal of a positive number of dual fermions that happens to occur at the beginning of the experiment, when the withdrawal must be from the vacuum. The dual of a particle with positive energy is a dual particle with negative energy. The antiparticle is a dual particle with energy reversed, so that particle and antiparticle can both have positive energy. Dual particles are usually pictured as holes in a sea of particles, but this does not seem obligatory. A positive net number of input quanta and a negative net number of output quanta result in a negative transition probability. This may well be forbidden, but it is not meaningless. The need for such credit can arise when relativity transformations couple a system into its dual system while conserving the norm, much as Lorentz transformations couple space coordinates into the time coordinate while conserving the proper time. This is easier to formalize for finite-dimensional fermionic systems than for infinite-dimensional bosonic ones. This outrageous application by Dirac of quantum concepts to the metasystem again 70 CHAPTER 1. STRATA OF ACTUALITY violates a tenet of canonical quantum theory, but that does not guarantee that this one is correct too. In the Gupta-Bleuler interpretation, a vector with negative norm represents an impossible input operation, like the input of a scalar photon. Some transitions cannot even be tried, let alone succeed. The number of trials of such an input and the number of transitions are then both 0, and so the transition probability is never negative but merely 0/0, undefined. An indefinite norm is natural on a vector space W that is a Clifford algebra over a vector space V . For elements of grade 0 or 1, the norm is the square, up to a constant. This can be extended to any grade by defining the bmean-square normc

Tr γ t γ Tr γ2 kγk = = = Grade γ2 (1.28) Tr 1 Tr 1 0 for a Cliﬀord algebra W of linear operators γ on a spinor space Ψ. Cliﬀord elements of negative square have negative mean-square norm, and are inevitable. This norm is invariant under SL(Ψ). This includes SO(V ) but becomes much larger than SO(V ) as Dim V grows.

1.3.9 The representation of symmetry Classical set theory has no symmetries; every set is special. Queues are as asymmetric as classical sets but for a discrete overall sign-reversal symmetry

R : S → S, I 7→ −I, (1.29) which induces an isometry of S. All the symmetry groups of present physics must be approximated by non-symmetry groups of operators on S. This shortage of exact symmetries does not eliminate the theory, since there are no observable exact symmetries in nature either, all symmetries being broken by their very observation [84]. The problem at hand is to allocate the small and large energy changes in a system to changes in the surface or deep structure of vectors like those in Table 1.1.

1.3.10 Finiteness More important for the present study, such simplicity also permits finiteness. The simple groups have a rich spectrum of finite-dimensional representations, with indefinite probability forms for the non-compact versions. If the group of physical transformations of a quantum system is a simple group of matrices, which are then finite-dimensional, the variables of the system are also such matrices, with finite discrete bounded spectra. A quantum angular momentum is a familiar example. This algebraic simplicity was the compass for the second stage of the present exploration. 1.3. QUANTIZATION 71

The superlative of “quantum” is “simple”. Non-commuting variables may make a system quantum but they may still have continuous spectra. All variables on every stratum of a fully quantum theory have discrete bounded spectra, finite both in the large and the small. The move from classical theory to canonical quantum theory exchanges an allegedly complete theory for one that less complete but is closer to being simple; but not close enough. Canonical quantum theory is infested with infinities because certain of its groups are compound: the bPoincarégroupc, the bcanonical groupc defined by the canonical commutation relations, and such functional groups as the bdiffeomorphism groupc of general relativity, and the functional bgauge groupcs of the standard model all require infinite- dimensional representations, in which most operators have undefined products. The semisimple case reduces to the simple. A direct sum of simple matrix algebras represents a statistical mixture of simple possibilities. One measurement suffices to determine which of the simple possibilities is the actuality, and then the others can be discarded. Therefore the semisimple case is usually omitted here. Non-simplicity of a Lie algebra is indicated by its bradicalc.

Terminology: For any Lie algebra a, the bderivedc Lie algebra Da is that generated by the n Lie products {x × y : x, y ∈ a}. A Lie algebra a is bsolvablec if ∃n : D a = {0}. The bradicalc of a Lie algebra is its maximum solvable . A Lie algebra is bsemisimplec if and only if its radical is {0}.

The existence of a radical implies action of one group element on another without reaction. It implies an bidolc in the senses of Francis bBaconc and bNietzschec. The canonical quantum theory shrank the radical of the kinematical Lie algebra enormously but not entirely, and left radicals on other ranks untouched. Here the radicals of all the Lie algebras in the formulation of gravity and the standard model are reduced to 0. The first group to simplify is the bkinematical groupc of the system, consisting of all reversible physical operations on the system. For a spin this is already simple; for a particle on a line it is bcompoundc. In classical physics filtrations as infinitesimal generators belong to the radical. Clearly a system with a simple kinematical group is a quantum system in the contemporary sense that its filtrations do not commute. A system with a simple kinematical group is also quantum in the old sense of being granular, since all its physical quantities have discrete bounded spectra. The superlative of “quantum” is “simple”. Often a singular limit of such a simple theory is convenient, for example to bring the powerful tools of the differential and integral calculus into action, but this limit generally introduces unphysical infinities too. The proposal is that the divergent quantum theories of current physical interest are approximations to simple finite theories, though usually some variables of the simple system have to freeze for the singular approximation to work. Moreover, the noxious compound case is never required by any experiment. If one 72 CHAPTER 1. STRATA OF ACTUALITY admits the possibility of frozen degrees of freedom, as in ferromagnets and crystals, the compound groups are rather arbitrarily chosen singular limiting cases of simple groups that are just as consistent with experiment. Therefore this arbitrarily introduced non-simplicity, the root of structural instability and the infinity syndrome, can be gently corrected by a small variation of Lie algebraic structure. Every compound system has simple systems in its neighborhood. They can be reached by adding a few frozen variables and slightly varying the commutation relations. Recovering a simple quantum theory from a singular theory in this way is called bsimplification by quantizationc. The bdeformation quantizationc of Flato, Fronsdal, and coworkers [9, 41] simplifies one group on one stratum. The continuum of the complex matrix elements of quantum theory, seems harmless so far. These continuous variables are probability amplitudes, and integrals over them never appear, and so cannot produce infinities. Assuming that the probability amplitudes and the probabilities that they define can take on a continuum of values, amounts to assuming that an experiment can be repeated as often as desired, and that the limit as the number of experiments approaches infinity has meaning. This assumption of unbounded resources is clearly unphysical but does not seem to have outlived its usefulness for theoretical physics, at least if the allowed volume is also large enough to avoid black holes. There is no physical difference, actually, between fully quantum theories based on real, rational, or integer values for the relative transition-probability amplitudes, due to the finiteness of the number of experiments we can carry out. Measured probabilities are always rational, the ratios of counts. One could begin with a vector module over the integers, and develop the vector space and its coefficient field recursively, but this would seem to be a presently pointless exercise. Simplicity is still not a sharp enough guide for this project, however. Each of the singular groups of the present theory abuts several quite different simple groups, in the way that the Galilean group abuts both SO(3, 1) and SO(4). One simple group with the proper canonical limit must be selected by other criteria. This is done here by the choice of statistics.

1.4 The groups of nature

Nature seems to require a description with stratum structure; for example, classical field theory has at least a stratum F of field histories f(x), a lower stratum E of space-time events (xµ), and a still lower stratum D of differentials dxµ. Canonical quantum physics tacitly inherited the classical stratum structure. For example, quantum fields occupy a higher stratum than the space-time events on which they depend. But ordinarily only one stratum is described with a non-commutative non- deterministic quantum theory, and other strata are assumed to be classical, hence singular. Canonically quantizing one physical stratum and leaving the lower ones classical necessar- 1.4. THE GROUPS OF NATURE 73 ily broke long-standing constitutive relations between strata. If a theory has explicitly classical strata it cannot be simple or structurally stable. Each stratum has its own groups to be simplified. Finiteness, structural stability, and true simplicity require us to quantize all the strata used in a theory and the relations between them. Even if each stratum were simplified separately, the entire structure would still not be simple. Quantum superposition must operate across strata for total simplicity. A stronger strategy is needed to choose how to simplify each stratum and also the totality of strata. Classical set algebra provides infinitely many mathematical strata of assocation related by a common statistics, the classical analogue of Fermi-Dirac statistics: An element must be in a set or not, the occupation number is 0 or 1, multiple occupation is forbidden; and the order of elements in a set is ignorable. The stratum counter is set to -1 for the empty stratum. Stratum 0 has the empty set 1 = {} for its sole element. In general, stratum L + 1 is the exponential of stratum L. Peirce, Peano, and von Neumann put the natural number n on stratum n. Here we take all the real numbers as given on all strata, for use as quantum probability amplitudes. The operator I maps each stratum into the next. In bfull quantizationc, the quantum strata are exponentially constructed by one universal generator of strata I. About the association symbol I: Braces {s} and the bar s are more familiar synonyms of I but clumsy to iterate; Peano’s functional notation is more powerful. The quantum variant of his ι = {...} is designated by I to avoid confusion with i. Bracing is also called bassociationc, bmonadizationc, and bunitizationc. Bracing disrupts associativity: a{bc} differs from {ab}c. The finite strata of classical set theory support only finite groups. To support a continuous group, a classical state set must belong to an infinitely high stratum, where undesired infinities are then endemic. In contrast, strata ,..., 7 of the queue algebra S carry the Lie groups used in present-day physics or simple approximations to them. And fully quantum theory, like classical set theory, is fractal in that an isomorph of the entire family tree sprouts from each node of it. The groups of the root recur in the fruit. Since the state strategy worked as well as it did, the canonical quantum strategy made but a small change in it, and specifically in the associated algebra of random variables. Since the canonical quantum strategy works even better, the least change in it is made here that results in a simple algebra of quantum variables for every stratum. The heuristic process of bfull quantizationc. replaces the classical random-set algebra S by a bqueue algebrac bSc whose Grassmann Relation expresses the exclusion principle of quantum physics as well as the limitation of classical sets to single occupancy. Like the classical random set algebra S, the bqueue algebrac S constructed and used D here is constructed of nested strata tied together by one generator I and its adjoint I = −1 T H I H. Like the Fermi algebra of electron theory, S is a Clifford algebra over S as well as a Grassmann algebra over S. The strata of the queue algebra S have simple Lie algebras. Indeed, the strata 2 – 7 74 CHAPTER 1. STRATA OF ACTUALITY support all the continuous groups of current physics or passable approximations to them, with room to spare; fewer strata may suffice. An individual system with input vector space V is designated briefly by I[V ] and called a V quantum. A full quantization replaces the compound classical space-time continuum by a queue with simple groups, and specifies the kind of simple group to choose at each fork in the road. At the same time it represents existing experimental symmetry groups as closely as necessary. It greatly reduces the number of possibilities, but still leaves a great many, mostly connected with the quantum bstratumc structure of physics. To proceed constructively requires some alignment of physical strata with mathematical. The mathematical strata have absolute stratum numbers L = −1, 0, 1, 2,.... The physical strata are given floating addresses A, B, ..., G so that their alignment with mathematical strata can be adjusted to experiment. The stratum assignments used in this study are summarized in the next two sections. Canonical quantum field theory too has several strata of association. Macroscopic variables belong to the higher strata, while some quantum variables proper are already found in the lower. Three higher strata D, E, and F have counterparts in canonical quantum theory: .

1.4.1 Canonical strata The event stratum E of the canonical quantum theory is conveniently assigned to the events x of classical space-time, with both long “external” and short “internal” dimensions in Kaluza-type theories. The differential stratum D supports space-time differentials dx whose aggregation is x = R dx. It is not always immediately clear to what stratum in this hierarchy a construct belongs. A space-time is used in the metasystem of field theory to locate field meters and targets. To measure the field at a point, one puts a field meter or quantum detector at that point, a metasystemic operation. The space-time event then seems to belong to the metasystem, a higher stratum than the field system. But on the other hand, in the mathematical theory the field variable is a set of pairs of events and field values, putting space-time events into a lower stratum than the field system. The resolution of this puzzle seems to be that an isometry of one stratum can binducec isometries of every higher stratum (§3.3.2). In particular, if there is a discrete spectrum of time and space coordinates with spacing bXc in the event stratum E, then the spectrum of time in the metasystem stratum G, due to parallelism, has much smaller spacing ∼ X/N where N is the number of event elements in a metasystem element, a cosmologically large number. The time read on the wall clock is not the same time that drives an atomic vibration in the system; its spectrum of values is much finer and larger. Presumably both testify to an underlying set dynamics that drives them both. 1.4. THE GROUPS OF NATURE 75

Thus the metasystemic time can be bquasicontinuousc even though the system time has a conspicuously discrete spectrum. This can lead to a serious underestimate of X if it is not taken into account.

1.4.2 Metric forms Two familiar metric forms pervade the standard model, one for chronometrical structure and one for statistical structure. The bcausality formc g deﬁnes causality and proper time through

2 µ ν dτ = −g dx dx = −gνµdx dx (1.30) and is used to construct action principles at least in the canonical limit. In ϑo, the classical gravitational potential tensor gνµ is a singular limit of a dynamical variable igh on stratum F, the bgauge potentialc, that in the canonical limit generates gauge bosons like the graviton or the standard model gauge vector bosons. The bprobability formc iHh defines the hits and the relative transition probability amplitude A = φiHhψ (1.31) between two vectors. Its canonical limit is the Hilbert-space metric form of canonical quantum theories. The probability form transforms any input process hi into an output process ih = iHhi for which the transition ohi is assured, called the (total) b reversalc of hi. In the real positive definite case iHh is the real Hilbert-space metric form of the real quantum theory. Kaluza-Klein theories provide a clue to the origins of the chronometric form. In those theories the components of the event coordinates consist of four commutative space-time differentials dxµ and N infinitesimal elements dξ of an internal Lie group. The chronometric form therefore consists of three parts, the external part gµ0µ, the gauge vector boson gµξ, and the internal part gξ0ξ. The last of these is a quadratic form on the Lie algebra and can be identified with the Killing form kξ0ξ of the Lie algebra. In fully quantum theory all the differentials belong to a simple Lie algebra. The first two parts of the chronometric are absorbed into the third, which is all that remains. At least on one cell of the appropriate stratum, one has:

The chronometric form is the Killing form.

1.4.3 Fully quantum strata

Ranks are tentatively assigned as follows in model ϑo. 76 CHAPTER 1. STRATA OF ACTUALITY

Notation: bSL[V ]c is the uni-determinantal automorphism group of the linear space V . It is generated by second-grade elements of the Cliﬀord algebra 2V , acting as inner automorphisms of 2V , and restricted to act on the ﬁrst-grade subspace of 2V .

bSO[L]c is the automorphism group of the vector space S[L]. SO[L] is inﬁnitesimally generated by second-grade elements Grade2 S[L + 1], whose commutator Lie algebra is isomorphic to so[L]. The lowest stratum with enough dimensions for the events of cosmic history is stratum 5, which allows for 2216 events. Therefore tentatively E ≈ 5. This permits F ≈ 6 and D ≈ 4. Rank 3 has SO [3] = SO(10, 6). Rank 2 has SO(3, 1), the Lorentz group of special relativity. Therefore in ϑo strata C ≈ 3 and Bapprox2 have been assigned. Stratum C provides the group of the unit cell of the cosmic dome. Therefore the probability form of stratum-B quantum theory is isomorphic to the Lorentz causality form of space-time. This seems to be the germ of a uniﬁcation of quantum theory and gravity. With 4 generators set aside for the Lorentz group, the remaining 12 generators of stratum C are available for internal dimensions and i. This seems just enough. Ranks −1, 0, and +1 have trivial Lie algebras of dimensions −1, 0, and +1.

1.4.4 Fully quantum self-organization Self-organization is required to happen on or below stratum F, producing central event µ coordinates xb , the central constant imaginary i, and the Higgs field. This sets a lower bound on the number of subquanta composing these quantum entities; there must be at least enough to organize themselves. The 216 basic monads of stratum 5 could only form a near-condensation, with fractional fluctuations as large as ∼ 2−8. Rank E could reduce fractional quantum fluctuations to ∼ 2−216 . Fully quantum variants of the Dirac and Maxwell equations may then apply to one E quantum, since they are both real, and so do not require the organization of i. Tentatively, the event stratum E is chosen to be stratum 5 and the organization of i and the Higgs variable is assigned to a field stratum F = 6. A large number of elements is necessary but not sufficient for the organizations hypothesized. The choice of statistics is also critical. In the canonical limit, the operator i is to become central and its square is to become fixed at −1. This suggests that i is the suitably normalized resultant spin vector of a polarized aggregate of spins of an orthogonal group on stratum C beyond the Lorentz group, and implies bosonic-type statistics either for the constituent spins of i themselves or for the quanta of some strong field that polarizes them. µ On the other hand, the event cooordinate operators xb are not only central in the canonical limit but also exhibit a dense quasi-continuum of values of macroscopic or even cosmological extension. 1.4. THE GROUPS OF NATURE 77

Once the physical meaning of coordinates is defined, as by radar, the fact that the event coordinate values in the vacuum form a quasi-continuum without macroscopic voids is not conventional or tautological but experimental, like the absence of large voids in the depths of the Earth. For example, radar coordinates stop at event horizons. The manifold assumption of the canonical theory needs theoretical explanation in the fully quantum µ theory. Since i and the xb are different basis elements for the same Lie algebra, it must µ also be explained why i is effectively constant while xb is effectively a free variable. The easiest explanation in both cases is based on Fermi statistics. If events have fermionic statistics, then the absence of voids in the position spectrum in the vacuum may, like the absence of voids in a planet, be a consequence of the self-gravitation of the mass density of the vacuum, measured by the bcosmological constantc. The canonical quantum concepts of bself-organizationc, spontaneous symmetry breaking, degenerate vacuum, or off-diagonal long-range order, have been important for the standard model, so it is natural to seek their correspondents in a fully quantum theory (§4.4). In quantum mechanics systems ordinarily have singlet ground projectors. A random Hamiltonian almost always has a singlet ground projector. But the energy of a spin s in the vacuum is 0 for all 2s + 1 basic vectors. For spins with appropriate symmetry, ground multiplets are ordinary; and a fully quantum system is composed of spins.

1.4.5 Full quantization tactics Fermi quantiﬁcation is deﬁned and iterated here for use in full quantization. In quantifying the fermion an extension of the one-fermion theory is useful, uniting the fermion and its dual into one bduplexc fermion of double vector dimension. If V is a one-fermion vector space, vectors v ∈ V are usually associated with input operations and dual vectors u ∈ V D with output operations for the same fermions. One may also interpret both v and u as input operations for one duplex fermion, whose vectors are the linear combinations of the usual fermion vectors and their duals, making up the D bduplex vectorc space W = V ⊕ V (§3.4.1). The duplex fermion can be either the fermion or its bdualc. For any vector space V there is a natural hermitian norm k ... kDup on the duplex D space W = V ⊕ V =: Dup V , here called the bduplex normc:

D ∀v ∈ V, u ∈ V : kv + ukDup = 2w.u, (1.32) in which w.v is the value of w on v [62]. A form of signature 0, and its bilinear space, are called bneutralc or bKleinianc. The duplex norm kwkDup is bneutralc. Its polarization is the bduplex formc HDup. A neutral norm on any space W is isomorphic under GL(W ) to its own negative:

kwk =∼ −kwk. (1.33) 78 CHAPTER 1. STRATA OF ACTUALITY

The duplex norm kwkDup requires no metrical form on V for its construction; its invariance group goes beyond the usual isometry group of a quantum theory with vector space V , and includes the linear group SL(V ). D The original vectors of V and V are bsemivectorsc of W . The orthogonal group for the duplex norm kwkDup is taken as the relativity group of the duplex fermion. This relativizes the customarily absolute distinction between input and output, and between quantum and dual quantum. Now a fermionic quantiﬁcation V → 2V , W → 2W is constructed. When V is the V vector space of an individual quantum I V , 2 is the vector space and algebra of a Fermi assembly of such individuals, and 2W is the Fermi operator algebra of that assembly. 2W is still the Grassmann algebra of polynomials

2W := Poly(W : w2 = kwk) (1.34) identiﬁed modulo the Grassmann relation, as indicated. The familiar bFermi operator algebrac over V , Fermi V , is the Cliﬀord algebra over D the duplex space W = V ⊕ V with the duplex norm k ... kDup. This is

D D Fermi V := 2V ⊗ 2V = 2V ⊕V := Poly(V,V D), (1.35) the ﬁnite-grade polynomials in V and V D modulo the Fermi-Dirac relation

∀v ∈ V, w ∈ V D :(v + w)2 = 2w.u. (1.36)

Fermi V is itself the operator algebra of 2V := Grass V , the Grassmann algebra over V , which may be taken to be a vector space for the Fermi assembly. But any isomorphic space such as Grass V D = 2V D is just as good for this purpose. The step from the vector space of the individual to that of the Fermi assembly is represented by the functor Grass. This takes care of the incidence relation ø defining the misses, but provides no probability form H defining the hits. Iteration of Fermi starting from the empty Clifford algebra leads to the bself-Fermi vector algebrac S = 2S. (1.37) The Fermi operator algebra of a Fermi assembly of individuals with vector space V is Alg 2V , the algebra of linear operators on a Fermi vector algebra V , or an isomorph. It is mathematically possible to interpret any Clifford algebra 2V as a quantum set algebra. V holds the vectors of the quantum elements, and 22V holds the vectors of a quantum set or queue of such elements. The queue may have any number of elements up to the dimension of V . If one specifies a basis and suspends superposition the queue reduces to a classical set. The bqueue strategyc is also a simplification strategy, because queues have simple kinematical groups. 1.4. THE GROUPS OF NATURE 79

A physical system represented by vectors in the power space over a given vector space V is thereby endowed with a graded Lie algebra, that generated by vectors of V , the bgenerating vectorsc of the system. The usual Fermi and Bose statistics are not simple as (graded) Lie algebras, due to the 0’s in Fermi law ψ2 = 0 and the Bose law [1, q] = 0. But the generating vectors of these algebras provide familiar examples of generating operators. bCliffordc called his algebra a bgeometric algebrac and restricted it essentially to the classical differential stratum. bFermi statisticsc (§1.4.5, (1.35)) uses a Clifford algebra not as a geometric algebra but as a q algebra, to represent a quantum power set. The elements of this Clifford algebra are vectors, as in Dirac’s Clifford algebraic formulation of Fermi-Dirac statistics [26]. The geometric interpretation is then a classical vacuum of the quantum interpretation. Assumption 3 implies that physical systems have Fermi statistics. Since bFermi statisticsc covers Palev statistics as well this is not immediately absurd, especially in view of the re-analysis of the bspin-statisticsc correlation in §3.4.3. Dirac constructed Clifford algebras that unite a Grassmann product with a causality form, as in the Dirac equation, or a probability form, as in Fermi-Dirac statistics [26]. The bmean-square formc on S[2], stratum 2 of the self-Grassmann algebra S, is Minkowskian, and remains indefinite for all strata ≥ 3. It is easy to see from the matrix representation of the Clifford algebra that the signature σL of the mean-square form of stratum L is, for L > 2, √ σL = 2L. (1.38) This leads to the chronometrical hypothesis (Assumption 8): that the probability form (a pseudo-Hilbert space metric) iHh for vectors of stratum F and the causality form igh for space-time vectors of stratum E have a common origin in the probability form of a lower stratum B, where they coincide (§1.4.3). C is the lowest stratum whose isometry group SO[C] includes a Lorentz group, an approximation to the Poincarégroup, and the unitary groups of the standard model. The stratum used for this purpose will be called the bcharge stratumc C. In the model ϑo C ≈ 3. S is used in a heuristic process of bfull quantizationc, which converts non-simple Lie algebras into simple Lie algebras of queues by small variations, inferring frozen variables when necessary. This is recapitulated in §4.3 and then tested on bgravityc. In field theory, the space-time manifold is part of the metasystem and is used to fix where and when we make our field determinations. Its continuity produces infinities in field theories and has no experimental basis, one can only say that if space-time has atomic elements then we have not yet wittingly resolved them; though in the present state of physical theory we might trip over them without recognizing them. This leaves open the possibility explored here, that just as quantum spins of so(3) are quanta of bPenrose spacec, quantum spins of a higher so(n) are quanta of space-time-matter. Here we extend quantization five strata down from the field stratum F with a simple Lie algebra on every stratum. 80 CHAPTER 1. STRATA OF ACTUALITY

1.5 Fully quantum regularization

To simplify the groups is to regularize the theory; to fully quantize the theory is to simplify its groups. A canonical quantum theory may have a single-quantum stratum and a many-quantum stratum, related by Fermi or Bose statistics. For a gauge theory like the standard model a third stratum is essential, for differentials dxµ and infinitesimal displacements, which the gauge vector field represents. This stratum is supposed to be lower than the single- and many-quantum strata, but classical nonetheless. In the fully quantum theory the strata are related by the quantum bracing operator I.

1.5.1 Fully quantum dynamics The fully quantum kinematics is founded on a vector space for quantum histories, which are queues of quantum events. A fully quantum dynamics is defined by one specific dynamics vector hD in this space. A maximal experiment is then represented by a dual vector Eh such that the transition amplitude for the experiment is EhD. In canonical theories an experiment vector h(x) can be defined by a sequence of points (x) = {x1, x2, . . . , xN } approximating a classical path, and a dynamics vector can be cast into the action form R hD = ei (dx) L/~ (1.39) with action density L. In a fully quantum theory classical space-time is replaced by a fully quantum space. The basic form survives: There is a dynamics vector, and its phase has a contribution from each cell of the quantum space, now combined not by integration but by addition (3.20). Without the time continuum, one can no longer assume that vectors must obey a differential equation that contains time differentiation only linearly, as canonical quantum theory does. This can at best be a property of the canonical limit of classical time. There- fore one should expect observable corrections to this widespread assumption. Since the time continuum is an artifact of a singular limit, small corrections involving second or higher time derivatives, or integral transforms, can probably improve the approximation of continuous classical time but cannot make it exact.

1.5.2 Gauge A c gauge theory is a dynamical theory whose dynamical variable is a connection on a fiber bundle F → B → S whose base space S is space-time, whose fiber F is the space of field values at a point [?], and fiber group G : F → F is the gauge group. The prototype, general relativity, has gauge group Diff = Diff(S), the diffeomorphism group of the space- time manifold. Its Lie algebra is

µ diﬀ = {Poly (x)∂µ}, (1.40) 1.5. FULLY QUANTUM REGULARIZATION 81

µ linear combinations of ∂µ with coeficients polynomial in x . It has not been possible to say exactly what a q gauge theory is, due to the great singularity of the c concept, which combines the algebraic singularities of the base space and of the space of connections. It is in the q spirit is to preserve the group Diff exactly, on the grounds that it operates on a differential stratum D < F, deeper than the field stratum F. Diff is part of the concept of a classical history, and q sums over classical histories without modifying them. We may specify the connection by a gauge differentiator Dµ(x). We avoid some singularities by taking G to be simple. Some of the infinity of commutation relations responsible for the other singularities are, schematically,

[D, x] = 1, [x, x] = 0, [1, x] = 0, [D,D] = F, [F,F ] = 0, [F, x] = 0, . . (1.41)

In a bcanonical gauge transformationc, an isomorph of a gauge group G acts on a space fx of field variables at each space-time event x. It is proposed here that gauge groups arise as the representations within the relativity groups of higher strata of the quantum relativity groups of lower strata. In vartheta(2008) the field variables and space-time coordinates work on an event stratum E, two ranks of association above a charge stratum C whose relativity group G[C] is the gauge group “of the first kind,” here called the C gauge group. In a canonical gauge field theory there is also a much bigger gauge group “of the second kind” acting on the field stratum F. If X is the space-time manifold, the gauge group of the second kind is GX , the space of (differentiable) maps g : X → G. GX acts on the space F = f X of differentiable fields, which are maps X → f. Here the gauge group of the second kind is the relativity group of stratum F, or the F-gauge group. In a fully quantum theory there is thus a kind of gauge group for every stratum L, namely the relativity group of that stratum, called the L-gauge group. Some are shown in (4.33). Gauge theories are particularly attractive to theorists because they enable us to deduce a dynamical interaction from an assumption of symmetry, under plausible ancillary assumptions. A gauge theory produces a most-favored action principle for study, determined up to several parameters by the gauge group and general principles like maximal locality and minimal coupling. Such a gauge action can be fitted to experiment with relatively minor surgery. Fully quantum gauge theories are as fertile for action principles as canonical ones, but still require ad hoc assumptions, such as self-organization like the Higgs phenomenon. 82 CHAPTER 1. STRATA OF ACTUALITY

Some experimental data must be used up in defining these organizations. It remains to be seen whether there will be any data left over to test the theory. The most obstinate singularities encountered are those of gauge theories, especially general relativity. Gauge field theory has been so fruitful that it seems worthwhile to fully quantize it and eliminate its infinities. The argument by Einstein from gauge invariance to singularity is one of the beautiful arguments of theoretical physics: A gauge transformation could be the identity in the past and present and not in the future and still be infinitely differentiable. It would then change some future variables but no past variables. Therefore gauge invariance implies that the past variables do not determine the future ones. Therefore it must be impossible to cast the variational equations into canonical Hamiltonian form, which determine the future by the past. Therefore it must be impossible to express the velocities in terms of the momenta. Therefore the Hessian determinant of the Lagrangian with respect to the velocities must vanish: ∂2L = 0. (1.42) ∂q∂˙ q˙ This is clearly a structurally unstable condition, but it is a consequence of general covariance. Therefore it must be concluded in any fully quantum theory that general covariance – and any other canonical gauge invariance, since they are all singular — is an approximation to a fully quantum gauge invariance, whose gauge group is regular. The usual gauge theory breaks down when the canonical limit does, for example at the highest energy. The singularity of present gauge theories also shows up in the divergence of integrals over histories of the gauge field. If one integrates over all histories then each history is accompanied by a wildly infinite set of all its gauge transforms, having the same physical meaning. This redundancy causes the integral to diverge badly, by an infinite factor that counts the redundant occurrences of a given physical field. It has been possible to eliminate this infinity by the BRST method ([80], Chapter 15) which selects — or at least favors — those gauge potentials that obey some gauge-fixing condition and excludes — or at least attenuates — redundant copies of each physical field configuration. This method introduces bghost particlesc that are designed never to appear in experiments. The theory remains structurally unstable but is renormalizable with infinite renormalization factors. This is generally understood as indicating that the renormalizable theory is the singular limit of some finite theory. Here a queue theory provides that finite theory. In it the gauge group and its representation become finite-dimensional, and the divergent integration over gauges becomes a finite trace over histories of the system. A fully quantum gauge group is not a function group but a classical group. In the theory ϑo it is an SO(n+, n−) of some stratum. The usual functional gauge group is a singular approximation to the fully quantum gauge group. The redundancy factor is merely the dimension of the representation space of 1.5. FULLY QUANTUM REGULARIZATION 83 the fully quantum gauge group. There is no need to fix the gauge and break the gauge invariance, or introduce ghosts. One simply does the trace. The condition that the present determine the future is already weakened in canonical quantum theory, but a quantum determinacy survives in the canonical quantum theory: The dynamics defines a unitary transformation that maps present vectors into future vectors. This quantum determinacy allows us to hit with certainty any one desired target in the future that we choose to, telling us exactly what input will accomplish the desired output. If after such an input we change our minds and insert some other output target, the quantum determinacy does not predict the outcome with certainty as classical theories do. Since vectors do not determine what happens in all possible experiments, the quantum determinacy is a weak statistical kind of determinacy. In canonical gauge theory this weak determinacy is weakened further. The past gauge potential does not determine the future gauge potential, because it does not determine the experimenter’s free choice of gauge frame. This reduced determinism is still too much for structural stability, and is further reduced in a fully quantum theory. Any fully quantum time variable is a finite matrix, although usually a large one, Newton’s concept of time derivative does not apply. It is impossible to increase the time coordinate of one frame above a certain maximum eigenvalue. Different time values even have eigenspaces of different dimensionality, smallest for the extreme time values and greatest for the middle time value. Therefore the passage of time does not define a one-parameter group on the vector space, although there is a regular, fully quantum correspondent for the canonical diffeomorphism group. General relativity describes how the matter in space-time influences the causality form in analogy to how electric charges influence the belectromagnetic fieldc. Einstein’s bgravitational fieldc and Maxwell’s electromagnetic field have both matter-determined (Newtonian, Coulombian) and matter-independent (radiative) parts in each frame. The matter-determined part, however, is determined by generalizations of the Gauss Relation −∇ · E = ρ. These are differential bconstraintsc arising from the existence of field variables whose rates of change do not appear in the dynamical action principle. These constraints result from the singular nature of the relevant full gauge group. A gauge transformation Dµ(x) 7→ Γ(x)Dµ(x)Γ†(x), x 7→ x (1.43) couples space-time coordinates x into the gauge coordinates in D without a reverse coupling of gauge coordinates into space-time coordinates, creating a radical in the commutator algebra. A fully quantum theory, we have seen, can eliminate the fundamental distinction between field and space-time variables; they are both coordinates of the event, possibly with different scales of organization. It thereby regularizes these singular full gauge groups and thus eliminates the gauge constraints, resurrecting them only in a singular organized limit. The photon and graviton presumably have small masses, though possibly cosmologically 84 CHAPTER 1. STRATA OF ACTUALITY small.

1.5.3 Spin What remains of the classical distinction between matter and field in the canonical quantum theory is the separation into integer spin bosons, which can organize into classical fields, and half-odd-spin fermions, which can organize into classical matter. In a fully quantum theory, it seems, half-odd-spin is anomalous, a result of vacuum organization of integer-spin elements as seen in a singular limit. 080530 ...... [Merge above with following, eliminating redundancy] Canonical gauge theories have not only global conservation laws for some kind of charge but also local conservation laws: not only is total charge constant but its current is continuous, the charge increase in any region is the charge influx through the boundary of the region. This brings in the topological concepts of the boundary and current-continuity. In canonical quantum theories a finite-dimensional gauge group on the stratum D of differentials induces in higher strata both a finite-dimensional representation connected with gauge charge conservation and an infinite-dimensional enveloping group connected with gauge current continuity by bNoetherc. These are called gauge groups of the first and second kind by Dirac. To leave room for the groups of other strata, they are called D and F (stratum) gauge groups here. Charge conservation can survive in a fully quantum theory as a commutativity relation

[E,e Qe] = 0 (1.44) between skew-symmetric operators of charge Qe and energy Ee. The non-locality of the theory suggests that there is no exact fully quantum correspondent of current continuity. The first grade of stratum D of ϑo supports an so(7, 5) and the second-grade represents it. The Lie algebra of space-time-momentum-energy is modeled in this Lie algebra. The stratum D operations ωα5 ∈ so(7, 5) are not local in space-time in the sense of the corresponding partial derivations ∂µ, but in a weaker sense. Consider for example the transformation ω45 ∼ i that represents energy in stratum D. ω45 replaces each vertex s4 of a cell by another, s5. Two cells differing by one vertex contact each other on maximal subcells or faces, and are contiguous in the first degree. Briefly, ω65 couples cells of the queue that are contiguous in the first degree. A gauge system is generally believed to result from microscopic organization, with gauge fields as order parameters. Sommerfeld, for example, noted that the electromagnetic field can be reinterpreted as a vibrating relativistic elastic solid [70]. The bVolterra-Burgersc theory represents bdefectsc in a crystal organization by a gauge field, the Burgers vector. This theory has been extensively developed in a classical framework. Can queue theory describe quantum defects in the vacuum organization, in a quantum version of Volterra- Burgers theory? 1.5. FULLY QUANTUM REGULARIZATION 85

For any stratum L, the orthogonal group SO [L] is the automorphism group of the quadratic space S[L]. The groups of classical gravity and their proposed fully quantum correspondents are:

1. Lorentz and Poincar´eGroups ←SO[C].

2. Diﬀeomorphism Group ←SO[E].

3. Canonical Group ←SO[F].

Any gauge theory has analogs of this triad of groups in its D, E, and F stratum gauge groups and its kinematical group. The self-Grassmann algebra S includes the stratum-F vector space for the queue. This makes possible a simplifying assumption that accommodates gauge theory.

Assumption 4 (Fully quantum gauge) The C gauge group is the orthogonal group SO[C] of the ﬁrst grade of stratum C. The E gauge group is the orthogonal group on the event stratum E induced (in the sense of §3.3.2) by the C gauge group. Gauge invariance of the canonical quantum theory and the classical theory are singular limits of an organization of the q vacuum on stratum F .

This organization of the set stratum F is described in Chapter 6 and (4.33).

1.5.4 Real quantum theory The Lorentz Lie algebra is both the real matrix Lie algebra so(3, 1) and the complex matrix Lie algebra sl(2C). It is not clear yet whether the fully quantum theory that best replaces the complex canonical one is better chosen to be real orthogonal or complex unitary. The use of i in the canonical theory is to provide an absolute connection between operations and observables; that is, between the Lie algebra of inﬁnitesimal isometries, which are skew-hermitian, and the Poisson Bracket Lie algebra of classical observables, which are hermitian. The real theory is harder than the complex theory because it lacks this connection, and because in it one cannot diagonalize all normal operators. On the other hand, the real theory virtually compels one to account for the i of the canonical quantum theory as a self-organization, which can have important physical consequences. Moreover, complex Lie algebras are not structurally stable within the manifold of real Lie algebras, due to the instability of the centrality condition [i, a] = 0. The canonical quantum i is already non-central in the weak sense that it does not commute with time reversal T . Ultimately one should expect a real quantum theory, therefore, in which i and time t result from self-organizations, perhaps the same one. Nevertheless, the complex imaginary i seems to be responsible for no inﬁnities, and so could be tolerated. One could postpone the real theory and work within the manifold of complex Lie algebras. 86 CHAPTER 1. STRATA OF ACTUALITY

Here instead a real quantum theory is studied from the start. The main purpose for this choice is to explore new gauge phenomena arising from the bquantized imaginaryc i, which seems to be a candidate for a bHiggs fieldc. The main goal of bfull quantizationc is still to regularize present field theories and eliminate infinite renormalizations. But hopefully once again a quantization will produce a theory that also fits experiment better than its singular ancestor, and has greater conceptual unity. One can account for the success of complex quantum theories up to now by exhibiting a physically plausible candidate for a bquantized imaginaryc i that has the classical i as a singular limit, and constructing a vacuum in which i is frozen (§6.4). Since the dynamical i is not central, the quantization relation between classical Pois- son Brackets and quantum commutators must be weakened and generalized. To make a symmetric operator S from an antisymmetric one A using an antisymmetric i, the product iA does not always work. It is replaced by an anticommutator:

Assumption 5 (Full quantization rule) Canonical commutation relations of the forms

[A, B]+ = i~C or [A, B]− = C are limits of fully quantum commutation relations of the forms

~ [A,b Bb]+ = {i, Cb} or [A,b Bb]− = Cb (1.45) 2 respectively, in the singular limit X → 0, Ab → A, Bb → B, Cb → C.

1.5.5 Root vectors and quanta

Heisenberg’s path from discrete atomic spectra to bquantum non-commutativityc was ele- gantly explained by bConnesc ([21] page 37). In the classical theory of integrable systems, P −iωt the observables are almost periodic: q(t) = qωe , and the algebra of observables is the group algebra (convolution algebra) of the discrete spectral group — the multiplicative group of the system frequencies e−iωt). Both the algebra and the group are commutative in classical mechanics. In atomic spectroscopy, however, the frequencies have two indices, as in e−iωnmt labeling the input and output to the transition. If E1 < E2 < E3 are atomic energy strata, the transition 2 → 1 is found immediately after the transition 3 → 2 but not immediately before. The spectral group is thus empirically non-commutative: Therefore its group algebra, presumed to be still the algebra of observables, is non-commutative. A discrete spectrum of transition frequencies practically implies non-commutativity. The essential point for this non-commutativity is not the discreteness, however, but the fact that any observed frequency belongs to a transition of the system, not a state of the system. Non-commutativity alone does not guarantee a discrete spectrum. In Heisenberg’s original quantization, qp − pq = i~. This guarantees that neither p nor q are quantized. 1.5. FULLY QUANTUM REGULARIZATION 87

They are continuous, not discrete. The canonical quantum theory implies no granularity of either p or q, but leads to granularities of energy further along the road depending on the choice of a Hamiltonian operator. From the viewpoint adopted here, the canonical quantum strategy is only a ﬁrst step in the direction of a fully quantum strategy. A deeper kind of bnon-commutativityc that necessarily leads to granularity in a quantum theory was formulated by bKillingc and bCartanc before the quantum theory. This is the bnon-commutativityc of a bsimplec (or bsemi-simplec) Lie algebra. If a is a simple Lie algebra and H ⊂ a is a maximal commuting independent (Cartan) Lie subalgebra of a, then there is a basis of simultaneous eigenelements Ψn ∈ a of all h ∈ H:

h × Ψn = ΨnEn(h). (1.46)

This says that Ψn increments each h by the quantum En(h), a broot vectorc of a. The h are therefore granular properties and En(h) is the size of the granule of h in the Ψn dimension. Quanta correspond to root vectors of a Lie algebra interpreted as a quantum statistics. The canonical Lie algebra lacks eigenelements Ψ with non-zero granule sizes, since among its h ∈ H is the scalar 1, and 1 × Ψ = 0 for all Ψ ∈ a. When the Lie algebra a is bsimplec (or semisimple), there is a complete set of eigenelements Ψ, deﬁning granule sizes of all the Hα. The strata of fully quantum theory are all quantum but are analogous to those of classical set theory, which are all classical. One or possibly two strata of canonical quantum bﬁeldc theory are quantum, and all the lower strata are classical. Here all of these strata are quantized:

1.5.6 Full Fermi quantization bFull Fermi quantizationc is a heuristic process that presents a singular physical theory as singular limit of a regular fully quantum theory whose vector space is a self-Grassmann algebra within S. The present version approximates the Lie algebras of the singular theory within linear-group Lie algebras sl(n) . The resulting fully quantum theory is simple and finite, and hopefully works better than the singular theory. A bfull quantizationc is set up for the bstandard modelc and bgravityc, where all observables have bounded discrete spectra. The correspondence principles of fully quantum theories state that the fully quantum theory agrees with the canonical one in a certain limit X → 0. This provides simple regular variants of present-day singular gauge theories but is not strong enough to define the fully quantum theory completely. Further specification is needed to define the various charges, groups, and masses that are built into the fine structure of the fully quantum set theory. Analogously, the correspondence principle did not lead to definite predictions in the domain where ~ is not small. The possibility of arbitrary terms of order X must be excluded, like Pauli magnetic moments in the Dirac equation. Pauli moments were excluded by the 88 CHAPTER 1. STRATA OF ACTUALITY

bminimal couplingc principle and by specifying a formal expression of the classical theory on which gauging, the replacement ∂ → ∂ + A, could then be carried out. The gauging process that led to general relativity is similar, and is referred to as bgeneral relativizationc. In general relativity, the relativistic dynamics was defined up to one cosmological constant by the assumptions that the dynamics was gauge invariant and local of bminimal differential orderc. The condition of minimal differential order can be regarded as a condition of maximal locality, permitting the action to couple only events in the smallest possible cluster of nearest neighbors. Analogous assumptions are made in fully quantum gauge theory and define an action function up to a coupling constant. To make predictions of experiments in the current ambient vacuum, however, additional parameters describing the organization of the vacuum may be required.

1.5.7 Physical Lie algebras We ask here which physical Lie algebras require structural stabilization, as in full quantization. In the following listing, Lie algebras are allowed to be graded unless otherwise stated, and so the list includes Grassmann and Cliﬀord algebras, deﬁned aby anticommutation relations rather than commutation relations.

1.5.7.0.1 Statistical Lie algebras A bstatistical Lie algebrac is a Lie algebra generated by input-output operators of elementary systems and their graded commutators, representing the statistics of the systems. It is used to generate associative operator algebras whose elements represent physical actions and transformations. Example 1: The three-dimensional Fermi Lie algebra generated by one 0-grade element ∗ ∗ ∗ 1 and two first-grade elements a, a , with [a, a ]− = 1, [a, 1]+ = 0 = [a , 1]+. Example 2: The three-dimensional Bose Lie algebra generated by one 0-grade element ∗ ∗ ∗ 1 and two first-grade elements a, a , with [a, a ]+ = 1, [a, 1]+ = 0 = [a , 1]+. Example 3: The three-dimensional Clifford Lie algebra generated by one 0-grade element 1 and two first-grade elements a, b , with [a, a]− = 1 = [b, b]−,[a, b]− = 0, ∗ [a, 1]+ = 0 = [a , 1]+. In the present work all the other algebras, Lie or associative, that occur in physics are built out of statistical Lie algebras.

1.5.7.0.2 Kinematical Lie algebras By a bkinematical Lie algebrac is meant a Lie algebra of the inﬁnitesimal isometries of a vector space with the commutator as Lie product, used to represent inﬁnitesimal operations on the system under study.

1.5.7.0.3 Generating Lie algebras By a bgenerating Lie algebrac is meant one that generates a kinematical Lie algebra. Examples: The three-dimensional canonical Lie algebra h(1) with [p, q] = i~ generates the kinematical Lie algebra of operators on the Hilbert 2 space L (R) with the commutator as Lie product. 1.6. GRAVITY AND OTHER GAUGE FIELDS 89

1.5.7.0.4 Invariance Lie algebras are subalgebras of kinematical Lie algebras leaving invariant some structure of the system such as the Hamiltonian. Example: The Poincar´e Lie algebra of special relativity.

Full quantization derives all these Lie algebras from one source, statistics. It uses statistical Lie algebras as generating Lie algebras, uses these to form kinematic Lie algebras, and finds the symmetry Lie subalgebras within the kinematic Lie algebras from experiment. This replaces the old assumption that “everything mathematical object is a set” with a quantum version: “Every physical system is a queue.” To define a queue it remains to choose the statistics. The Clifford statistics is selected for study here for the following reasons. A set contains each element with occupation number 0 or 1, and is unchanged by interchanging two of its elements. bFermi statisticsc has these properties, but is structurally unstable, Grassmann’s Relation a2 = 0 being the singular limit λ → 0 of Clifford’s Relation a2 = λ. Clifford statistics also includes a structurally stable Palev version of Bose statistics as well. Clifford algebra is used in classical physics to describe the chronometrical structure at one space-time point, and in quantum physics as the spin algebra of leptoquarks; it is assumed here that these are vestiges of the deeper Clifford statistics of the elements of quantum space-time. So it is Clifford statistics that is iterated here to form the fully quantum theory. In the best outcome, the operations of a fully quantum theory would have a uniform physical interpretation so that every polyadic would have a physical meaning that could be read from its structure, with no redundancy. This would make queue theory a universal physical language in a strong sense, but one without self-checking. This is considered desirable here, but is not assumed. Dynamical symmetry algebras like the PoincaréLie algebra of space-time, or the unitary Lie algebras of the bstandard modelc, were traditionally treated as absolute, given once for all. Set theory has no intrinsic symmetry, however; all its sets are different. Sets (finite!) are generated from the empty set 1 by Clifford multiplication (x2 = 1) and the D operation of unit-set formation or bbracingc I : x 7→ {x} (§4.1.4). Since there can be no “fundamental symmetries” in a queue theory, where all vector rays are intrinsically different, it is assumed here that all interesting symmetries are contingent symmetries of some fully quantum structure like the vacuum that can be represented as a queue. For example the PoincaréLie algebra is presented as a singular limit of a simple Lie algebra fixing a quantum organization that has Minkowski space-time as a singular limit.

1.6 Gravity and other gauge ﬁelds

If physical space-time is indeed a fully quantum space, then bgravityc is likely a vestigial quantum effect, in that the non-commutativity of the infinitesimal translations of Riemann can be vestiges of the non-commutativity of swaps of quantum events in the near-classical 90 CHAPTER 1. STRATA OF ACTUALITY limit. The causality form introduced by Einstein is well-known to be the Clifford form of the Dirac Clifford algebra with generators γµ(x), and these can be a higher-stratum organization of the Clifford vectors sα of elements below the event stratum E. These ideas lead into the following familiar conceptual developments.

1.6.1 History as quantum variable Heisenberg, emulating Einstein, set out to work solely with observables, and ultimately encoded operations of observation in single-time operators Q of his quantum theory. But his dynamical equations d ∂ [ − i −1H(t) − ,Q(t)] = 0 (1.47) dt ~ ∂t concern not his alleged observables at one instant but observable-valued functions of time Q(t). In a relativistic theory the construct of instant is relative to the space-time frame and it is convenient to formulate dynamics for histories, extending over time, rather than for instants. In a q theory of gravity this is all the more convenient, since quantum measurement at one instant demands infinite energy resources of the experimenter, and therefore should not be taken seriously, except as a singular limit ~ → 0. In a Q theory most experimenter frames do not even diagonalize time. Therefore the present full quantization is based on history vectors, not instantaneous ones. It is diachronic, not synchronic. A bhistoryc of a variable q in the usual synchronic quantum theory might be understood as an operator-valued function of time {q(t)}. This Heisenberg history is not the history that is meant here. If variables at different events are to be assigned values, they must commute, they must be independent variables, and the dynamical “laws” that relate them in the synchronic theory must be overridden. Indeed, processes of control of a system, by an experimenter, like observation, override the system dynamics. A Heisenberg history q(t) obeys the dynamics. The quantum history studied here overrides the dynamics. It is a pre- dynamical history, represented in each frame by a Dirac-Schwinger-Feynman probability amplitude vector hE] for histories. In Q theory these form a finite-dimensional history vector space, designated below by S[F]. One then express dynamics not by a relation among operators but by a subsidiary condition, restricting the vectors hE] to a subspace of S[F]. A one-dimensional subspace will usually serve. Then a typical unit vector of this ray is selected and called the dynamics vector hD]. Fourier transformation figures prominently in the q theory. For example, generating functions are history probability amplitudes Fourier-transformed from field-variables to the dual variables called sources. The Fourier transform is as structurally unstable as the canonical commutation relations on which it is based. It makes no sense in a finite- dimensional algebra of variables. Fortunately, the Grassmann Fourier transform is based on canonical anti-commutation relations of a Clifford algebra, and is structurally stable. Q 1.6. GRAVITY AND OTHER GAUGE FIELDS 91 theory uses the Grassmann Fourier transform wherever q theory uses the classical Fourier transform. Operators of a Q theory operate on history probability vectors hF ] describing the course of an experiment. Vectors hF ] need more than vector-space structure to describe histories. Naturally we use a stratification structure for this purpose and assume that a Q history is a catenation of Q events as a c history is a catenation of c events. The space S[F] of Q history vectors hF ] is assumed to be one stratum higher than an underlying vector space S[E] of Q events described by event vectors hE]:

S[F] = 2S[E]. (1.48)

In this Q is more faithful to the c structure than canonical quantization, which suspends this exponential relation. The singular sum over q histories is now a singular limit of a finite-dimensional trace over Q histories as the dimension grows. Finite quantum instants of the canonical quantum theory arise when a time operator bt is chosen, a singular limit bt ⇒ t is taken, and an eigenvalue of t is selected. In the fully quantum strategy of Assumption 3, all systems are catenations of simple systems with Fermi statistics. This permits the theory to be structurally stable. To permit replication of an experiment its metasystem must be protected. Therefore high-resolution quantum measurements are restricted here to small regions well removed from ourselves and our instruments. We will therefore assume that the usual macroscopic continuous space-time construct still works well enough in the metasystem, though not in the system. A dynamical law is represented as usual by a history probability dual vector [Dh ∈ DS[F], assigning a probability amplitude [DhE] to any experimental history vector hE] ∈ S[F]. One vector in a vector space does not give enough information to describe an arbitrary dynamical development, which calls for a one-parameter family of transformations. Here the stratification of S[F] supplies the missing information, in both q and Q theories, by providing the event construct. The dynamical vector [Dh entangles adjacent events in the development it defines.

1.6.2 Fully quantum equivalence principle

1.6.2.0.5 Classical equivalence principle The Galileo-Einstein classical bequivalence principlec states that the effects of gravity are locally equivalent to the effects of an baccelerationc. This means that to lowest order in the size of the experimental neighborhood one acceleration simulates gravity for all bodies and fields in that neighborhood. It is implied that the entities of the theory all have laws of transformation under acceleration, and that the theory is invariant under these transformations. Taken literally, the classical equivalence principle implies vanishing btorsionc at every point, since torsion is absent from special relativity and is not introduced by acceleration, but clearly a small 92 CHAPTER 1. STRATA OF ACTUALITY torsion cannot be excluded experimentally, and must be expected on grounds of structural stability. 1 2 Acceleration is a time-dependent translation x → x − 2 at , a special case of a gauge transformation of coordinates. Invariance under acceleration is intended to imply invariance under the group generated by accelerations. Accelerations have local Jacobian determinant unity at every event, in a preferred coordinate system, called bunimodularc. Such transformations are also called bunimodularc and form a group, the bunimodular diffeomorphismc group UDiff[M] of the manifold M. The unimodular condition is stronger than the bspecialc condition, which refers to a global determinant that does not exist for diffeomorphisms. The group that accelerations generate is surely not the diffeomorphism 4 4 group Diff(R ), but it can be the unimodular diffeomorphism group UDiff(R ) of differentiable transformations of local Jacobian determinant 1 in a unimodular coordinate system on stratum E. Einstein understood that UDiff invariance and Diff invariance have the same physical consequences; invariance under Diff he called “general” covariance; invariance under UDiff is called bunimodulat covariancec. UDiff transformations of events on stratum E induce transformations of the field on stratum F that form an isomorphic group designated by UDiff[F]. Here invariance under at least UDiff is assumed in the classical limit in order to satisfy the classical bEinstein Equivalence Principlec: Gravity is a fictitious force that is locally equivalent to a gauge transformation in the group UDiff[F]. UDiff, however, still requires transformed coordinates xµ0 to depend only on the co- µ ordinates x , though possibly nonlinearly. A transformation to bharmonic coordinatesc µ0 (de Donder coordinates) x , each of which obey the covariant wave equation x = 0, is an example of a more general coordinate transformation, in which the xµ0 depend on the gravitational potentials as well as xµ.

1.6.2.0.6 Gauge equivalence principle Modern gauge theory generalizes the Ein- stein equivalence principle: Gauge equivalence principle All forces are ﬁctitious forces derived from gauge potentials that are locally equivalent to gauge transformations. This reduces to the equivalence principle in the case of gravity, so it will be called the bgauge equivalence principlec.

1.6.2.0.7 Fully quantum equivalence principle The full quantization leading to ϑo replaces the unimodular diffeomorphism group UDiff by a finite-dimensional Lie subgroup of SO[E]. To avoid notational clutter the entire SO[E] will be used for now, subject to later reduction: UDiff ⇐ SO[E] (1.49) 1.6. GRAVITY AND OTHER GAUGE FIELDS 93

Invariance under this fully quantum correspondent of Diff is more general than general covariance even though it is finite-dimensional, in that it mingles space-time coordinates with momentum-energy coordinates of the event. Einstein described gravity with a symmetric tensor. Clifford algebra does not include tensors symmetric in two vector indices; the symmetric part of sνsµ is a mere scalar. The Minkowski bilinear form gνµ(x) is enough to guide point planets but not electrons, which carry significant spins. Quanta of spin 1/2 represent their spin in a Dirac Clifford algebra at each event x, with generators γµ(x), whose Clifford product defines and is defined by the chronometric g(x) at that event. This is a natural way to represent gravity in a Clifford algebraic fully quantum theory. One arrives at the construct γµ(x) by gauging the flat-space-time Clifford algebra, whose generators γµ do not depend on x. Now we formulate a fully quantum correspondent of this gauging process. The γµ originate on a deep stratum C, the event x on a higher stratum E. The structure γµ(x) transforms under the groups of both strata. [To be continued.] XXX It is supposed that a quantum bilinear form igh on vectors of a lower stratum C underlies the classical Minkowski form of special relativity. g(x) designates the Minkowskian form of unimodular relativity, a symmetric tensor field of determinant everywhere unity in special coordinate systems, reducible in any one reference frame to a fixed Minkowski form g0 by a local frame change Λ(x):

T g(x) = Λ(x) g0 Λ (x). (1.50)

Let

Dµ = ∂µ + Γµ (1.51) be the covariant differentiator for the differential manifold M of space-time according to general relativity. The term ∂µ is a translation generator. For a quantum moving in a classical gravitational field, ∂µ is the quantity that is conserved if translation invariance holds. Therefore it represents the total momentum(-energy). For a spinless quantum, the term Γ is missing. Therefore Γµ can be regarded as the spin momentum. What is left, Dµ, is then the orbital momentum. λ C For vector fields Γµ is of type Γ κµ(x). More generally, it is of type Γ µ(x) where C is a Lie algebra index, specifying the transformation that acts on the entity being transported in the direction µ at x. In the queue theory ϑo, momenta become higher-stratum representatives of underlying swaps. The infinitesimal displacements on M are singular limits of representatives of elements of so[C] in so[E]. The queue correspondents of the total, orbital, and spin momenta, represented by ∂, D, and Γ, can be called swaps of strata F, E, and C, represented by ω[F], ω[E], ω[C]. 94 CHAPTER 1. STRATA OF ACTUALITY

The queue correspondent of the Minkowski coordinate xµ is an E-stratum representative of a C-stratum swap ω[C]µ ∈ so[C]. C also indexes the complementary momentum(-energy) coordinates and the angular momentum of the quantum. The queue correspondent of a function of x is an operator on e0 0 stratum-E vectors, represented by a matrix of the type Me with two vector indices e, e labeling a basis se ∈ S[E] of stratum E. Therefore the presumptive queue correspondent of C e0 the total momentum ∂µ is a swap tensor ω C0 e representing how the cumulated swaps of stratum C act on the vectors of stratum E. e0 Now the total swap tensor ωC e corresponding to Dµ is to be decomposed into parts corresponding to borbitalc and bspinc parts ∂µ and Γµ. The basis vector he probably e0 consists of many D-stratum factor vectors, all contributing to the moment ωC e . Probably a logarithmically small number of these are C-stratum factors. Their contribution to the total moment will be identiﬁed with the bspin partc; the rest with the borbital partc. To consider whether the spin and orbital parts of the usual connection can be singular limits of the stratum-C and stratum-E swaps we must examine how they transform under the queue transformation that corresponds to a diﬀeomorphism of the event manifold M. General relativization includes the familiar replacements

g0 → g, ∂ → D = ∂ − Γ, (1.52) designed to convert a Poincaré-invariant theory into a nearby UDiff invariant theory that is equivalent in the limit G → 0 of flat space-time. To fully quantize the process of general relativization one first fully quantizes these groups. The process of general relativization was already generalized within canonical physics to the heuristic process called bgaugingc. Gauging converts a theory invariant under a gauge Lie group of stratum C (“of the first kind”, as Dirac calls it) into one invariant under a gauge group of stratum E (“of the second kind”), indicated in (4.33). A gauge group acting on elements of stratum C will be called a bC-gaugec group, and similarly for other strata. There are two famous ways to gauge, here termed the Weyl (§1.6.3) and Kaluza (§1.6.4) gauge strategies. Both set out from general relativity.

1.6.3 Weyl gauge strategy

The bWeyl gauge strategyc adjoins to the space-time tangent space at each point an “internal” gauge vector space V , on which the gauge group G acts through its defining representation G → V D ⊗ V . This enlarges the fiber of the physical bundle but not its base G space of events. It accounts for the gauge vector field Cµ (x) — here G is an index for a G basis of the Lie algebra dG — as a connection form Γµ (x) defining the parallel transport of new gauge variables of various kinds from event to event:

G G Cµ (x) = Γµ (x) (1.53) 1.6. GRAVITY AND OTHER GAUGE FIELDS 95 carries new things in V in old directions labeled by µ. The Weyl theory of electromagnetism, the Dirac theory of electromagnetism, the bYang-Millsc theory of the isospin connection, and the standard model gauge theories of hypercharge, electroweak, and color gauge ﬁelds, are instances of the Weyl strategy.

1.6.4 Kaluza gauge strategy In the Kaluza gauge strategy, four-dimensional space-time manifold M is directly multiplied by a Kaluza space that is isomorphic to the n-dimensional gauge group manifold G itself. This supplements the old space-time coordinates xµ by new group coordinates xG — here G indexes the group parameters — on the group G. Their diﬀerentiators ∂G(1) at the origin 1 ∈ G span the Lie algebra dG. For Kaluza the tensor C that multiplies the currents in gauge interactions is not a connection at all, but a sector of the causality form,

CµG(x) = gµG(x). (1.54)

G µ Cµ (x) is the element of the metric tensor that couples the external differential dx with an internal differential dxG. In Kaluza’s original model, the internal Lie algebra was the one-dimensional so(2) = u(1), so the index G had only one value. Now the gauge vector field appears as an off-diagonal block gµG in the chronmetric, coupling the usual macroscopic or external space-time dimensions of M with new microscopic dimensions of space-time forming the manifold of a Lie group G in an enlarged space-time M 0 =∼ M × G (1.55)

The same metric also has a gravitational block gµ,ν, and a block gG0G, the Killing form of the internal Lie algebra. Kaluza strategy enlarges both the base space and the fiber of the physical bundle. Its connection carries new things as well as old to new places as well as old. Since Einstein calls gνµ the gravitational potential, its bordering block, the gauge vector field C, can be called the gauge potential in the Kaluza strategy. The Kaluza strategy builds the strong bgauge equivalence principlec into the foundations of the resulting theory as part of an extended Einstein equivalence principle. The bKaluza-Kleinc, bDe Wittc [22], bMacDowell-Mansouric [54] theories and many others use the Kaluza strategy. One defect of the original Kaluza strategy is that for the extension to quantum field theory the internal manifold must be a compact Lie group, closed on itself, so that integra- tions over it converge, and huge energy is required to close it, but none is provided by the theory. This is the bcompactification problemc. It arises in Kaluza theory because, being classical, this theory can account for a Lie algebra only by providing a physical continuum for the algebra to act on. 96 CHAPTER 1. STRATA OF ACTUALITY

1.6.5 Queue gauge strategy

Yet molecular biologists do not see a compactification problem in DNA, because they understand its quantum structure and do not invent internal continua. In a queue theory the internal Lie algebra can arise from a finite-dimensional vector space instead of an infinite-dimensional function space over a classical manifold. To sum or average over such quantum degrees of freedom, one does not integrate over the group, one traces over the Lie algebra. The non-compactness of the group is irrelevant. In quantum theory regularity — convergence — is assured by simplicity and the ensuing finite-dimensionality, not by compactness of the group. In the theory ϑo, the Kaluza classical internal manifold is therefore replaced by an internal finite-dimensional vector space supporting the simple gauge Lie group. One problem of quantum physics today is the non-simple nature of the external part of the system, which gives rise to infinities. In a queue space-time, the external manifold of Kaluza too is replaced by a vector space, whose Lie algebra includes the quantized coordinates, momenta, and angular momenta. That is, instead of compactifying the internal space of Kaluza, a fully quantum theory quantizes both the external space of Einstein and the internal space of Kaluza. The external and internal groups are both simple Lie subgroups of one larger simple group, acting on unified vectors for the combined space. Since one does not integrate over vector spaces but merely traces over them, the compactification problem is replaced by the bgrowth problemc: to account for the great growth of six dimensions relative to the others, and the freezing of two, that results in a crystal film. It is natural to use the large ratio between the multiplicities of the low and high strata of the queue to account for the Large Number. Full Fermi quantization represents the system as an organization of a great many simple unit cells without any external imbedding space or manifold. What are usually regarded as internal dimensions are represented by the thickness of the crystal dome, perhaps one cell thick. Cosmic cortication presumably results from anisotropic crystal formation, like that of a snowflake. In the Weyl gauge strategy, the internal and external groups act on spaces of supposedly separate origin and nature; in the Kaluza and fully quantum strategies the spaces are united. The canonical quantization process worked well enough for photons but poorly for classical point electrons. The spin of photons is already represented in classical physics by the tensorial nature of the electromagnetic field, and so does not need to be invented from scratch during quantization, but electrons were represented in classical physics as point particles, with neither spin nor tensorial nature. This problem is evaded here by starting from the standard model and its spins, which are readily accommodated within the self-Fermi algebra S. 1.6. GRAVITY AND OTHER GAUGE FIELDS 97

1.6.6 Fully quantum gauge group Here a fully quantum gauge strategy is executed. The equivalence principle of Einstein and its gauge generalization are local. An infinitesimal acceleration generator, for example, is an infinitesimal translation generator 1 2 with time-dependent parameter, 2 at ∂x. In full quantization, space-time coordinates and momenta become binducedc representations on stratum F, of rotation generators, spins, ωβα ∈ so(n) on stratum C.. The idea of a function of non-commuting variables is not well defined, but the construct of ordered polynomial expression in non-commuting variables is unproblematical. This will be called a bpolynomialc for short. A conspicuous gauge algebra that has bdiffc (the Lie algebra of Diff) as a singular limit is the Lie algebra of all polynomials in the E matrix representatives of the C gauge Lie algebra. This is a sub-Lie algebra of so[E] := so(SE). For simplicity it is taken to be the entire so[E]. That is, the fully quantum correspondent of diff := d Diff is taken to be so(S[E]). This symmetry ignores the deep structure of the vectors in S[E], associated with the general event. This makes it possible to fully quantize the process of general relativization, and of gauging in general. In ϑo six quantum strata A – F , each an exponential of the previous, are used for this. Three quantum strata D – F correspond to the familiar classical strata D – F. The bC gauge groupc is so[C] = so(10, 6), the kinematic group of stratum C, and has induced representations in so[L] of every higher stratum L = D, E, F. The F gauge group is the representation of so[C] in the orthogonal group so[F] of the vector space of stratum E. General relativity is a special case of a gauge theory, and suggests the following definitions: A bgeneral coordinatec of an event is a normal operator on the event vector space S[E]. Therefore the queue correspondent of the group Diff(M) is presumably the special orthogonal group of the vector space of the event: SO(S[E]) ⇒ Diff(M) (1.56)

The bqueue gauge groupc of the F stratum is then GF := Π SO[E], (1.57) the group on stratum F induced by SO(S[E]. It is to be shown that the gauge transformations of the standard model too are approximated by elements of the group SOE, modiﬁed by the assumed vacuum organization. GE is necessarily non-local; simple quantum space- times do not admit the construct of locality. Full quantization of the local equivalence principle, however, provides a quantum bequivalence principlec: Assumption 6 (Quantum equivalence principle) The interactions of nature are e- quivalent in each queue cell to a fully quantum gauge transformation. 98 CHAPTER 1. STRATA OF ACTUALITY

Assuming Fermi statistics, the low-stratum quantum gauge group is a linear group. The gauge group of the standard model with gravity is

Gg,sm = ISO(3, 1) × S(U(2) × U(3)) (1.58) taking into account a discrete central correlation [62, 63]. This is not even a simple group, let alone an orthogonal one, so in the canonical theory one assumes that Gg,sm is a singular limit of an actual simple group GE of the event stratum E. A gauge transformation is now a group action on the internal coordinates by an amount that depends nonlinearly on the external coordinates. In the classical limit this does not affect the external coordinates and so the theory is singular. In full quantization the external coordinates are affected too and the theory is regular. The usual Dirac Clifford algebraic representation of the causality form through the µ spin form γµ puts a Clifford algebra on the tangent vectors at each event. The γ (x) encode the causality form as the square of a Clifford vector. Therefore they also encode a classical gravity field. At the same time they are spin operators of a spin 1/2 quantum at the point x, and so are already quantum variables from birth. They therefore need not be quantized in the usual sense but can merely be quantified, aggregated, an easier problem. The spin system at an event has only four dimensions to its vector space, those which in Dirac one-electron theory give rise to spin up or down, energy positive or negative. The quasi-continuity of gravity arises from the many events supporting such four-valued spin variables, and this number already becomes finite when only the xµ are fully quantized. µ Segal and Vilela-Mendes quantized the coordinates and momenta x , pµ on one stratum within an orthogonal Lie algebra (§5.2.5, §5.2.8). A similar one-stratum quantization within a Clifford algebra is described in §5.3. This is imbedded in a full quantization in §6.3.

1.6.7 The space-time truss The spectral spacing for the quantized time coordinate can be regarded as a quantum of time or bchronc X, with the understanding that times add in series but average in parallel, leading to finer spectra. It is sometimes heuristically useful to picture an binputc vector of n a quantum mechanical system as a “bcellc of size ~ ” in a classical phase space. It may also be useful to picture binputc vectors of the individual quantum event as n-dimensional “cells” of size Xn in a classical tangent bundle to event space, with a preferred vertex of each cell as its origin, and elements of a lower stratum as edges or struts at the origin. Similarly, the vectors of the quantum space bsetc are cells with events as their edges. The cell of 16 vertices has the kinematical group SL(16), which accommodates the stratum-C gauge groups of gravity and GUT. The fully quantum ambient space-time would then have a highly asymmetrical structure like that of a fullerene or a btruss domec: long in some directions, short in others. The so-called external and internal dimensions of the standard model are the longitudinal and transverse directions of the cosmic shell, the blongc and bshortc dimensions. The 1.6. GRAVITY AND OTHER GAUGE FIELDS 99 long dimensions have many struts extending along the dome hypersurface for macroscopic distances, and for many purposes can be handled classically, because they contain many quantum units. The short dimensions are transverse spreaders only one or several cells long. Therefore they require description at the quantum stratum of resolution. There are well-known problems arising when classical and quantum systems are coupled. The validity of such a semiclassical description is limited. The standard model is of this kind. Here we look a little beyond the semiclassical stratum. Possibly the short dimensions of the set belong to lower strata but the long dimensions certainly require higher strata. For example, one might consider a btrussc composed of uniform basic quantum cells, each composed of 16 independent struts. One such cell is first found on stratum 4, whose Grassmann algebra S[4] has 16 monadic generators. A truss of such cells can then be 216 constructed on stratum 6, which accommodates about 16 such cells. The SL(16) would then provide both a bGUTc group SO(10) of the short dimensions [43, 42] and a simple Lie group SO(3, 3) of the long dimensions [66]. The truss is best not assembled cell by cell, however, but by the groups that act on its swaps and monads. Gauge theory is singular due to the unstable commutation relation between the gauge differentiator and the space-time coordinates:

µ µ µ µ λ [Dν, x ] = δν , [δν ,Dλ] = [δν , x ] = 0. (1.59)

The assumption of zero torsion is also unstable. Like the commutativity of space-time coordinates, these commutativities must be artifacts of the classical space-time limit, and do not hold exactly in the fully quantum theory. Full quantization eliminates these singularities. Curvature, the classical non-commutativity of the covariant diﬀerentiators, is then a classical vestige of the quantum no-commutativity that results in a discrete uniformly spaced spectrum for basic coordinates and set variables. This classical non-commutativity is the gravitational ﬁeld.

Gravity is a quantum eﬀect.

1.6.8 The gravitational and gauge potentials Fermi algebras accommodate antisymmetric tensors readily but not symmetric ones like µ µ5 the gravitational potential gνµ(x). However the first-grade differential dx becomes dx in quantum space-time, part of a swap dxα2α1 . This can be assumed to be anti-symmetric in its two indices. In ϑo, the space-time tangent space of stratum D corresponds to stratum 4, the lowest with enough dimensions. This vector space supports the Lie algebra so(10, 6) and its associated swaps 1 1 ω = [γ , γ ] =: [γ , γ ] (1.60) βα 4 β α 4 β α + 100 CHAPTER 1. STRATA OF ACTUALITY and metric form 1 1 g = {γ , γ } =: [γ , γ ] (1.61) βα 2 β α 2 β α − This spin serves as a prototype quantum space-time cell. Each such cell carries its own Clifford bilinear form. Beneath it lies stratum 2, with a Minkowskian vector space and a Lorentz Lie algebra so(3, 1). The operator-valued causality form gνµ(x) of canonical bgravityc is understood to F F D µ correspond to a single operator gb ∈ S ⊗ S . The space-time coordinate x becomes a representative on stratum E of a grade-2 swap γµ5 of so(6; σ), part of a form γα2α1 .

The tensor gνµ(x) becomes part of a form gα4α3α2α1 (x), symmetric with respect to the interchange of collective indices α4α3 ↔ α2α1 because it is totally antisymmetric. The x-dependence becomes a large number of further indices, approaching ∞ in the limit of classical space-time. This means that the bgravitonc, as well as and the gauge vector bosons of this model are not true bosons at heart but bpseudo-bosonsc with fermionic cores, analogous to two- neutrino models of the photon and four-neutrino models of the graviton considered by de Broglie, Feynman, and others. There need be only one space-time coordinate in gb, while two or four neutrinos would provide two or four coordinates. The catenation that forms this graviton occurs at stratum E, a deeper stratum than the set stratum F where fermions are found. This core structure might show up at high energies but it may be invisible at low energies.

1.6.9 Vacuum bNewtonc and bFresnelc reasoned that their vacuum, the betherc, is crystalline because it propagates transverse modes of light with great transparency, unlike fluids. This argument applies to the vacuum today. Nothing stiffer than the vacuum is known in nature if one judges stiffness by the speed of waves, and also nothing more transparent, judging by the mean free paths of photons and neutrinos. These observations indicate that the vacuum is highly organized and quite cold at the ambient temperatures common today, compared to its melting point. On the other hand, it seems plausible that space-time meltdown occurs near central quasi-singularities of black holes. The vacuum of the standard model is a bhigh-temperature superconvectorc of color and weak currents in that some of its gauge symmetry is broken, such as the symmetries between the electric and weak gauge dimensions. According to GUT there is a similar gauge symmetry breaking between the hypercharge, electroweak, and color gauge dimensions of GUT. In a Kaluza-style theory like ϑo there is another symmetry breaking between gravity and the other gauge dimensions. Only the gauge invariances of electricity and gravity are ultimately considered to be unbroken in the present vacuum. In the classical limit with finite c and with ~, X → 0, the vacuum experimentally defines a light-cone and an electromagnetic axis at each point. On a laboratory rather 1.7. UNIFICATIONS 101 than cosmological scale, Poincarésymmetry also survives. In the even more singular limit c → ∞ the vacuum defines a Galilean instant, still without absolute rest.

1.7 Uniﬁcations

Canonical quantization unified energy and frequency, and other ~-related pairs, almost as an incidental by-product. Further simplification results in further unification. The following units of this section summarize unifications that arise in this work. One unification guided the construction, that of

D • the bracing I of classical set theory, Peano’s ι, • the bracing of spin (§3.3.4),

• the Dirac construction of the space-time Cliﬀord algebra

• the Fermi-Dirac construction of fermion algebras

• hopefully, the rank-raising operator connecting the three successive strata of leptons and quarks are uniﬁed here into one bbracingc operator I. The other fusions that this requires are somewhat unexpected outcomes and are listed in advance to alert the reader.

1.7.1 Being and becoming There are well-known unifications of being and becoming, or essence and existence, in both relativity and quantum theory. In classical thought, space is a pattern of relations between states of being and time is a pattern of relations between becomings. In special relativity that partition between space and time is relativized, and relations that are purely spacial for one experimenter have a temporal component for most others. In classical physics, again, the points of the coordinate space describe states of being and tangent vectors represent modes of motion, becoming. In quantum physics each operator is both a coordinate and a generator of a mapping, describing both being and becoming. In quantum space-time both unifications are unified. The idea of a space-time point is unified with that of a tangent vector into one construct called here an event or E which decomposes into coordinates and momenta only as the result of a symmetry-breaking organization. In this respect the event E resembles a point of phase space more than a point of space-time. The Lorentz group that mixes space and time is enlarged here to an orthogonal group on the event space that mixes space and momentum, time and energy like the canonical group. 102 CHAPTER 1. STRATA OF ACTUALITY

During cosmogenesis, one of the early phase transitions was a quantum organization of events that formed the space-time quasi-continuum, centralizing the position variables and freezing momentum variables 0 by a quantum organization. Local space-time meltdown must be expected in suﬃciently hot ﬁreballs today.

1.7.2 Gravity and quantum theory bPeanoc introduced a bsuccessorc operation ι, first in his theory of the natural numbers, to generate the natural numbers from 0, and then into his set theory, where it generated all sets. The name suggests that ι determines a temporal succession, though perhaps only metaphorically. His ι was soon drowned out by the weaker notation {s} := ιs, {s, t} := ιs ιt, . . . , (1.62) resulting in the inconvenience of a function with a variable number of arguments and no function symbol. I generates a Clifford algebra S of fully quantum probability amplitude vectors in much the way that Peano’s ι generates classical set states. S includes Dirac’s other Clifford algebra too, the Clifford algebra of Fermi-Dirac creators, as a limit. Dirac quantum spin theory and the classical ι of Peano are unified in one construct here, the brace I. The adjoint of I is the bde-bracec D H T −1 I = I = H H (1.63)

It is posited that for vectors v of a certain bcharge stratumc C, any spacelike v is unfeasible and any timelike v is feasible. as Hvv > 0 or < 0 (Assumption 8), and so the probability µ ν form serves as a causality form. The quadratic causality form v ighv = gµνv v = g v v of Einstein gravity theory is to be a singular limit in stratum F of a form induced by the fully quantum probability form v iHhv of stratum C. The existence of a three-fold hierarchy of bstratac or bgenerationsc of quarks is still an outstanding puzzle. Perhaps queue theory can generate such a hierarchy in an ad hoc way, using the operator I invented for other purposes.

1.7.3 Products Canonical quantization unifies two seemingly independent products of classical mechanics 0 into one of quantum mechanics: the classical commutative product is the ~ term in a 2 power series, and the classical Poisson Bracket is the ~ term. Similar product unifications occur during full quantization and help as guides along the way. The bClifford productc famously unifies the bGrassmann productc and the scalar or inner product of vectors. Full quantization uses this Clifford balgebra unificationc process to unify the Grassmann product of Fermi statistics with both the inner product of Hilbert space and the scalar product of space-time vectors on a deeper stratum than space-time events (§3.3.3). 1.7. UNIFICATIONS 103

1.7.4 Non-commutativity and granularity

Canonical quantization expands the classical construct of random variable to a later construct of quantum variable. These variables need not be numerical; the system itself is referred to as a random variable in probability theory. A random variable is described ontologically, by saying what it can be. The space of the possible states of a random system is called its bstate spacec (or bphase spacec, from its use by Gibbs in the statistical mechanics of phase transitions).

An bontologyc is a a theory of what it means to “be”. Here the term is applied to an analysis of Nature into beings, things that exist. Calling something ontological is an emphatic way of saying that it exists as a physical object, in contexts where there also things under consideration like probability distributions, that have a more abstract nature. It is explicitly understood that these “beings” can be associated with symbols that completely deﬁne them, on the grounds that this belief is implicit in most ontologies. The integers of mathematics and the coordinates of classical mechanics are symbols in this sense. Thus “bontologyc”, like “reality”, is used here somewhat pejoratively, in a naive pre-quantum sense relevant to classical physics, where it works well enough. By bontologismc is meant the belief, tacit or explicit, that such an ontology exists.

Theories that work in the quantum domain do not deal with pure unobserved being but with experimental procedures. They describe a quantum not by its possible states of being but by actions that can be carried out with it, especially how to prepare, sort, and register it, by input, throughput, and output processes. This formulation in terms of feasible operations, such as ﬁltrations and sortations, makes possible kinds of non-commutativity not easily imaginable within classical mechanics, which left ﬁltrations out of its early formal structure but tacitly assumed that they commute.

One may therefore speak of quantum theories, say of geometry, simply as non-commutative theories. Mere non-commutativity is not what this means to convey, of course. There are a great many non-commutative groups in classical theories too. The name “quantum theory” originally meant a corpuscular or granular theory, and noncommutativity does not mean granularity. It was a remarkable intellectual leap to go from the grains of light and the discrete atomic energy strata of the bBohr atomc to the non-commutative complex algebra of Heisenberg. This is not a mere change in a theory, but a change in what is understood by a theory, in what the theory is about, and in the theorist’s bstrategyc and philosophy. What is non-commutative in non-commutatuve physics or non-commutative spaces are ﬁltration actions representing predicates, or combinations of predicates and numbers representing coordinates. When that non-commutativity is of the kind that can be represented by a simple Lie algebra, as assumed here, it leads to total granularity, and to a uniﬁcation of the strata of physics. 104 CHAPTER 1. STRATA OF ACTUALITY

1.7.5 System and metasystem

The bmetasystemc has degrees of freedom relating one experimental reference frame to another. Such degrees of freedom of the metasystem are represented in the system as well. For example, in the 19th century, when space was flat and canonical mechanics reigned, every system in nature had to have a momentum operator p generating the changes in the variables of the system generated by a uniform translation of the experimental reference frame in the metasystem. Such operations in the metasystem will be called ımetasystemic. System operators that represent metasystemic operations have long been used. In the last century they grew enormously in mathematical complexity and theoretical importance. The story can begin with Einstein’s treatment of time in special relativity, which assumes that the experimenter can assemble a rigid lattice of clocks and rulers in the metasystem. Such frames were supposed to form a 10-parameter family: 4 coordinates for the origin, 3 for its velocity, and 3 orientation angles. In general relativity the rigid lattice is replaced as framework by a fleet of experimenters smoothly distributed over space-time in an otherwise arbitrary way. Such a frame is specified by an infinity of parameters, defined by 4 arbitrary smooth functions of time-space. In canonical quantum theory this time frame is accompanied by a fact frame, each experimenter having a maximal collection of mutually exclusive input or output possibilities for the system, related to others by a unitary group. The metasystems in gauge theories, including general relativistic spin theory, where independent transformations of the gauge frame are permitted at each space-time event in the metasystem, have the most complex structure so far. The relation between system and metasystem is not a symmetrical or transposable one. The metasystem belongs to a higher stratum than the system in several senses. A person can know more about an atom than the atom can know about the person. The metasystem is thus on a higher epistemic stratum than the system. When the bmetasystemc inputs a system it can be represented as a source by a class or virtual set of systems, from which one is withdrawn by any input, and into which any output is deposited. A class belongs to a higher stratum than its elements, and the same stratum as its subclasses. Under closer inspection, if necessary by another experimenter, any part of the metasystem always reveals its own quantum structure. In the small we are as quantum as any system we study. bBohrc suggested that the canonical quantum strategy might need to be revised to take into account the quantum structure of the experimenter [13]. This would seem to require extending the quantum theory from the system stratum to a higher stratum. In this work an exponential algebra S with an infinite hierarchy of nesting strata is constructed, of which a finite number are used. The interface between quantum system and metasystem is not located or shifted by a mere mental decision or shift of attention. It is a controllable physical gate, closed during throughput and open during input-output. Its construction may require vacuum Dewars, lead walls, superconducting cages, or physical cuts. To move it may be a major enterprise. Nature sometimes provides the insulation needed for the system interface. At the clas- 1.8. OUTLINE 105 sical level of resolution, the planets of the Solar System are surrounded by natural vacuum isolation, fortunately making each nearly a one-body problem. In the Malus experiment the photons in the beam are effectively insulated from each other and the apparatus while they are in flight. In particular, dispensing with the interface, regarding the metasystem as part of the quantum system, seems absurd at the quantum level of resolution. This is not the unification discussed here. Some processes cannot be watched. A maximal observation overrides the dynamics of the system with a system-metasystem interaction, greatly changing the behavior of the system. An observation process that is observed closely cannot observe reliably. To be sure, sufficiently small and isolated chunks of the metasystem can be transferred to the quantum system without making the experiment impossible. This may be a way to infer the dynamical law of the metasystem, piece by piece, Then one can require that the determination processes on the system conform to the dynamics of the metasystem. This is the extent of the unification contemplated here.

1.8 Outline

The job at hand has at least four interlocking phases, taken up in as many Parts in what follows:

Syntactic Set up the queue algebra.

Semantic Translate the concepts of the standard model and gravity into the queue algebra as closely as possible, including the dynamics.

Logistical Deduce experimental consequences of the regularized theory and compare them with those of the previous theory and as far as possible with experiment.

The succeeding Parts of this treatise undertake these respective stages. Chapter 2 introduces probability vectors and takes up their addition. It reviews the praxics of random or quantum individuals, including single-quantum kinematics. Chapter 3 adjoins the operation of vector multiplication. It reviews the classical and quantum polynomial praxics of a random set or queue of individuals, often called second quantization or ﬁeld quantization. Chapter 4 adjoins the operation of vector bracing. It develops classical and quantum exponential (power set) praxics, the theory of a random set of sets of ... sets of individuals; in which any assembly of one stratum can serve as an individual of the next stratum. The brace I generates a self-Grassmann algebra S, a Grassmann algebra over itself:

S = Grass S = 2S. (1.64) 106 CHAPTER 1. STRATA OF ACTUALITY

Chapter 5 reviews increasingly quantum space-times that have been proposed, which guide and inspire the present fully quantum constructions. Chapter 6 constructs queues in general and corresponds them to variables of the standard model and gravity. Chapter 7 specializes fully quantum dynamics to bgravityc, giving a quantum origin for the chronometric...... [080331: Update] Chapter 8 sums up. Chapter 2

Linear praxics of hits and misses, superposition and complementarity.

2.1 Praxics in general

This chapter deals with entire systems. Grade structure is added in Chapter 3 to describe composite systems, and stratum structure is added in Chapter 4 to describe stratified or ranked systems. Boole characterized the classes of a logic by elective actions, a kind of mental filtration, and represented serial action as a product and parallel action as a sum [14]. He left system sources and targets implicit. It suffices to introduce an ideal universal source and universal counter, for then the most general source and target can be made by catenating filters after the source or before the target. Praxics treat of physical filtrations rather than mental elections. The filtrations commute, AB = BA, for classical logics and not for quantum praxics. One critical difference between classical and quantum theories is the quantum superposition principle: a quantum physicist sums complex transition probability amplitudes in situations where classical thinkers like Pascal, Boole, and Markov would sum positive transition probabilities, as in a two-slit experiment. Another well-known difference is that classical filtration operations or predicates all commute, while no non-trivial quantum filtration commutes with all the others, leading to quantum complementarity. The superposition principle leads to a matrx algebra, implying the complementarity principle and more. For example, linear polarizers X, Y , and Q oriented normal to the z axis with polarizing directions along the directions of x, y, and x + y filter photons, and one can see with the naked eye that one order XQY passes some photons, one by one, and another order XYQ stops them, one by one: XYQ = 0 6= XQY . (2.1)

107 108 CHAPTER 2. LINEAR PRAXICS

Classical probability theory deals with both state probabilities (like Boltzmann’s) and transition probabilities (like Markov’s). Quantum probability amplitudes like QHX are always for transitions like X → Q, from an input to an output, never for states of being. Tables of such transition amplitudes are square tables or matrices that almost never commute. That the squares of the vector components αhψ are relative probabilities follows from the quantum law of large numbers. The quantum principle implies the integrity of the system under measurement. When v0 ø v does not hold, the theory does not predict whether the output process v0 counts a system or not. It predicts, however, that the count will be 0 or 1, not a fraction, just as in classical probability theory. The quantum does not divide when the vector does. The quantum principle can be stated succinctly, if cryptically: A quantum praxic is a projective geometry. The elements of the projective geometry are the elements of the Galois lattice of the relation ø (or of the relation Hv2 ◦ v1 = 0) [40]. For example, the line uv determined by two inputs u, v is deﬁned as the set of all inputs w that miss whatever u and v both miss, and corresponds to the plane of the two vectors u, v. Its elements are the quantum superpositions of u and v. It is then empirical whether any two points of a line determine the same line, an axiom of projective geometry. The quantum superposition principle — that distinct input processes have a quantum superposition distinct from both — is then the postulate of projective geometry that there are at least three points on every line, another empirical matter.

2.2 Heisenberg and von Neumann praxics bBohrc and bHeisenbergc introduced non-commutative bpraxicsc when they invented the quantum theory. Even in classical physics, assertions about a physical system are binary variables of the system with the value 0 for “False” and 1 for “True”. The matrix product BA represents doing B after A. In classical mechanics predicates are binary functions on the state space of the system, and commute. In quantum theory they are projection operators and mostly do not commute. Later bvon Neumannc extracted a bquantum logicsc of ∩ (and) and ∪ (or) from the matrix praxic of Heisenberg, using a blatticec of projectors within the matrix algebra. His background uniquely qualiﬁed him for this work. His 1925 doctoral thesis had reconstructed logic and set theory as a theory of mappings rather than propositions. It was quite unwieldy but its functional foundations foreshadowed both non-commutative quantum observables and categorial algebra [77]. Then he had been assistant to David Hilbert, whose program to axiomatize physics still required axioms for quantum mechanics. Thus bvon Neumannc was singularly prepared to invent a logic whose assertions are functions, now from vectors to vectors. In at least one discussion he spoke of a “quantum set theory”, without constructing one. 2.2. HEISENBERG AND VON NEUMANN PRAXICS 109

On the other hand, bvon Neumannc concerned himself less with the operational meaning of quantum theory than with the mathematical structure. His bquantum logicsc were blatticesc with mathematical operations ∪ and ∩ designed to resemble the ∪ and ∩ of earlier logics, not any laboratory operations. ∪ and ∩ have the advantage over + and × of being invariant under unitary transformations of their arguments and so being implementable operations on projectors, at least as limits. But they are further from laboratory operations than Heisenberg’s ×. It takes an infinity of ×’s to make one ∩. They are also structurally unstable. And they are also unwieldy for theorists, no practical physical theory, classical or quantum, has ever been invented in the lattice language. Only the language of linear algebra has been fertile for new quantum theories that work. We use it here. In the language of linear algebra, a predicate is a binary-valued observable, represented by a symmetric matrix obeying the projector condition (2.12), which restricts eigenvalues to 1 and 0, standing for true and false. The physical operation it represents is a filtration . for individual systems with the value P = 1. The logical complement is ¬A := 1 − A. The fundamental dyadic operation is the matrix product BA, representing B after A. Predicates are not closed under this operation however. The praxic algebra is best regarded as the entire matrix algebra, provided with the probability form H. The famous bnon-commutativityc of physical variables is equivalent to the bnon-commutativityc of the predicates, and the filtrations they represent. Quantum praxic is non-commutative logic as quantum geometry is non-commutative geometry. It is related to the usual commutative logic in the way that Galilean relativity is related to special relativity: the older theory is a singular limit of the newer that is an adequate approximation in much of ordinary experience. Matrix addition too can be carried out in the quantum laboratory, though only as a limiting case. The problem was solved in principle when projective geometers of the 19th century showed how to pass from the axioms of synthetic projective geometry, which use ∪ and ∩ only, to those of analytic projective geometry, which concern vectors defined by a sequence of coordinates, numbers in some field, and use + and ×. If a vector has N coordinates, the ray through it has only N − 1. To represent the vector itself by a ray, one uses a ray in an enlarged vector space of N + 1 dimensions [4]. Physically, one may increment the number of possibilities of the quantum system by adjoining a vacuum mode. In an alternative thoretical approach based on the Lie group of the system instead of the predicates, one defines the addition of operators by multiplying operators near the identity in one-parameter groups: d A + B = lim eAeB . (2.2) →0 d The same arguments that had been raised against the relative time and non-commutative boosts of relativity theory were also marshalled against the relative states and non-commutative predicates of quantum theory. The reconstruction of the theory of physical truth has proceeded more slowly than the reconstruction of our theory of physical time, 110 CHAPTER 2. LINEAR PRAXICS even though it had the the earlier reconstruction as a guide and enormous technological advances as a by-product. Kant said that Euclidean geometry is a necessity of thought, given apriori, and so could never be revised. Poincarésaid that Euclidean geometry is a linguistic convention, and so need never be revised. Einstein cut through this fog by talking about two experimenters and their experimental operations in plain language.

Similarly, some said that bclassical logicc is a necessity of thought, so that it can never be revised. Others said that bclassical logicc is a linguistic convention, so that it need never be revised.

bMalusc ignored such philosophical arguments and talked about transition probabilities between two polarizers in plain language, although the epochal significance of the Malus Relation was not realized until long later. First bPlanckc, bHertzc, bEinsteinc, and bComptonc had to re-establish the existence of the photon that Malus had accepted on the authority of Newton, even though Young had just demonstrated the interference fringes of light. Nor can classical logic be an absolute necessity of human reasoning. If it develops that we are biologically constrained to prefer bclassical logicc, as is plausible on Darwinian grounds, we nevertheless communicate our measurements with symbols, and so we can reason about them with classical symbolic logic, without straining our brains more than by learning matrix algebra. In this way we use bquantum praxicsc for systems by using bclassical logicsc in the metasystems; just as we use Einstein relativity for fast electrons and Galileo relativity for the electron accelerator. This is not a contradiction, only a useful approximation. Nor can a physically useful logic be entirely conventional. To be sure, there is much convention in any language, considering all the words that could be invented and are not. But in practice the domain of discourse imposed important constraints too. For example, once we establish a language to represent physical filtration operations, and define the product AB = A after B among these filtrations, whatever we call them, the non- commutativity AB 6= BA is learned by experiment, not imposed by convention. One could say that only polarizations that commute are predicates, thereby resurrecting classical logic by casting out quantum superpositions, but this would break the experimental symmetry between the X and Y of one experimenter and those of another, and so violate rotational relativity. Filtration non-commutativity is about as factual as sunrise whatever we call it. Similarly for the non-distributivity of the ∩ and ∪ of quantum praxics, which represent much the same facts about polarizers as the non-commutativity of after. It seems clear to many, including founders of the quantum theory, that the problems of present-day quantum physics, especially the infinities, call for a post-quantum physics that departs even further from classical physics in its praxical structure than the theories of Bohr, Heisenberg, and von Neumann, and is less commutative. This is the directive to which the present work responds. 2.2. HEISENBERG AND VON NEUMANN PRAXICS 111

2.2.1 The system itself Quantum theory is an exercise in humility. It is the ﬁrst mathematical physical theory to accept that the entities of nature have no mathematical physical model. As Bohr put it, quantum theory is not about nature, it is about what we can say about nature. Put in more detail, quantum theory does not model quanta mathematically, it models processes performed by huge aggregates of quanta like ourselves that create, transmit, or register quanta without determining all their properties. Mathematical modeling sometimes becomes possible for quanta organized in masses even though it is impossible for the individual. The minimum condition for a mathematical model is that the physical entities should be distinguishable when their symbols are. If A and B are symbols, it is assumed that the transition probability for writing an A and reading it as a B is 0 or 1:

BhA = δBA. (2.3) On the other hand if b < and ha are processes for “reading“ and “writing“ a quantum, like polarizing and analyzing processes, then they can be represented by vectors so that the transition probability is |bha|2 = cos2 θ, (2.4) which ranges continuously between 0 and 1. It is easy to accomplish the yes-or-no transmission probability for symbols and impossible for quanta. On the other hand, there is no physical obstacle to distinguishing quantum vectors bh and ha, and so they can be mathematically modeled in quantum theory. Just as one can explicitly distinguish between a numerical variable and any one of its values, an individual quantum system can be distinguished from any one of its states or actions. This rather nondescript kind of quantum entity admits a mathematical model. In probability theory, the entire state space S, not one of its points, serves as a mathematical model for the random variable, the individual system itself, abstracted from its contingent features. This is the conventional zero-point of information about the system, or the predicate of mere existence of the system, without further speciﬁcation. The unnormalized constant probability distribution function, the bunit functionc

1S : S → {1},S 3 v 7→ 1, (2.5) represents mathematically the mere existence of the system. 1S is also used here to represent the individual system with state space S. Similarly a quantum individual is the general case where the special case is what is created or destroyed by the process represented by any one binputc or output ray in an associated vector space. The entire binputc vector space V , or the associated ray space, not one of the vectors or rays, can be used as a mathematical model of the hypothetical system itself, the individual that the system counter counts, abstracted from its contingent features. Therefore the unit operator on V , which is the projector

1V : V → V, v 7→ v, (2.6) 112 CHAPTER 2. LINEAR PRAXICS can also be used as a representative of the quantum individual (system) I[V ]. The operator 1V corresponds to multiplication by the unit function 1S in the classical theory. The quantum individual can be modeled in this way because the model says so little about the quantum. Example: Since 2 := {0, 1}, I[2] is the binary random variable, with values 0 and 1; 2 is a two dimensional space with basis 2 = {0, 1}; and I[2] is a hypothetical quantum with vector space 2.

The bmultiplicityc of a random system I [S] counts mutually exclusive possibilities for the individual, and is the cardinality of its sample space: Mult I [S] = Card S.

The bmultiplicityc Mult I[V ] of a quantum individual I[V ] is the dimensionality Dim V of the binputc vector space of the individual and the number of mutually exclusive, together exhaustive, possibilities for the individual. D The bdual systemc I[V ] to the system I[V ] is the hypothetical system whose input D vector space is the output vector space V of the system IV . Therefore an binputc of a I[V ] is an output of a I[V ]D. C The banti-systemc I[V ] is the dual of a system of the negative energy, deﬁned more fully in §6.4.2.

2.2.2 The equatorial bulge in Hilbert space

∞ Almost all the area of the unit sphere S in Hilbert space H = ∞ · C is at its equator. Almost none of the area of S∞, proportionally speaking, is near its North Pole. More exactly put: Let δθ > 0 be any fixed angle, no matter how small, As the Hilbert space dimension D → ∞, the area of the belt about the equator of SD−1 of width δθ approaches the area of the entire unit sphere, while the equator of a zone of width δθ about the North Pole approaches zero in relation to the total area. The probability of finding a random vector within δθ of the equator approaches 1 as D → ∞. This geometrical phenomenon is the “equatorial bulge” in Hilbert space. If we combine N systems of multiplicity M, we need a vector space V of dimension D = M N to describe the composite. In any vector space of high enough dimension, however, almost every pair of vectors is as close to orthogonal as desired, due to the equatorial bulge in Hilbert space, and their projectors, together with all the variables that can be formed from them, are as close to commuting as desired. Therefore two directions chosen at random in infinite-dimensional Hilbert space are almost always arbitrarily close to being orthogonal; their projectors are almost always arbitrarily close to commuting, and as predicates are almost always arbitrarily close to obeying Boolean logical laws. It is highly improbable to see noncommutativity among typical variables of macroscopic systems. Classical commutativity emerges for sufficiently complex quantum systems as a result of this equatorial bulge in Hilbert space. 2.3. STANDARD SEMANTICS 113

2.2.3 Commutative reduction The process that converts a quantum system I[V ] into a random object I [S] whose state space S is the set of rays of the vectors of some orthonormal basis B ⊂ V is called bcommutative reductionc here. The predicates of a bcommutative reductionc are all represented by diagonal matrices with respect to the chosen basis B for the quantum system. The operator methods of quantum physics also apply to classical physics in a singular limit. One can reduce a quantum system to a classical one by reducing the operator algebra A of the system to a Cartan (maximal commutative) subalgebra Ac ⊂ A. This is a commutative reduction. Classical logics are diagonal parts of quantum praxics. Changes being oﬀ-diagonal operators, it is paradoxical that change still occurs in the classical limit, where all variables are diagonal. The well-known resolution of this paradox is discussed in §2.4.

2.3 Standard semantics

By the semantics of a physical theory is meant here mean the two-way connection between its operators and laboratory actions. Almost all working physicists use the quantum semantics presented in the rest of this section. It includes a correspondence principle: It reduces to the classical semantics under a commutative reduction of the quantum theory. By a “bcompletec representation” of a system is meant here one from which answers to all meaningful experimental questions about the system can be computed. This usage is consistent with the logician’s, and in this sense quantum theory and number theory are both incomplete, one physically and the other mathematically, though in different ways and for quite different reasons. Bohr called a quantum representation “complete”, however, to express something else: that even though a quantum representation of a system leaves most questions about the system undecided, merely assigning probabilities to the possibilities, it does not follow from a complete representation of some possibly larger system by deleting some information. Such representations of a system are called “maximal” by von Neumann and here. The term “maximal” too should be used carefully. A quantum theory often gives birth to a more informative theory when new degrees of freedom are discovered in nature, and then both mother and daughter can have maximal representations. The ontological quantum mispresentation leads to misunderstandings of the dynamical process. Some infer from it that the quantum correspondent of Newton’s equations, or of their Hamiltonian form, is the bSchrödingerc equation. Actually it is the Heisenberg equation; the Schrödingerequation is the correspondent of the Hamilton-Jacobi equation, not Newton’s. The system changes in time, its vectors represent actions that we simply do or not. They are not measured, they measure. To speak of them changing or not changing with time during the development of the system confuses the system with an input-output operation for the system, a photon spin for a polarizer, a product for the 114 CHAPTER 2. LINEAR PRAXICS process. When a vector ht is understood to be a process carried out to produce a system, the equation ht0 = h∆tht does not mean that ht is changing, it means that as far as any later output operation oh is concerned, the process ht is indistinuishable from the later process ht0 = h∆tht, the predicted transition probability amplitude being oht0 = oh∆tht, where ∆t = t0 − t labels the dynamical transformation h∆th. Similarly the unitary operator h∆th is not what actually happens to the system nor what the system actually does. Its matrix elements are transition probability amplitudes for what might happen, not what does happen, but nevertheless determining the probabilities for all possibilities. It is a coherent quantum analogue of the table of transition probabilities of a Markov process, which is also not what actually happens but only a statistical description. What actually happens is not completely describable in quantum theory, any more than what actually is. Nevertheless it works.

2.3.1 The orthogonal group is used to projectively represent the kinematical group of I[V ], the group of the reversible transformations of the system, including both coordinate transformations, which act in the bmetasystemc alone, and dynamical transformations, which are actions on the system. Example: Passing a photon of monochromatic visible light through a cell of sugar water eﬀects a dynamical transformation of its spin, a rotation through an angle θ about the beam axis. Rotating the binputc polarizer by −θ eﬀects a coordinate transformation represented by the same operator. The identity operator I[V ] on V represents the quantum system better than V itself, since I[V ] = I[V D], simply acting from the left or right on the two spaces.

2.3.2 Probability vectors The contraction [o < i] is a relative transition probability amplitude. This is expressed by

Assumption 7 Malus-Born Probability Principle There is a bprobability formc | > hh on probability vectors deﬁning the probabilities for the transitions oh × hi by

ohi iho P = , where ih = iiHh, ho = hh−1 io. (2.7) oho ihi

D Example: The two-dimensional space 2R and its dual 2R are used to represent binputc and boutputc (processes) for photon spins using linearly polarizing filters at the binputc and output ends of the experiment. If the filters are oriented along vectors hi ∈ 2 and oh ∈ 2D 2 then (2.7) becomes the bMalus Relationc P = cos θ. In classical thought an input-output process of maximal resolution defines a state of the system. If Alf prepares one system in a state s, then Bea can determine that state in 2.3. STANDARD SEMANTICS 115 a single measurement on that same system. The state is observable in that sense. A state can be determined from a single system. A vector that inputs a system cannot be found from that system, which it describes only statistically. If Alf inputs one system with a vector hi then Bea can never determine hi, or its ray, by any measurements on that system whatever. In order for Bea to learn hi with arbitrary precision, Alf must supply arbitrarily many systems with vector hi. In this regard a vector is like a probability distribution function, not a classical state. The classical correspondent of quantum theory is not classical mechanics but statistical mechanics [64]. Variables of the system that may be observable in a single measurement are represented by operators, not vectors, with the possible values as eigenvalues. The classical state is a variable of the classical system. One classical system has a state just as it has (say) a position or a momentum; one quantum system does not. The term ‘bstate vectorc seems to facilitate a misunderstanding of the quantum theory on this point. It leads the unwary to suppose that every quantum system has a state vector as every classical system has a stste, an error of level. Heisenberg’s term “probability function” avoids this reification but also omits the quantum aspect: the components of the vector are probability amplitudes, not probabilities. Dirac’s terms “bra” and “ket” remind us appropriately of how these vectors figure in expectation values, designated with brackets by Dirac; unfortunately expectation values here have the non-bracket form ψhQhψ, so the bra-ket terminology is no longer a helpful mnemonic. On the other hand, there are quantum constructs that can well be called states without violating the correspondence principle (§2.9). A probability vector has a dual role, with both a macroscopic and a microscopic aspect. On the one hand its ray may be a macroscopic state variable of the experimental apparatus, as the polarizing angle is for a polarizer. On the other it represents statistical information about the quantum system. This double role is familiar from thermodynamics. The piston position for a steam engine also has two such roles: it is a macroscopic variable of the engine and a parameter in the distribution function of the steam molecules in the cylinder of the engine. A major difference is that the piston’s position does not give maximal information about a water molecule, but the polarizer’s angular position gives maximal information about the photon polarization. It is the custom in physics to keep the pre-evolutionary terminology as long as it can be made to work. We still speak of the time of an event in special and general relativity, knowing that a reference frame must be specified to give this term meaning; in principle we can do the same with state in quantum theory, but it has not worked as well. The evolution from classical to quantum physics required a relativization of state that seems deeper than the relativization of time and simultaneity in special relativity. The quantum theory not only relativizes but also dualizes the classical concept of state. Each quantum experimenter defines and is defined by a class of input actions and a dual class of output actions, proper to the experimenter, for which classical logic works. The probability vector is relative to 116 CHAPTER 2. LINEAR PRAXICS the experimenter in that the class of accessible actions varies with the experimenter, and a probability vector is determined by examining the experimenter, not the system. It is dualized in that quantum theory has both input and output vectors, used in mutually dual ways. In a general experiment one experimenter carries out the input process and another the output process; the experiment is a quantum communication channel. Vectors of V and V D represent the actions chosen by the two experimenters, not state variables of the system, which are represented by operators.

2.3.3 The probability form A probability vector space V requires a symmetric form iHh : V ↔ V D to represent assured transitions and system similarity. iHh then also induces experiment transposal, transposing the order of all operations and interconverting input and output processes. H is also the form used to compute transition probabilities. The transition from an input hi to an output oh is bassuredc, has transition probability 1, if oh = iiHh. A form H with these physical interpretations is called the bprobability formc. It is, to be sure, completely determined by the inclusion and occlusion relations for the system input-output processes, which mention probability 0 or 1 only; but these relations are still determined by statistical sampling. Queues require a probability form for each stratum. It follows that if h i h i oh := ho †, hi := ih †. (2.8) then ohi = iho. (2.9) (In a complex theory a complex conjugation C is required on one side.)

2.3.4 The linear operators The dynamical development from one time t to another t0 deﬁnes an isometry ht0, th in quantum mechanics as in classical. In a relativistic theory this extends to a representation of Poincar´etransformations T ∈ ISO(3, 1) by isometries hT h and one postulates invariance under ISO(3, 1). This postulate is structurally unstable and implies that time is not simple. Since it has worked rather well it is not dropped here but changed slightly, perhaps undetectably, just enough to make its group simple and structurally stable. It is possible to hide the coordinates and work with invariants. Then instead of vectors one speaks of points of a bprojective geometryc, which correspond to rays of vectors, and instead of operators one speak of projective transformations, as in perspective theory. Vectors and matrices are more powerful tools for theory construction, however. Every useful physical quantum theory was discovered in the language of matrices or linear operators; none in the language of projective geometry or lattice algebra. 2.3. STANDARD SEMANTICS 117

Spectral linear operators on V → V are used to projectively represent dynamical variables, ideal multichannel sortations of the system by the bmetasystemc. Linear operators are also used as inﬁnitesimal generators of one-parameter groups of system transformations. Sometimes one treats matrices themselves as vectors in a higher-dimensional vector space. The matrix hAh can be written with both brackets on one side, representing a composite index: h hAh ≡ A i. (2.10) The contraction of two such vectors is the trace of the operator product:

h Tr hAhBh = A iB. (2.11)

2.3.5 The projectors A predicate or class is deﬁned as Boole did, by a ﬁltration of the system under study. Projection operators, or projectors, of V , the idempotent symmetric operators P with

P = P 2 = P †, (2.12) represent such ﬁltration operations on I[V ]. The lattice they form corresponds to the bclassc or bpredicatec logic of a classical system. They are partially ordered by the inclusion relation P1 ≤ P2 := P1P2P1 = P1. (2.13)

The supremum P1 ∨ P2 and inﬁmum P1 ∧ P2 of projectors in this partial order represents the Boolean operations P1 or P2 and P1 and P2 on predicates. The product operation P2P1 represents an and then operation; the predicate algebra is not closed under this operation. If P1, P2, and P3 are projectors then P3 is called a bsuperpositionc of P1 and P2 if for all projectors P ,

P1P = 0 and P2P = 0 : implies P3P = 0. (2.14)

The above list of standard usages is redundant. With reasonable deﬁnitions, the usage of §2.3.1 probably entails all the others. It excludes superselection (centralization) laws and bcompoundc — non-semisimple — Lie groups; but these still arise as singular limits. The semantics of the quantum predicate algebra can be considered known. Its commutative reduction to the classical theory requires a special limit (§2.4). The system under study is here a set, either c (sea) or q (queue). The bracing operator I on probability vectors, together with addition and multiplication, results in a bquantum set theoryc, or bqueuec theory, in that the resulting theory has classical set theory as a D commutative reduction. While some combinations of I and I are given empirical meaning, further clariﬁcation of their meaning is still needed. 118 CHAPTER 2. LINEAR PRAXICS

2.4 Change is a quantum eﬀect

The commutative reduction of quantum predicate algebra is classical predicate algebra but that of quantum kinematics. is not classical kinematics. A quantum theory uses the same non-commutative algebra to house its projectors and its dynamical processes. Classical theory is somewhat weird in that its predicates commute and so all its variables commute yet its dynamical processes do not. Diagonal operators generate no change in diagonal operators. There should be no dynamical evolution after a commutative reduction to classical physics, because the the rate of change

dQ i = [H,Q] (2.15) ~ dt is a commutator and all classical variables commute. Motion remains in the classical limit because the limit is singular. The time-development operator U(∆t) = e−iHt/~ (2.16) involves 1/~. This ampliﬁes the eﬀect of the dwindling commutator so that change can −1 still occur as ~ → 0. The commutators [q, p] → 0 vanish but −i~ [q, p] does not; instead it approaches the Poisson Bracket. All change, according to canonical quantum theory, is a vestige of quantum change that survives the singular classical limit. The split between observing and acting is an artifact of the classical limit and does not occur at the quantum level of resolution.

2.5 Simple systems

By a bsimple systemc is meant one with a simple kinematical Lie group. Nowadays one recognizes several simple systems, such as the spin, isospin, and color of a quantum. A simple kinematics is quantum, without superselection laws. No classical space-time is simple, since its coordinates are central. By the simplicity strategy Cartan’s Table of the Simple Lie Groups is also the Table of Simple Systems. Here the possibility is explored that actual systems have SO(n; σ) kinematical groups, in the D family of the table. The usual quantum theory represents the dynamical development as a one-parameter group of isometries with time as the parameter, applying the first construct in the above list. In bqc theory, however, time enters more deeply, as one of the coordinates of the space-time events to which the field is attached. The continuity of time leads to an infinite density of field variables and to all the important infinities in present physical theory. Usually the event of space-time is treated as a classical random object E with state space S[E] composed of space-time events. Here the event is treated as a physical system in its own right, with a vector space S[E], though each event is probably strongly entangled 2.6. PROBABILITIES 119 with other events, and the fully quantum strategy is applied (Assumption ??): The generic event is represented with a simple vector space V [E] = Grade1 S[E]. A coordinate of the event I[SE] is then a normal operator in the operator algebra Alg(V [E]). A quantum coordinate transformation is an automorphism of this algebra, which is inner, generated by an operator in SO[E]. Then a fully quantum time cannot have a continuous spectrum and cannot be the parameter of a Lie group. It is an operator on the vector space V E associated with the event, with a discrete bounded spectrum. This time construct, like most quantum constructs, is relative to the partition into system and metasystem. Since i generates a radical, on the grounds of simplicity we use a real quantum theory here in the sense of bStückelbergc, who also showed how to reconstruct a complex quantum theory within the real one [72]. It is likely that Stückelberg’s study of real quantum theory influenced his early theory of the massive vector boson [?], as it did later work [?].

2.6 Probabilities

Any variable of a system is represented by an operator on a vector space for the system. To input a photon (say) through some chosen polarizer is an input action hi. To output a photon through some chosen analyzer is an output action oh. The input and output actions can be chosen independently by the experimenter at each end of the optical bench. There is no question of one developing or collapsing into the other; that idea is a vestige of the non-quantum theories of de Broglie and Schr¨odinger. Then the Malus-Born Principle of Assumption 7 and equation (2.7) gives the transition probability. The transition probability is invariant under the group SO(V ) that respects H.

2.7 Mixtures

Each bvectorc and its negative represent the same input process but give diﬀerent results in superpositions. Therefore quantum superposition does not represent a physical operation on input processes alone. The invariant operational content of the phase of the vector and of quantum superposition is discussed elsewhere, for example in [40]. The process of random bmixturec of the inputs from two separate input processes is physical, though it loses information; It is also called bincoherent superpositionc. In contrast, vector addition is then called bcoherent superpositionc. To represent mixing one associates with each vector hi an input bprobability tensorc (“bdensity matrixc”) and a dual output bprobability tensorc

hi ⊗ ii hI i := , iOh := io ⊗ oh. (2.17) iii

The common term “bdensity matrixc” is rooted in ontology. The ontologist reads its eigenvalues as material densities, although they are actually probabilities. Such mispresen- 120 CHAPTER 2. LINEAR PRAXICS tations of physical interpretation are avoided here. The rule in quantum theory is to name operators after the physical meaning of their eigenvalues. The diagonal elements of one of these tensors are probabilities, so it is properly termed a bprobability operatorc. Input processes represented by vectors hi are called bsharpc or coherent. Those represented by a projector of any dimension are called bcrispc. The most general input process is called bdiﬀusec and is represented by a more general input bprobability tensorc hI i =: hhI, operationally deﬁned by the condition that the transition probability for an output vector oh is P = ohI io (2.18)

i Output probability tensors iOh = O h are deﬁned dually. The btransition probabilityc when both input and output processes are mixtures is

P = Tr iOhI i. (2.19)

The definitions have been chosen so that the bdualityc H is not brought in until it is needed. The empirically forbidden transitions alone, with ohi = 0, determine the vector space structure, including the dimension and the ring of coefficients. The empirically assured transitions then determine the bprobability formc bhHh c so that if oh = hHhi, then the transition ohi happens in every trial. A more common convention converts these second-grade symmetric tensors hI i and iOh to linear operators hIh and hOh by appropriate factors of H = iHh. Then there is no difference in appearance between the input and output process symbols, and for a time people spoke ontologically of “the” bdensity matrixc when there are two in every experiment. The explicitly operational analysis of bGilesc restored this duality [45]. Then bsuperpositionc adds vectors, ha+hb = hc, while bmixingc or bincoherent superpositionc adds probability tensors, iAh + iBh = iCh.

2.8 Transformations

A general bexperimentc can be idealized as a three-stage transaction oh ×hth ×hi consisting of input, throughput, and output (or emission, transmission, and admission) of a system. This usually employs two experimenters, one at each end of the experiment, each with an experimental frame that includes the action performed. The expression

A = ohthi (2.20) for the btransition probability amplitudec is also a diagram of the experiment. If both oh and hi have unit norm then A2 is the transition probability. Sharp experiments require a bmetasystemc much more complex than the system studied. Merely recording the outcomes requires the bmetasystemc to have a separate subsystem for each possible system input process; for example, a photosensor for each channel of a 2.9. STATES, PROPER AND COORDINATE 121 spectroscope. Even if these parts are merely binary, the number N of independent possibilities for the bmetasystemc must then be at least exponential in the number n of possibilities for the system: > N ∼ 2n. (2.21) Determining all the properties of a quantum system together is impossible, according to the quantum theory, since determining one property is found to undetermine others, in the way described by quantum bnon-commutativityc. This makes the quantum representation of experiment bincompletec. It is still possible to extract statistical information about practically all the properties of practically all the systems from a given source of statistically independent systems, however, by determining the mean of each property on a small but not too small sample of all the packets from the source. Operations on an individual do not change the averages over such an ensemble by much.

2.9 States, proper and coordinate

The bstatec of a classical particle moving on a line is a point in a phase space with a coordinate pair (q, p) of a position variable q and a momentum variable p. It provides a complete description of the particle for the purposes of mechanics. The classical state (q, p) has a quantum correspondent, the Manin state (q, p) [56, 57]. The Manin state clearly satisfies the bBohrc bcorrespondencec principle. Since q and p do not commute, the Manin state is not observable. In classical physics it is common to use the same name for a variable and one of its values. For example, one can say that the momentum is p or that the momentum is 22 kg.m/sec. The former declares a variable, the latter gives a value. At least four kinds of thing can be called the bstatec of a classical particle on a line without solecism: the pair of variables (q, p), a pair of values (q0, p0), the variable point of phase space with coordinates (q, p), and the fixed point of phase space with coordinate values (q0, p0). Generally the correspondence principle is used to fix our quantum terminology, but in quantum theory the variable and the value have such different commutation relations that it is helpful to distinguish them more clearly than classical usage. The classical pair (q, p) is both a maximal set of independent variables and a maximal set of commuting independent variables. In quantum theory, (q, p) is a maximal set of independent variables while q by itself is a maximal set of commuting independent variables. (q, p) is covariant under canonical transformations but not observable; q is observable but not covariant. Which shall be called the quantum bstatec? One condition that is imposed here is that it is something that can be determined from the individual system, or is a collection of such things. A vector is not such an entity; it is found either by a single inspection of the metasystem — a polarizer, for example — or from statistical studies of many experiments on like systems. 122 CHAPTER 2. LINEAR PRAXICS

This recalls the relativistic choice between proper time, which is invariant but not integrable, and coordinate time, which is integrable but not invariant. Indeed, the proper time is not an observable coordinate of the particle, but of a history segment. Both time constructs are useful and are used. This relativistic policy is adopted here for the following two state constructs also:

2.9.0.1 Proper state

A bproper statec of a quantum system is a maximal set of independent generating variables. Example: (q, p) is a bproper statec. This is the bManinc concept of bstatec. It is bcompletec, in the sense that its bcommutantc ((6.59)) is trivial, but it is not observable, since its variables (q and p in the example) do not commute.

2.9.0.2 Coordinate state

A bcoordinate statec of a quantum system is a maximal set of independent commuting generating variables. Example: q is a bcoordinate statec of the same system. A quantum bcoordinate statec is an observable like the classical state; but it is not complete or unique. It defines a frame, and is defined relative to that frame. Notice the trap that awaits one who calls the vector a state vector: A property of a system is a variable whose different values correspond to disjoint sets in classical phase space, or orthogonal subspaces in quantum theory. The state is a property of the classical system, whose values correspond to distinct points of phase space. If the state vector were mistaken for a property of the quantum system, as the term suggests, then its different values would correspond to orthogonal vectors of the vector space. This would imply that all vectors that are not parallel are orthogonal; which actually holds for classical systems in a vector description.

2.10 Praxiology and its singular limit bHeisenbergc called quantum theory “non-objective physics” presumably because his operator formulation worked not with nouns standing for objects but with verbs standing for actions, especially on quanta. “Action physics” would also have been a reasonable term, but action painting was still two decades in the future. A theory of what exists is 1 called an ontology, a theory of physical actions can be called a bpraxiologyc. Then the

1Not to be confused with praxeology, a general theory of human actions from the viewpoint of economics [76]. The praxeology of von Mises has elements in common with the present concept, but diﬀers substan- tially. For example, it explicitly assumes that we have apriori knowledge and in particular that only one logic is possible for the human mind, and it does not undertake to analyze objects into actions. Praxiology seems consistent with the pragmatism of Peirce and the process philosophies of Whitehead [?] to the extent 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 123

Bohr-Heisenberg theory is bpraxiologicalc, not ontological. Since the state space of every object has a transformation group and semigroup, which characterize it and can be used to represent operations on the object, every ontology implies a praxiology, but one of a special kind, in which the sorting actions associated with observable properties of the system under study commute, as Boole noted [14]. Almost no praxiologies in the general sense derive from ontologies in this way. In generic terms: An ontology is a singular limit of a praxiology. The proposed fully quantum theory is praxiological in that it does not assume absolute objects as ontological theories do. There is no reason to suppose that this theory is the end of the line, however. It gives its actions an absolute identity in that it allows one to speak of the same actions acting in a different order, but the empty set associated with the vector 1 may be relatively rather than absolutely empty, and all other vectors are defined relative to 1. Some of the founders of the present quantum field theory spoke in operational terms, praxiologically, and others attempted to interpret the theory ontologically. Much of the language in current use was formulated and promulgated by ontologists. As a result, agreement is excellent about how to use the quantum theory in the laboratory, but not about how to present it. Sometimes such presentations are called interpretations of quantum theory, as if quantum theory did not already have one. By the interpretation of a theory we mean rules about how to use the theory. But a physical theory less its interpretation is not a physical theory but a formalism. Quantum theory already has its interpretation, discussed in §2.3. Bohr and Heisenberg do not posit systems with complete representations but concern themselves with operations on systems. A presentation that accurately describes their quantum theory is necessarily praxiological. An ontological mispresentation strives to fit such theories into the classical state strategy, which contradicts them. Today the name “Copenhagen interpretation” is often used for various mispresentations of the quantum theory that attempt a classical ontology, avoiding the quantum concept of physical entities that have no complete mathematical representations. These mispresentations attempt to remain within the domain of classical logic and probability theory, while quantum theories modify the predicate algebra and the probability algebra of the system. Ontological mispresentations all share one tactic: A construct that is known only statistically, from many experiments, and is used to compute probabilities of future experiments, is nevertheless given ontological status, and said to be physically present in each experiment. Being statistical, the chosen construct is not appreciably affected by its determination, which can be carried out on a small subsample, and so it can safely be treated as a classical object. On the other hand, the statistical construct is no more present in the that these focus on processes rather than their products, but is applied here just to the interpretation of quantum theory. 124 CHAPTER 2. LINEAR PRAXICS individual case than a probability. Furthermore, the attempted ontological theories never close, in that they do not explain how an object that is alleged to be physically present in the individual experiment determines probabilities as if it were a probability distribution across many experiments. Some of the terms that are found in ontological mispresentations are “the vector of the system”, “the collapse of the vector”, “many worlds”, “the vector of the universe”, and the “quantum potential”. The purpose of this section is to prepare students for the mutually inconsistent presentations of quantum theory found today. This section is not necessary for the formal deductions in this work, nor for those already aware of the inconsistencies in current usage. The earliest such ontologization is incorporated in the terms “wave function” and “state vector”. Heisenberg called such vectors “probability functions”, not state vectors, and properly they are probability amplitude functions, but these names are unwieldy. The term “vector” used here makes it plain that the meaning intended is statistical, not ontological. Some say that every individual system has a state vector, just as every individual system has a state according to classical physics. They imagine a random variable I [V ], the wave function, which ranges over a vector space V . The praxiological presentation, on the contrary, concerns a quantum system I[V ] (both defined in §2.2.1). A vector of V represents a quantum process and I[V ] designates the variable quantum produced in such processes; ontologists confuse the process and the product. Both praxiological presentation and ontological mispresentation accompany the same statistical practice, but one presents the practice accurately and praxiologically, and the other reifies probability distributions. This discrepancy does not affect the application of quantum theory in atomic physics, which mainly ignores the ontological presentation and carries out the praxiological one, but it affects the research strategy, and hinders attempts to extrapolate quantum theory into new domains. Ontological mispresentations also lead many to see mere consequences of this mispresentation as difficulties of the quantum theory. The ontological mispresentation of the non-commutative quantum theory is so absurd that it makes it seem that we must have gone astray, and must go back and start over. This diverts valuable research resources from the well-known genuine problems of quantum theories, and spreads public misinformation about nature. The purpose here is not to enshrine the canonical quantum theory but to go forward into physics that is even more quantum rather than less. The term “Copenhagen quantum theory” seems mired in ontological mispresentation, and permanently associated with a concept of wave-function collapse alien to what was taught in Copenhagen. Its use is therefore avoided here. One may speak instead of the canonical quantum theory; and of its praxiological presentations and ontological mispresentations; and of the fully quantum theory worked out here. Ontological theories proceed as if the theorist has a representation of what things really are, perhaps by seeing them as they really are, perhaps by gnosis. Leibniz gave a source of 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 125 such ontological knowledge explicitly in his monadology: Monads are informed about each other and the universe directly by God. While the canonical theory is praxiological, ontological presentations continue to suppose the main purpose of science is ontological, to mirror nature, to represent the world and its history and relations by sets as faithfully as possible. Even quantum theories allow statistical distributions or collective variables of macroscopic apparatus to be completely represented, at least in a useful limit. Ontologism assigns to these statistical constructs an ontological status that they do not actually have, so as to limit discussion to constructs that have complete descriptions. Ontologism still allows relativity, different observers can make different complete representations of the same object, but it assumes that there is a 1-1 transformation relating these representations, and that representations can be complete, answering all well-formed experimental questions about the object or its history. The common method is to imagine that the statistical construct is present in the individual experiment, ignoring how it is actually determined experimentally. A photon cannot be completely represented by symbols but a photon vector can, for example, so an ontologist might teach that “the vector of a photon” is the state of the system, or is even the system itself. When ontologism effectively mispresents every system as a random object of the form I[S] for some state space S, it represents every transformation of the system as a permutation of the points of S. It tells us what “is”: a point of S. The quotation marks are to remind us that this existence is supposed to include unobserved existence. Presum- ably ontologists think that knowing what “is” can help us to understand and control our experiences with nature, but actually a statement of what “is” is useless for predicting experience unless it is backed up by theories of perception and dynamics, relating what “is” to to what we experience. From an operational viewpoint, the assertion that something exists is meaningless without such a context. The context may be a naive theory of perception, claiming that what we see is what there is. Since the process of perception was often unmentioned in the past, some may have believed that somehow we see things as they “really are”, as if by a form of mystical gnosis. It is not easy to believe this today. A typical one-mol-kg object has on the order of 1024 molecules. The eye, however, has only 108 receptors, so we have access to but a tiny fraction of the properties of such an object; the rest is missing but not missed. For celestial bodies our ignorance is even greater. The construct of complete knowledge is useful when a high organization among all these variables leads to a few collective variables whose future behavior can be usefully predicted from their present value, leaving the vast majority of variables relatively unknown but permitting effectively complete knowledge about the collective variables. The success of macroscopic ontology, furthermore, tells us nothing about the validity of some submicroscopic ontology. We know few of the variables of an “object” system, and we also change some of them as we perceive others. The argument is based on a principle already known to bBerkeleyc: There is no fundamental physical difference between 126 CHAPTER 2. LINEAR PRAXICS perception and other physical interactions. Light reaching us from a system causes changes in us, since we perceive it. The common sense that action accompanies reaction implies that there must be comparable reactions in the object from which the light comes. Because we see so little of the system under study the changes produced by light leaving it are usually ignored. The changes occur in relatively few variables, leaving the collective variables effectively constant. For sufficiently small systems, the total number of variables is small enough that changes in them cannot be neglected. Most of the cosmos is outside the system under study and almost unobserved. The effects of its many active elements on the system is small but certainly exists and appears as random noise. To be viable, a theory must not be too unstable against such noise. As a result of all these effects, the idea of an exact complete description of a system is not useful for quantum physics. As bNicolas of Cusac taught, we can know mathematical objects exactly because we make them, but not physical ones because we do not. The quantum strategy takes the main purpose of science to be praxiological or functional knowledge as opposed to ontological knowledge; know-how as opposed to know-what. Each quantum theory starts from a repertory of actions represented by operators, and de- duces their experimental relations from their algebraic relations. Quantum physics omits the hypothesis of the unobserved object and the state space, and cuts directly to the processes that they were invented to handle. In doing so, Bohr and Heisenberg intentionally applied the practice that Newton and Einstein promulgated: Omit the unobservable. Quantum theories represent a quantum system by a class of acts that we can carry out on it, rather than a class of states of being, unobservable by postulation. Namely — to repeat D — it represents each system as an I[V ] with a bvectorc space V , a dual vector space V , and an operator algebra V ⊗ V D. Its presentation is the minimum required to express the quantum semantics (§2.3): The probability amplitude for the experiment consisting of the actions oh × hth × hi is ohthi. This principle is not an addition to the classical theory but a viable descendant of it, it actually holds in the classical theory, only superpositions are forbidden in the classical theory, so interference terms are missing, and the probabilities add as well as the probability amplitudes. It is therefore not necessary to explain this probability amplitude formula in the quantum theory any more than one explains the corresponding probability formula in the classical; we must only explain why classical theory omitted superposition, and this is clearly a matter of large numbers, an economic limitation. The quantum theory does not describe our experience with the system by a state in a system state space and a naive theory of perception, but by an operator in a system algebra, representing the entire process, including the perception and recording of the result. Since this distinction between the praxiological and ontological is often understated, it is exaggerated in the next two sections, leaving out many moderating overlaps and variations. 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 127

bNewtonc (in Query 34 in one edition of his Opticks) supposed that light consisted of a corpuscle stream accompanied by transverse waves. The companion waves were supposed to influence optical surfaces and gave rise to apparently random “fits and starts” of transmission or reflection of the photon at these surfaces. He postulated a crystalline ether to provide a medium to carry these transverse companion waves for the photon. Since he recognized both wave and particle phenomena in beams of light, I think it fair to regard him as the initiator of quantum mechanics as well as classical mechanics, though bMalusc did not discover the Malus Relation, the prototype for the bBornc probability law basic to quantum theory, until about 1805. When Newton introduced force — a push or a pull – into natural law, he was proceeding praxiologically. When he assumed that a light beam consists both of photons and guide waves, however, he was still ontological. Malus did not model the photon itself but only polarizers, and discussed only the transition probabilities between polarizers, not the trajectory of a polarization. Boole explicitly based bclassical logicc on an algebra of human actions of mental selection from a population. He postulated that these actions commute, but declared at the same time that this evidently praxiological postulate was an empirical conclusion, and envisaged with some drama a logic of a more general kind [14]. This puts Malus and Boole at the head of a procession of praxiological quantum theorists that includes bBohrc, bHeisenbergc, bPaulic, bFeynmanc, and bSchwingerc. Both parades can validly claim to follow Newton. Quantum theory was invented twice, some say, once by Bohr and Heisenberg, and again by bSchrödingerc. This version of history is part of the problem today. It obscures vital differences between two theories, differences which were small at the time compared to the new achievements of the theory, but which seem to have grown during the intervening decades. These differences, it should be understood, have nothing to do with the later difference between Heisenberg and Schödingerpictures, which are equally valid when both exist, and equally praxiological. Rather, the quantum theory of Bohr and Heisenberg was praxiological and worked while the attempt of Schrödinger was ontological and, strictly speaking, did not. Heisenberg and Bohr assumed that the atomic domain was granular and obeyed non- standard probability laws. Matrix mechanics replaced the multiplication and addition of transition probabilities that occurs in a Markov process by the multiplication and addition of probability amplitudes. bSchrödingerc originally assumed that the electron in a hydrogen atom was a wave running around the nucleus, implicitly retaining classical probability laws for states of the wave. Since his wave equation was of the first degree in the differentiator with respect to time, the state of his wave was a bwave functionc evaluated at one time. His theory is as ontological as the Newtonian particle model of nature, it merely made the atom a wave instead of a particle. This lead to the following immediate difficulties for the bSchrödingerc theory that are not problems for the bBohrc-bHeisenbergc theory. Some of these trouble 128 CHAPTER 2. LINEAR PRAXICS ontologists to this day:

2.10.1 Ritz combination rule. bSchrödingerc found a well-known discrete spectrum of possible vibration frequencies fn = 2 f1/n for his hypothetical wave. Had his wave theory been right, these would also have been the frequencies that the hydrogen atom radiates, as a vibrating cymbal radiates sound of the same frequency as the vibration. Instead the bSchrödingerc frequencies agree with the empirical spectroscopic terms for the hydrogen atom; the atom radiates some of the differences fn −fm between terms and not the terms themselves. This disconfirms the wave theory.

2.10.2 Probability Principle.

Moreover the electron does not divide when the bwave functionc does, as in electron diffrac- tion. In the ontological theory this definitenesss of events is a problem because according to the bSchrödingerc equation the bwave functionc usually spreads out during the dynamical development and never comes back to one point, while the entire electron is often registered by a detector much smaller than the alleged wave. The praxiological theory has no such problem. The classical correspondent of the bSchrödingerc equation, the Hamilton-Jacobi equation, also describes a principle function that may be concentrated at one point of configuration space initially and then spreads out, generally never converging to a point again, and this is no problem for classical physics because the principle function is not a physical object. The Boltzmann equation has similar properties; its probability distribution does not reconcentrate on one point either. Nevertheless in these theories there is no problem deducing that definite events happen; one simply assumes it from the start as a matter of definition. A probability is the probability for a definite occurrence. A probability p = 1/2 does not mean that half an event occurs; it means that the event in question tends to occur about half the time in the long run. Quantum praxics make the same assumption for a probability amplitude. A probability amplitude ohe = 1/2 does not mean that 1/4 of an event occurs. Nor does it give the amplitude of some physical wave. The experiment acts on a quantum, not on a bwave functionc. A probability amplitude is by definition a probability amplitude for a definite occurrence to the quantum. It therefore cannot be measured with one quantum but requires many trials and a fixation of a frame, like a probability. One cannot learn the probability interpretation from the equations governing the probabilities but must provide that interpretation at the start, in both classical and quantum theories. In the quantum theory, the definiteness of experience was even made explicit, as the Quantum Principle: The quantum remains whole when the probability amplitude divides. Here the name “Quantum Principle” is a monument to early ontological confusion. A probability distribution for Mars also spreads out, and Mars too does not split, but we do 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 129 not call this the Classical Principle, it is part of the operational meaning of probability. Likewise the Quantum Principle is merely part of the operational meaning of probability amplitude. It could better be called the Probability Amplitude Principle. There is no more possibility for events to split or “collapse” or become “indefinite” or fuzzy in quantum theory than classical. A fuzzy prediction is not a prediction of fuzz. There is still another parade in town beside the praxiological one. Following Newton is a second procession of many ontologists including Einstein, de Broglie, bSchrödingerc, bWignerc, and bGell-Mannc, who believe that we can do better than a theory of ineffable entities, with no complete description in symbols. Ontology can be attractive.

2.10.3 System catenation

Again, if the bwave functionc were the state of a wave, in the Helium atom it would still be a function of one point of space. Instead a probability amplitude distribution for a Helium atom depends on the coordinates of all its electrons, just as a probability distribution does. When we combine several uncorrelated electrons we multiply these probability amplitude distributions, each with a separate argument, just as we combine planetary probability distributions, except that the product is anticommutative for electrons. If one has maximal information about a solar system one also has maximal information about each planet, while one can have maximal information about an aggregate of electrons and know nothing about any variable of any electron in the aggregate. The electrons of such an aggregate are said to be entangled.

2.10.4 Relation to probability

Finally, if the “bwave functionc” were indeed a wavy object, it would not be a probability amplitude. A probability for an object is not an object. An object does not have a probability as one of its properties. Objects and transition probabilities belong to different levels of discourse. Unlike classical probabilities, quantum probability amplitudes can interfere. Since material waves like sound can interfere, and probabilities cannot, ontologists, who insist on classical probability theory and logic, are liable to regard probability amplitudes as material waves. Then, however, to close the theory, they must account for how allegedly material waves can give correct probabilities for experimental transitions. No ontological formulation seems to have such closure. The wave theory has the right terms, but it has the spectrum, the Quantum Principle, the rules for composing systems, and the probabilities wrong. When we call a vector a bwave functionc we memorialize Schrdinger’s original missing of the center of the mark. bHeisenbergc blocked such bontologismc by using the term “probability function”, not bwave functionc. The term “transition probability amplitude function” would have been more accurate but longish. It is understood that probabilities do not exist in the way that 130 CHAPTER 2. LINEAR PRAXICS pretzels do; they cannot be eaten. The theory of collapsing wave-functions is often called the Copenhagen interpretation, but it is opposed to the interpretation of Heisenberg and Bohr. The term vector has an appropriately praxiological root sense, that of carrier: “1704 ... A Line supposed to be drawn from any Planet moving round a Center, or the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy, called the Vector; because ’tis that Line by which the Planet seems to be carried round its Center.” Oxford English Dictionary. Here it is used in the praxiological sense too, indicated with the modifiers “input” and “output”. The praxiological bBohrc-bHeisenbergc theory of non-commuting variables automat- ically gets these four matters right from the start. Its primary vocabulary consists of matrices representing operations. Vectors are optional luxuries; and they do not represent the system under study in the way that a bwave functionc represents a wave, they projectively represent ways to input or output a system. by assigning probabilities to all ways of outputting the system. They are praxiological, not ontological. This would seem to settle the question of praxiology versus ontology. Then Dirac showed that Schrödinger’sequation was mathematically equivalent to Heisenberg’s. The two differed merely by a choice of representation of the invariant formulation set up by Dirac, a quantum relativity transformation. This made it possible to transfer the Bohr- Heisenberg semantics to the Schrödingermathematics. It is a poor road that cannot be followed both ways. The Dirac dictionary made it possible to translate the praxiological quantum theory of Born-Heisenberg in the ontological language of Schrödinger.One could speak bSchrödingerc and practice bBornc-bHeisenbergc, despite the serious mismatch between the two theories, by permitting as many non-standard uses of words as this requires. What Bohr-Heisenberg called a bprobability functionc, to indicate how it is measured and used, the ontologist must call a wave, and consider it to be present in the individual experiment, and subject to changes by measurement. The normal switch in probability distributions when one transfers attention from one ensemble to another, and in particular, from input source to output sink in a quantum experiment, the ontologist must rename a “collapse” of this wave during measurement, and may then criticize the spookiness thus created.

2.10.5 The von Neumann ambiguity The ontological mispresentation already appears in the seminal work of von Neumann [78] side by side with the quantum logics of non-commuting projection operators. Von Neumann introduced formal quantum logics closer in form to Boolean algebras than the matrix praxics of Bohr-Heisenberg but also less practical. He also permitted some ontologisms to enter his formulations of quantum theory that have been canonized and stilll haunt us today, and are therefore discussed here. 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 131

The book of von Neumann is ambivalent about the semantics of the theory. In the following excerpt U designates a probability operator, not a unitary one:

“We therefore have two fundamentally diﬀerent types of intervention which can occur in a system S or in an ensemble {S1,..., SN }. First, the arbitrary changes by measurements which are given by the formula

∞ 0 X (1.) U → U = (Uφn, φn)P[φn] n=1

(φ1, φ2,... a complete orthonormal set, see above). Second, the automatic changes which occur with the passage of time. These are given by the formula

− 2πi tH 2πi tH (2.) U → Ut = e ~ Ue ~

(H is the energy operator, t the time; H is independent of t). ...

([78], page 351). The count “two” in the first line of this excerpt is inexact. There are three modes of physical intervention in a quantum experiment, input hi, throughput hth, and output oh. Each has its own large building in many particle laboratories, making it harder to miss one today, and the accelerator itself, part of the input process, dominates them all. But all three are present in quantum experiments since Newton’s prisms and polarizing crystals. The miscount suggests that the attention of the writer was on the mathematics, not the laboratory. Evidently U itself was taken as a given, when actually it describes the first intervention. In astronomy, to be sure, the input planet seems given. In the quantum domain, however, input is as drastic an intervention as output, and U itself represents that first intervention. In an accelerator laboratory, U includes the whole accelerator. If there is an external target, U describes the beam extractor too. For colliding beams, U includes both beams. Moreover, although the author speaks of interventions in the system, he actually at- tends only to changes in U. U represents an ensemble, not an individual system under study. In this quotation the author seems to take the probability operator U as the system, ignoring its statistical nature. This may be because the author seeks to deal with what is completely representable, and U is completely representable while the system is not; but that is an ontologism. The “or” reveals a significant ambivalence. One atom is physically very different from N atoms and yet only one theory is presented, allegedly applying to both N = 1 and N = 1028. The “or” seems to mean that the difference between the system and the ensemble is immaterial here. The author has elsewhere made it plain, however, that U is statistical and concerns ensembles, although quantum determinations – interventions — are performed on each individual in the ensemble, one by one. 132 CHAPTER 2. LINEAR PRAXICS

This inferred ontologism seems to recur. (1.) is said to describe the “changes by measurement” in S or {S1,..., §N }. But it what it gives is a change in U, not in S. It is a property of probabilistic ensembles that a change in one element does not affect the ensemble distribution. Thus despite the preamble, (1.) cannot describe the change in S but only the change in the ensemble suggested by {S1,..., §N }. The changes in S are described, incompletely but maximally, by the diagrammatic formula ohthi giving the input, throughput, and output process, and by stating whether this transition actually occurred. Besides this recurrent ambiguity there is a significant lapse in (1.) itself, undoubtedly pointed out many times. A measurement results in a value of what was measured, but U 0 is an unsorted ensemble of many different values, so the process producing U 0 is as much a non- measurement as a measurement. Newton’s measurement of the physical colors of photons with a prism suffices as an example to illustrate the difference between (1.) and a measurement. There are two typical ways to use a prism to measure color that must be distinguished, and neither is represented by (1.): bfiltrationc produces a beam of photons of one color, as in a monochromator, and bsortationc produces a plurality of distinct beams, each with one beam color and one beam direction, with a 1-1 correlation between color and direction, as in a spectrograph. A filtration is represented by a projector, and a sortation is conventionally described by a normal operator, a sum of orthogonal projectors, one for each output channel, with numerical coefficients giving the results of measurement in that channel. If the momentum of a photon is p and the velocity vector is v then the photon mass is their ratio, p = mv. The photon mass m is the photon color: blue photons are about twice as heavy as red photons. The prism accepts a collimated beam with one v and many m’s, passes them through a glass that couples m and v, and delivers a beam in which m and v are correlated. The prism does not change the color of the beams that pass through it, Newton showed, but it changes all their velocity vectors, and the velocity vectors serve to register the colors. If photon color is the system measured in this example, therefore, the direction of v is a registering variable in the metasystem, like a meter needle. No such correlation exists in U 0; all the beams of different colors have been mixed. Measurement is supposed to increase information of the experimenter about the system; U 0 has less information than U. The imprecise description (1.) should be replaced by two more precise descriptions, of filtration and sortation. These are given elsewhere in the same work, however: a puzzling inconsistency until one is given to understand that while the rest of the book was written by bvon Neumannc, the famous paragraph quoted was contributed by bWignerc [82]. It was suggested that the existence of “two fundamentally different types of intervention” is peculiar to quantum theory and problematical. In classical physics too, however, experiments have a distinct beginning, middle, and end as they do in quantum physics; and the input and output processes, in which the system is open, are not different in kind from the throughput process, in which the system is closed. Galileo drops two weights 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 133 in the input process, they fall during the throughput process, and they hit the ground loudly in the output process. The input and output acts are often left out of the theory in classical physics because they often have no measurable effect on the system and can safely be treated as purely mental acts with no physical consequences. The Correspondence Principle points to important cases where measurement cannot be left out of the theory. The classical correspondent of the statistical operator U is a classical probability distribution function U. To make the discussion comparable to Wigner’s, let us omit the input process that produces U in the classical case too. Suppose U = U(n) is a function of a variable n with integer values, as if the system were a digital computer, and that classical closed-system development proceeds in a way similar to (2.), conserving the number of possibilities for the closed system. One typical measurement is a classical filtration for the value 0: 0 U(n) → U (n) = U(0)δn0. (2.22) The experiment starts with a probability distribution function U spread over many values of n and ends with a distribution function concentrated on the value 0 with probability U(0), which may be 0. This process reduces the number of possibilities for the system, unless U happened to be concentrated on the value n = 0. Therefore closed-system development cannot carry U into U 0, and a filtration process is required separate from the dynamical development, classically as well as quantally. Perhaps the misimpression that measurement does not need separate mention in classical physics has to do with the misimpression that (1.) represents a measurement; for the classical correspondent of (1.) would be the identitity mapping U 0 = U. The two processes (1.) and (2.) — more accurately, the input-output processes and the throughput process — differ in that (1.) operates on an open system and (2.) on an closed system. Some accept the mispresentation of collapsing wave functions and attempt to account for the alleged collapse with mental activity. This is wide of the mark. There is no collapse phenomenon to explain, there are merely two vectors for two ensembles, one for input and one for output. The throughput process occurs in the closed system, after input and before output, and conserves the number of possibilities. The input and output process are informational and change the number of possibilities for the system. Compensating changes must occur elsewhere, in both classical and quantum physics. The major difference between the classical and quantum praxics is the emergence of filtration commutativity in the classical limit. This makes it seem that classical experiments can give complete information about the system, including the result of subsequent filtrations and sortations. Actually the classical filters can appear commutative only because they are far from maximally informative. The author invokes a meta-experimenter II who can regard the input-output processes of I on the system S as parts of one throughput process on the combined system I × S. For II the distinction can disappear between processes 1, 2, and 3 can disappear. But this merely shifts the distinction up one stratum. II still distinguishes between the throughput 134 CHAPTER 2. LINEAR PRAXICS and input-output processes of II. Here the author commits the grave error of imagining that the unobserved acts of I are somehow the same as the acts that I can carry out under maximal determination by II, which is generally lethal. This mis conception that acts of knowledge are merely mental and not physical recurs when von Neumann speaks of the bcutc between metasystem and system, here called the system interface. The cut is a physical information valve, open during input and output, but closed during throughput, when it acts as an adiabatic wall. A single uncontrolled quantum passing through the system interface during throughput can destroy quantum interference. For example, the interface for some systems might be a superconducting lead vacuum chamber with mirrored walls close to absolute zero, while for the spin of a photon in flight, clean air will suffice to maintain the isolation of the photon spin for some meters of flight path. Von Neumann speaks of placing the cut between the retina and the brain, so that the eye becomes part of the system while the brain remains in the metasystem observing the eye; as though such an operation were a mere change in point of view and normal eye function could proceed undisturbed. The problem is not merely that this is a difficult operation, but that it must interfere with the function of the eye in order to carry out its function. To measure the eye maximally is to freeze it. Much the same error arises when von Neumann assumes that the observer has a state and that “this state is completely known” ([78], page 439). The observer would then be in the cold state of Schrödinger’scat, discussed in §2.10.6. The physical process that the “state” represents has again been left out of the picture, as it is in classical physics. This frequent omission is good reason for discontinuing this misuse of the classical term “state” in quantum theory. Safer quantum state constructs are suggested in §S:STATES. bWignerc [83] called his ontological mispresentation of the quantum theory, with its collapsing wave functions, the “orthodox” quantum theory, and it is often called the “bCopenhagen quantum theoryc”, despite some sharp conflicts of formulation between it and the canonical theory of Heisenberg and Bohr. In view of this confusion, both terms “orthodox” and “Copenhagen” theory are avoided here. The ontological mispresentation made it possible to present the mathematics of the quantum strategy without dealing with its more challenging conceptual revisions in philosophy, probability theory, and predicate algebra, as in the above quotation. Nevertheless, it misuses ordinary physics language and creates diversions from the actual problems of quantum theory. It is widely written, taught, and discussed today, at the same time that the praxiological theory is widely used. The ontological mispresentation claims to describe what “is”, not what goes on. In an experiment where a photon is emitted, flies along an optical bench, and is counted or not, the ontological mispresentation says that in the real world a wave function propagates down the optical bench to the analyzer, and there “collapses” to another wave function or not. This is understandably considered to be problematical, even by the ontologist who introduced it. It is a position like solipsism, invented to attribute to others. A photon that we find in raw nature does not come with a sharp ensemble of like photons. 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 135

To speak of “its vector” is therefore operationally meaningless in principle and leads to confusion in practice. The system does not carry a vector as much as a vector carries systems. The vector represents a process, a system is one of its products. The term “the vector of the system” or “wave function of the system” confuses a process with its product, and is a reversion to the early wave theories of de Broglie and bSchrödingerc, in which the state of the hypothetical wave system was indeed a wave function. Today speaking of the “bcollapsec of the wave function” is a fortiori an ontologism. Despite the well-known uncertainty of quantum prediction, an ontologist might still assert that the dynamical law of quantum theory determines the future behavior of the system from the initial conditions [85]. He knows perfectly well that usually the orientations of two given polarizing filters do not determine whether a photon from one will pass through the other, so it must be the development of “the state of the system” that is being described as deterministic, not the history of the photon, and the ontologist must again be identifying “the state of the system” with some wave function, as if the system were a wave. The development of any statistical distribution is “deterministic” in this mathematical sense, even in classical statistical mechanics, where the development of the individual system is not determined. This common ontological mispresentation of a praxiological theory, not the transition from ontology to praxiology, is what causes many to suppose that quantum kinematics cannot be taken seriously even though it works. Here it is taken seriously, which means without ontological post-editing. To be sure, there are still troublesome infinities in quantum theory, but they arise from classical vestiges in the theory, were worse in the classical theory, and disappear altogether in fully quantum theories. In both classical and quantum physics an ideal filtration for a property of a system does not change that property. This is expressed by representing the filtration process by a projector, obeying P = P 2. This idempotency was one of the first facts that Newton verified experimentally for both refracting prisms and polarizing filters, exactly to convince himself that they did not change the color or polarization of photons but merely filtered or sorted photons by these properties. In the classical case, a sharp measurement operation is represented by a diagonal matrix P of 0’s and one 1. But the ontological mispresentation postulates that what the quantum theory calls a measurement P is actually a “collapse” process C : ψ 7→ ψ0. C clearly changes what the ontological mispresentation treats as the system, that is, “the wave function”. Whatever the wave function ψ was said to be before the measurement, if the system passes the test, “the wave function of the system” is said to become the eigenfunction ψ0 associated with the test. Such a collapse C is certainly not a measurement of ψ in the terms of the ontological mispresentation itself. A measurement of ψ would not change ψ. To be sure C : ψ → ψ0 is idempotent, C2 = C. But it lacks the symmetry required of both classical and quantum measurements. Since the alleged collapse C is a many-to-one relation, its dual C† is a one- to-many relation, and C 6= C†. To measure the ontologist’s “wave function” would require 136 CHAPTER 2. LINEAR PRAXICS a filtration process that passes a certain wave function ψ0 with certainty and rejects any other wave function with certainty. This process would violate the superposition principle of quantum theory. What the ontologist could call a true measurement is impossible according to quantum theory. Since the ontologist claims that the system is a wave function, and that quantum measurements change it, he or she must claim that what quantum experimenters call measurement is not truly measurement. Thus the ontologist’s mispresentation violates Standard Semantics for the term “measurement”. Nevertheless in practice even ontologists use the two vectors oh and hi of every experiment to compute the expected mean count just as the quantum theory does: as probability amplitude distributions, not as physical waves. And they find these vectors as the quantum theorist does: not ontologically, by measuring a wave function, point by point, but praxiologically, from an inspection of the measurement action that the experiment carries out on photons in general during input and output, repeated many times. Thus the ontologists speak ontology but practice praxiology. They speak of an imaginary random system I [V ], of which ψ0 is one value; but in practice they use vectors oh, hi to represent processes carried out on the actual quantum system I[V ], the photon, just like quantum theorists. By criticizing the ontological mispresentation of the canonical quantum theory, ontologists uphold the standards of physics. But a mispresentation is not good reason to modify the quantum theory; there are better reasons for that. Roughly speaking, the quantum formulation approaches a classical one as ~ → 0; this refers to the Correspondence Principle of Bohr. While Segal [66] examined both the singular limits c → 0 and ~ → 0 as examples of Lie algebra homotopy, the study of this mathematical process by bInönüc and bWignerc [47] significantly ignored the limit ~ → 0. By putting c and ~ on the same footing we explicitly accept that both relativity and quantum theory might be irreversible advances in physics. Some, however, expect that the quantum non-commutativity of filtrations will disappear and classical commutativity will re-emerge, for example when a suitable theory of consciousness is found. This leads to two conflicting teams in theoretical physics: a praxiological one working to eliminate residual commutativity from the canonical quantum theory, and an ontological one working to restore classical commutativity. Here we join the praxiologists. In the canonical quantum theory, as opposed to the ontological mispresentation, a quantum system carries no vector, any more than the planet Mars carries a probability distribution for the planet Mars. There is no vector present in the system to evolve or collapse during the experiment, what evolves is the system. The temptation arises to replace the system by “the” wave function because each wave function is a completely describable mathematical object, and the system is not. As long as we work with wave-functions they make no demands on us to change our logic, and they remain unchanged by our mathematical study of them. The quantum system is not completely describable and obeys a non-commutative 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 137 praxic instead of a commutative logic. To accept quantum theory requires us to revise our logic and probability theory. Many experience an urge to explain “why” we add probability amplitudes instead of probabilities in quantum physics. This too is a symptom of bontologismc. People did not explain the addition of probabilities back in the days of classical mechanics. They postulated it, presumably because it worked. Therefore, by correspondence, the addition of probability amplitudes in quantum theory must not be explained but postulated, because it works better. The laws of probability and logic are so basic that they defy deduction from something more basic, since the deduction itself would use them, but must be gleaned inductively from experience. If adding vectors — as in quantum interference — seems to require deduction, as opposed to induction from experiment, then one is attached to the classical laws of probability, and has not replaced them by the quantum ones. On the other hand, if our praxic includes vector addition, it is legitimate to ask how bclassical logicc emerges as an approximation. But this is clear. The difference between the classical and quantum kinematics is commutativity of the operator variables. Classical commutativity emerges from quantum non-commutativity through the a bLaw of Large P P Numbersc: If two quantities q and p are sums q = qi, p = pi of N independent terms from N independent systems, then in the limit N → ∞, the commutator [q, p] qp + pq becomes negligible compared to the product. Briefly put, if an experiment involves action S, an associated action quantum number S/~ becomes large as ~ → 0, and the bLaw of Large Numbersc takes over. Non-commutativity, however, is structurally stable, and can be verified with three polarizing filters, which also demonstrate the role of probability, granted the existence of photons. Quantum theories are innately stochastic as opposed to complete. The “determinism” of the Schrödinger equation resembles that of the Liouville equation, which also describes the time development in a stochastic theory. Non-stochastic mispresentations for the quantum theory abound. The ontological mispresentation reifies “the” vector and supposes it to be carried by the quantum, like the classical state. Then one accounts for the vector either by assuming two such objects, or by having the first “collapse” into the second, usually — for no good reason — during the output process rather than the input process. Such a belief in a “collapse” in the absence of experimental evidence is another symptom of bontologismc. In the Copenhagen theory, one probability-amplitude vector represents a certain ideal input process, another an ideal output process, in the way that probability distributions do in classical theory. They are needed to describe the two arbitrary choices, the input and output process, that we make in setting up an experiment. One does not develop into the other as the system develops. Nothing that could be called a “collapse”, with all its ontological implications, occurs in the experiment or in the canonical quantum theory. Classical probability theory also has a probability distribution for both input and 138 CHAPTER 2. LINEAR PRAXICS output processes, and few imagine that one collapses faster than light into the other. A consistent replacement of probabilities by probability amplitudes on the dynamical results in the quantum theory of Heisenberg and Bohr, equally free of “collapse”, and called canonical here. “Collapse” is an artifact of the ontological mispresentation. Attempts to eliminate a collapse that does not exist are symptoms of ontologism. For example, there is a bMany World Theoryc of bEverettc whose mispresentation terms a “world” what is merely a possibility in classical and quantum physics. This too is a reaction to the contradictions of the ontological mispresentation, not to the canonical quantum theory, which Everett may never have encountered, since it is rarely taught. Since the bMany World Theoryc attempts to work with one vector when there are two independent mutually dual processes, and attempts to avoid mentioning the experiment interface, it is yet another ontological mispresentation. Its “vector of the universe” is the state of the universe, regarded as an absolute ontological being. Since no one inputs or outputs the universe sharply from outside it, there is no physical need for a universe vector. If the Many World Theory theory were taken literally it could make no experimental predictions. Since the “vector of the universe” is taken as the system state, it cannot be a probability amplitude, which is not a system state. It is then inconsistent to use the “vector of the universe” to compute probabilities in the Many World Theory. Its exponents actually continue to use the canonical quantum theory in practice but describe their practice using a non-standard semantics.

2.10.6 Schr¨odinger’sfrozen cat

To indicate what he felt was an absurdity in the quantum theory, bSchrödingerc proposed a famous thought-experiment: A cat is caged with a lethal device that may be triggered by the decay of a radioactive nucleus. After a nuclear half-life, according to Schrödinger, “the wave function of the cat” would be in a quantum superposition of alive and dead states. This is presented as an affront to common sense, which tells us that a cat is either alive or dead and not in limbo waiting to be observed. Actually it illustrates two basic misunderstandings by the narrator: In the first place bSchrödingerc spoke of “the wave function of the cat”, as though every cat had one. This is an bontologismc, consistent with his original approach to the atomic spectrum problem. If the cat were a classical wave it would always have a state, and the state would indeed be a wave function. Actually the cat is composed of quanta, and what would a cat vector would actually describe is a coherent source of well-isolated cats. No comon-sense cat ever came from such a source. To be associated with an exact vector each cat in the emitted cat beam must be insulated from interaction with any other quantum and have entropy S = 0. Therefore the cat beam has absolute temperature T = 0. Its cats are frozen, not alive; unless some experimenter can produce an isolated live cat reversibly from its chemical elements at an absolute zero temperature. The bSchrödinger catc is therefore not the cat of common-sense every-day life as he suggests but a cryological 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 139 fantasy that will presumably never come true. Incidentally, the bAharanov limitc on coherent superposition, set by vacuum fluctuations in the gravitational field, requires anything that exhibits quantum interference to have mass much less than 10 micrograms, unless the preparation of the cat is accompanied by a special maximal preparation of the vacuum gravitational field, which is another fantasy that will presumably not be actualized. It would have been surprising if Schrödinger had known this limit to coherence. Similarly, the cat must be electrically neutral; a charge of 137 electrons would seriously damage the proposed interference. Superpositions of macroscopically distinct vectors are taken for granted today for quantum computers of much less than 10 microgram mass and less than 137 e charge, and are called “Schröedinger states” in his honor. They are indeed cold. In the second but more fundamental place, even if a cat is described by such a superposition vector, this does not mean that the cat is neither dead nor alive, or that it is both. Assuming as Schrdinger¨ did that these predicates correspond to projection operators L (for living) and 6 L (for dead), even in quantum theory L∪ 6 L = 1 is true, the cat is living or dead, and L∩ 6 L = 0 is false, the cat is not both living and dead. Every time we do the experiment the quantum theory predicts that we find a live cat or a dead one, just as in real life. The coherent superposition does not make a definite prediction about L but gives probabilities for L and for 6 L. An indefinite prediction, however, is not a prediction of indefiniteness. The probabilities provided by quantum physics are probabilities for definite outcomes. The quantum theory predicts that when we measure L for a cat from such a source, we will always find one of L or 6 L, never a fuzzy cat. The flaw that Schrödinger reveals is not in the quantum theory but in the ontological status he gave to wave functions. Heisenberg called vectors “probability functions” to forestall such ontologisms. The question is not a matter of convention or philosophy; it is a question of how one determines wave functions in actuality. Determinations of wave functions of the system under study are made through the Malus Relation for transition probabilities, not by measuring a wave amplitude. Normal practice is to reserve such ontological mispresentations of quantum theory for public or classroom occasions, not for doing experiments, not even thought experiments, where it is a hindrance. In this thought-experiment, for example, Schrödingerdoes not tell us how to determine the entity he describes as “the wave function of the cat”. If one were given a living cat rather than a frozen one and were required to find “its wave function”, much interpretation of this request would be required. It might be interpreted as a request to determine the values of a maximal set of observables of the cat. If the observables are chosen to be the positions of the particles in the cat, the result of such a determination would be an explosion of nuclear magnitude, by the uncertainty relation Most choices of complete sets of observables are similarly catastrophic. Schrödinger’sdiscussion shows that at least as of that writing he had not yet accepted the praxiological nature of vectors and the responsive nature of quanta but regarded “the 140 CHAPTER 2. LINEAR PRAXICS wave function” as a physical property of the cat, like the state of a classical object, giving it ontological status. bWignerc elaborated bSchrödinger’scatc to what has become known as bWigner’s friendc, who can be asked about what happened in the experimental chamber before the measurement. The treatment compounds both of Schrödinger’serrors. Again it is supposed that a living creature can be assigned a vector without disrupting its life processes; as though the friend actually carried a vector with his identity papers, and we merely had to read it. Again it is supposed that an indefinite prediction is a prediction of indefiniteness.

2.10.7 Fundamental law Quantum dynamics assigns probabilities to individual experiments, and in exceptional but important special cases the probability is 0 or 1. If quantum dynamics does not cause discomfort in a physicist then it likely has not been understood. Quantum dynamics does not fulfill the Laplacian dream of a fundamental law. For most experiments, no matter how ideal, it fails to predict the outcome except statistically and with large dispersion. Fur- thermore a theoretical probability statement relates what happens in an actual experiment to experiments of the future that may or may not be performed. Quantum dynamics gives an operator hT h that relates an early input action h to a later one hT hi, earlier probability to later probability. It never tells us if the experiment will actually be carried out, and it rarely tells us what will actually happen if it is. Therefore quantum theory does not control the future in the way that Laplace demanded of the law of nature. But clearly Laplace too could not actually predict such human phenomena, and his idea of a fundamental dynamical law that could was a great extrapolation beyond experimental evidence, and has failed, while the Malus Principle continues to work for us today without a viable competitor. Empirically, these conceptual evolutions seems easier for younger physicists than old, and some physicists regress to the classical metaphysics after productive years of quantum practice, with no experimental support for this decision. Two strategies can be suggested for reducing this conceptual dissonance between the quantum dynamics and the classical ideal of dynamics. First one might accept that the quantum theory does not deal with certainties alone. When we enter the quantum world we leave behind the ideal of fundamental universal law. This explains much of the pain that Bohr mentioned. Moreover, a quantum dynamics is not absolute but contingent on the metasystem, being disrupted by measurements. This makes it necessary to weaken the concept of fundamental symmetry too; if the dynamical law is contingent, then likely its symmetry is too. Both the dynamical law and its symmetries are contingent on the metasystem. Second, one might note that the classical principle of deterministic dynamical law emerges as singular limit of the Malus Principle and quantum dynamics or approximations thereof. This anticipates the many-quantum logics of the next section. When oi and hi are 2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 141 normalized and oii ≤ 1, then the many-quantum vectors (oi)n and (hi)n for a sequence of n trials of the experiment ohi are nearly orthogonal, and transmission is nearly forbidden. By taking n large enough this can be made as close to the deterministic case of probability 0 as desired. In this way one can deduce the full Malus Principle from the following Weak Malus Principle: For a given number of trials N, when |o < i| 1 is sufficiently small, the transition o < i almost certainly does not occur. This reduces the case of general probability to the case of arbitrarily small probability. The dynamical law of canonical quantum theory deals with vectors that depend on time. This goes outside the first-order language of linear quantum logics, since it uses the concept of function. It is taken up after the appropriate foundation is laid. 142 CHAPTER 2. LINEAR PRAXICS Chapter 3

Polynomial quantum logic which deals with pluralities of individuals. The system of the classical random set is discussed next (§3.1.1) to guide the discussion of the queue that follows (§3.1.2).

The terms bclassc and bpredicatec are reserved here for collections of possibilities for one system, represented by projectors in a space of monadics. The terms bsetc and bqueuec are reserved here for a classical or quantum system composed of lower-stratum systems.

3.1 Set algebras

3.1.1 The random set

Consider first the states of a random classical object, supposed to form a semigroup S of random sets. A space S of statistical distributions is introduced here that is in the first place a polynomial algebra. Its + is statistical mixing. Its × is direct multiplication, combining descriptions of n1 and n2 distinct bodies into a description of n1 + n2 bodies. These two operations suffice to express the most general statistical description of a random set of individuals as a polynomial in states of the individual. For example, if a and b are states of an individual, ab is a pair state, while a + ab statistically describes an object that is either in state a or ab with equal likelihood.

The number of factors in a monomial in S is called its bgradec g and is indicated by terms like (bmonadc, bdyadc, ..., bp-adc, ...), or bpolyadc, or by a subscript g. The 0-ad or cenad, the product of no factors, is 1, representing the empty set. The dimensionality of a subspace of S is called its bmultiplicityc and is also expressed by terms like (bsingletc, bdoubletc, ..., bm-tupletc). Thus the suﬃx -et indicates a count of terms, and the suﬃx -ad indicates a count of factors. The 0-plet, the sum of no terms, is 0 and represents nothing, certainly not the empty set, which 1 represents. The operation × distributes over addition because of the statistical meaning of addition.

143 144 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

To avoid multiple occupancy of a state, one posits that individual states obey a2 = 1 and commute. Lacking these postulates the polynomial algebra is that of a random sequence; with them, a random set. This algebraic language may not seem natural from the viewpoint of standard classical logics. It is a back-formation from quantum algebras in present-day use. To generate nontrivial objects or states themselves takes at least another operation. One monad-forming operation I of bbracingcwill do. For any monomial state p, of any grade, Ip = {p} is the state of a monad, different from p, whose sole element is p. Polynomials and bracing suffice to generate all finite abstract classical sets; the empty set, to start with, being the product of no factors. For example, a field state f(x) is a set of field event states {fx}, each constructed by multiplying a field value state f by a space-time event state x and bracing the product f × x. The random sets with no more than L nested braces form bstratumc L of S, designated by S[L]. Monomials may be classified by grade g and rank r. I is assumed to be additive because of the statistical meaning of addition: If a set is either a or b with equal probability, then its brace is either {a} or {b} with equal probability. The infinite-dimensional algebra of random sets generated by summation, multiplication, and bbracingc is designated here by bSc. The subspace of S consisting of homogeneous polynomials of grade g is then Gradeg S. Elements of S with no more than L nested braces form its stratum L, written as bS[L]c. In general, restriction of any construct to stratum L is shown by a superscript L. Numerical properties of a random set can be represented D by elements of the dual space S , maps v : S → R, the value v(p) being the expectation value of v on p. Thus I converts any polyadic set s of one stratum into a monadic set Is of the next:

[L] L+1 L+1 I : S → Grade1 S ⊂ S . (3.1)

In the present terminology, classical blinear logicsc represent random individuals using the linear operations +, ×R. Classical bpolynomial logicsc adjoin the operation × to represent random ﬁrst-order sets. Classical bexponential logicsc further adjoin I to represent random sets of all ﬁnite orders. The multiplicity of each stratum in the exponential logic is exponentially larger than the previous, so stratum multiplicity grows superexponentially with stratum number L:

L+1 Mult S[L] Mult S = 2 =: 2L+10. (3.2)

3.1.2 The queue The above classical logics have fully quantum correspondents. Canonical quantum theory revised the coefficient system and the two operations + and ×. The complete description by 3.1. SET ALGEBRAS 145 basis vectors in the algebra S turned into merely sharp but incomplete statistical descriptions by general vectors in the same algebra S.A bvectorc represents a sharp input-output operation, like a source or counter. Positive real probabilities became complex transition probability amplitudes. Real polynomial addition a + b representing mixture became complex polynomial addition α + β, representing quantum superposition. The commutative law ab = ba split into an anticommutative one αβ = −βα for fermions and a commutative one αβ = βα for bosons. The unipotency law a2 = 1 survived as an exclusion principle α2 = ±1 for fermions but was dropped for bosons. The real algebra S of statistical descriptions becomes an algebra S, here taken to be real. Thus the vectors in S do not correspond to states of a classical system, which do not add, but to statistical distributions, which do. §2.3 gives further principled reasons for replacing the common term “state vector” by “vector” in this work. One major difference between the classical and quantum theories is that the classical vector space S has a preferred set of basic rays S representing complete descriptions, states, while no vector in S gives a complete description of what actually happens, and there is no absolute preferred basis. There are no states in S; they are found elsewhere for quantum systems. Random set addition + and multiplication × are modified by the quantum evolution so that it is not possible for the brace I that they support to survive unchanged. To describe how I enters the quantum era it is helpful to review how + and × did, returning to I in §1.2.1.

3.1.3 Input-output processes Modern quantum theory began with a finitization, Planck’s prescription for regularizing a singular integral, one that that gave a cavity an infinite heat capacity. Since finitude is still far off it can continue to serve as our pole star. Yet effectively infinite groups of actions like translation and rotation occur everywhere and must be represented in our theories. Classically, the world cannot be both finite and round. Any circle has an infinity of points, except the trivial one of radius 0. One can turn a body through an infinity of angles. To realize such a continuous group faithfully a set of classical states must have infinite grade. But then it also supports unbounded variables. Classically, a potential infinity requires an actual infinity. Quantum theory passed between the horns of this dilemma. Queues of finite grade can still realize a continuous group. A finite quantum world can be round. For example the vector space of multiplicity 2, describing a quantum with two independent input modes, supports the defining representation of the infinite group SO(2R). Queues do not transform as random sets do, a different kind of relativity comes into play, the bDiracc btransformation theoryc, allowing a finitude of vectors to have an infinitude of frames. Instead of building from states, quantum theories build from ideal binputc and boutputc 146 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

processes of maximal resolution, such as sources and counters, collectively called bioc processes. An input process accepts a signal and puts a system with specified properties into the experiment chamber; an output process takes a system with specified properties out of the experimental chamber and puts out a signal. Unlike classical theories, quantum theories do not predict the result of every such experiment, but give the transition probability, the expected value of the ratio of output counts to input counts. Quantum theories represent input-output processes statistically by vectors in mutually dual vector spaces of the system, input and output, provided with a non-singular bprobability formc, that maps each vector into the other partner of an assured transition. The prototypes are the input and output actions of the Malus experiment (1805) with two linear polarizers. There the vectors in question can be etched on the polarizer and analyzer filters, and the probability form transforms any input action into the output action that uses the same polarizer as an analyzer. States constitute a preferred basis for S. S has no such preferred basis, until one is etched on the filter by arbitrary choice. The classical construct of state is relativized by the quantum theory. If the transmission probability is 0, as for orthogonal polarizing axes, the transition is called bforbiddenc; if it is 1, as for parallel, bassuredc. For orthogonal or parallel vectors the Malus Principle gives a transition probability 0 or 1, as does a deterministic law. Granted the existence of the bilinear space V , with definite probability form, V is completely defined as vector space by its forbidden transitions and then as a bilinear space by its compulsory transitions. This result of the 19th century culminated centuries of study of bprojective geometryc. It is presented more fully elsewhere [40]. Quantum theories represent bthroughputc processes including symmetry transformations, by square matrices, which accept an input vector on their right and an output one on their left. An operator defines a matrix in each reference frame and is defined by any one of them, so we need not pay much attention to the distinction. A matrix is a morphism of a vector space, and so is represented by a labeled arrow. The arrows

→A ≡ ←A ≡ hAh ≡ iAi (3.3) all represent the same matrix A, the arrows deﬁne their own directions. An input and an output vector combine into an arrow representing a throughput operation. Therefore they are best represented by semi-arrows like hi and oh. These combine to form the throughput arrow hi ⊗ oh. As matrices, an arrow is square, an input vector is a column, and an output vector is a row. The widely used symbols |ii (kets) and ho| ( bras) for for input and output vectors clash with the usual arrow notation for mappings or operators and are not used here. D Let V designate the bdual spacec to V . The Statistor Principle implies that in a real D quantum theory there is a ﬁxed symmetric bilinear form iHh : V ↔ V , the bprobability 3.2. CLIFFORD ALGEBRA 147

formc, such that when iiHhi > 0 the transition iiHhi from input hi to output ih := iiHh is assured. In complex quantum theory the form iHh is Hermitian symmetric and sesquilinear. Then hi and ih can be regarded as input-output processes of opposite polarity for like quanta. When oiHhi = 0, ho and hi have zero probability. A positive deﬁnite probability form is a real Hilbert-space metric form in its mathematical properties; its physical interpretation, is crucial for the development.

3.2 Cliﬀord algebra

The linear operators on a Grassmann algebra form a special case of Clifford algebra, Con- cepts of Clifford algebra are summarized next without reference to the quantum interpretation. A Clifford algebra C is an associative distributive linear algebra consisting of all polynomials — that is, finite sums of finite products — in given variables, with real scalar coefficients, subject to the Clifford Clause: The square of a vector is a scalar.

The real numbers then form a central subalgebra R = C0 ⊂ C. The linear combinations of variables are called the bvectorsc of C and form a real vector space C1 ⊂ C. 0 is the monomial with no factors; 1, the polynomial with no terms. The Clifford ring over a an arbitrary ring of scalars and module of vectors is an obvious generalization that is sometimes useful. It includes binary, integral, real, and complex Clifford algebras, with scalars 2, Z, R, and C, respectively . The Clifford Clause defines a bilinear form on the vectors C1 in terms of the Clifford product: uv + vu u · v = , kvk = v · v = v2. (3.4) 2 The linear transformations C1 → C1 preserving this bClifford formc and having determinant 1 make up a special orthogonal group SO(C1). If the Clifford form u · v is bregularc the Clifford algebra is called regular. A Grassmann algebra is a most singular Clifford algebra, with v2 ≡ 0 for vectors v ∈ C1. The Clifford algebra of queues constructed here will have a regular bilinear form. Every Clifford algebra has a natural Grassmann algebra structure as well, with Grass- mann product u ∨ v defined by setting

∀u, v ∈ C1 : uv = u ∨ v + u · v. (3.5)

One then deﬁnes a Grassmann derivative Dv : C → C with respect to a vector v relative to a given vector basis B ⊂ C1 as usual; and a basis-independent bgrade operatorcbGradec: C → C that acts on Grassmann polynomials P (v) ∈ C by

Grade =: sumv∈BvDv (3.6) 148 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

. Eigenpolynomials of Grade = g are said to have bgradec g and are called bg-adicsc. They form a subspace of C designated by Gradeg C. Those of odd (or even) grade form the subspace C− (or C+) and are called boddc (or bevenc, respectively). C+ is a subalgebra of C.

3.2.1 Cliﬀord semantics

Every Clifford algebra is isomorphic to a matrix algebra over some vector space, called a spinor space for the Clifford algebra.. Therefore a given Clifford algebra has at least two conceivable quantum interpretations, besides all the usual classical ones. Being a matrix algebra, it can be provided with an adjoint operation H and used as the kinematical algebra of a quantum system, to represent throughput actions, Being a vector space, it can be provided with a probability form H and used as the vector space of a quantum system, to represent input-output actions. Both interpretations occur in standard quantum physics:

The bDirac algebrac is the Clifford algebra over a Minkowski tangent space, alge- braically isomorphic to Alg(4R), used as the kinematical algebra of a spin 1/2. The Clifford algebra over V ⊕ V D is used as vector space for a Fermi catenation of replicas of the individual system I[V ], and is then called the Fermi (-Dirac) vector algebra (over the individual quantum system I[V ]). Any element of a Clifford algebra Cliff(nR) can be interpreted as a vector of a queue cell on n quantum vertices (§4.1.3). The Clifford algebra of a Fermi system is exhibited in more detail next.

3.2.2 Fermi Cliﬀord algebra

A classical bsetc can contain an element 0 or 1 times only and is unchanged by exchanging any two of its elements. bFermi(-Dirac) statisticsc defines a quantum assembly that also has these properties. The Fermi relations (1.35) define a bClifford algebrac over the duplex space W = V ⊕ V D with neutral duplex norm. In quantum logics it is ordinarily pointless to form the direct sum of an input and an output vector space like V ⊕ V D. Such a sum assumes a coherent phase relation between input and output vectors by allowing their quantum superposition. This assumption is dubious because each input vector refers to a prior interaction with a macroscopic source and each output vector refers to a later interaction with a macroscopic sink. Such macroscopic interactions destroy coherent phase relations. Yet a rather similar superposition is commonplace in canonical quantum quantification (§3.3). Quantification converts any one-quantum input vector hi into an input operator hih, and any one-quantum output vector oh into an output operator hoh, both acting on the many-body vectors. These operators are freely added even though they are linearly related to a vector and a dual vector, which ordinarily cannot be added. 3.2. CLIFFORD ALGEBRA 149

It seems that I and quantiﬁcation shift the system interface to enlarge the system. Acts previously regarded as input-output across the interface are now represented on the system side of the interface. Instead of saying that an electron is input from the metasystem, after quantiﬁcation one can say that the electron is produced by a certain beta decay within the system. This does not eliminate the input but enlarges it. It removes the electron input but replaces it, for example, with the pregnant neutron that gives birth later to the electron and its sibling. Maximal determination of the macroscopic external source is still inconceivable, but now a microscopic internal source, the neutron, can be maximally determined.

3.2.3 Fermi vectors

Queues, as correspondents of classical sets, are represented here vector spaces that are Grassmann algebras. Their operator algebras are Cliﬀord algebras over the duplex spaces. Evidently we come closer to the standard quantum physics if we use Cliﬀord algebras as operator algebras rather than as vector spaces. To make a kinematical algebra A like Dirac’s spin algebra from a Grassmann vector space G = 2V = Grass V , one merely “squares” G, forming A = G ⊗ GD.

Any vector s ∈ V deﬁnes linear operators of bleft-multiplicationc Ls, right-multiplication Rs, and commutation 4s = (L − R)s on G → G.

Commutation 4 defines a representation of the commutator Lie algebra of Grade2 C, called the bcommutator representationc (“regular representation”, “adjoint representation”). 4[s, s0] = [4s, 4s0] (3.7) is a form of the bJacobi identityc. The vector space V [F] of stratum F requires a probability form for quantum use. If a probability form H is given on V then it defines one on 2V in a well-known way, designated again by H when context prevents confusion. 2 There is a natural norm on R: krk = r . This can be propagated up the ladder of strata by imposing the canonical relation (1.15) and supposing that the norm of a product of orthogonal factors is the product of their norms. The resulting norm will be called the bmultiplicative normc. It is positive definite, making Lorentz invariance impossible. It seems that two indefinite forms are required: H on V [F] for the probability form, and g on V [F] ⊗ V [F]D for the causality form. If they are not to clash, then they must determine one another on some stratum. If v, w are vectors then the Clifford, Grassmann, and inner products are related by

vw = v ∨ w + v · w. (3.8) 150 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

3.2.4 Grade operator The Grassmann grade g deﬁnes a symmetric operator Grade : C → C with respect to the mean square form. Grade : C → C is the linear operator that when c is the Grassmann product of g vectors, obeys Grade c = gc. (3.9)

Such a c is called homogeneous of grade g, or g-adic. Cliﬀord elements of diﬀerent grade are orthogonal with respect to the mean square form. P∞ Grade has a spectral resolution Grade = n=0 n Graden with eigenvalues n and associated eigenprojections Graden. The operator Graden takes the grade-n part of its operand. Grade0 c is called the bscalarc or cenadic part of c; Grade1c is the vector or monadic part of c; Gradenc is the nth-grade or n-adic part of c.

3.2.5 Mean-square form Just as any Lie algebra has a natural bilinear form, its Killing form, any (ﬁnite-dimensional) linear algebra A has natural bilinear forms invariant under inner automorphisms of A. In the case of a unital algebra A, a unique invariant form is deﬁned for all α, α0 ∈ A by

Tr αα0 αHα0 := . (3.10) Tr 1

Its normalization was fixed so that the norm kvk := vHv on a Clifford algebra agrees up to a constant factor with the Clifford square Q(v) = v2 for first-grade elements. This will be called the bmean-square formc. It is designated by H, or iHh, because when H is definite it is a Hilbert space metric form. When the elements of A are used as vectors hα, the mean square form will be interpreted as a relative transition probability amplitude like the Hilbert space form. The symbol iHh accepts two vectors and delivers a real number. A vector hv = hα ·vα ∈ V is represented isomorphically in the Clifford algebra C = 2V α β by the element vb = sαv ∈ C. Similarly a linear operator hqh = hβ q α αh : V → V is represented isomorphically by the element

β †α D qb = sβ q α s ∈ Alg C = C ⊗ C . (3.11)

†α D where s ∈ C is the dual vector to sα with respect to the mean-square norm (1.28). A metric form on a Clifford algebra cannot be represented within the Clifford algebra but it can be represented within the algebra of linear operators on the Clifford algebra. Let α γα := Lsα stand for left multiplication by the basis element s ∈ C, and let the inverted comma ‘ stand for right multiplication: γα‘ = Rsα. Let us lower and raise the index α with the bmean square formc Hβα and its inverse. 3.3. QUANTIFICATION 151

α β α Then any bilinear form g : C ⊗ C → R, v = sαv 7→ gβαv v , with coeﬃcients gβα, can be faithfully represented by the element

β α g = gβαγ γ ‘ ∈ Alg C. (3.12)

3.3 Quantiﬁcation

The passage from a yes-or-no property of a individual to a how-many property of an aggregate of such individuals will be called bquantificationc. A quantum theory of quantification is called a bstatisticsc because it first appeared as the problem of how to weight the terms in a statistical average over all possibilities of a quantum catenation.

3.3.1 Choosing a quantification We choose a candidate quantification for a fully quantum theory as follows. For an individual with vector space of D dimensions, Fermi statistics leads to a complex Grassmann algebra of 2D dimensions as vector space of the catenation. This is structurally unstable, however, due to the central i. The prime candidate for its stabilization is a real Clifford statistics, based on these inconclusive indications:

1. The central quantum i of the quantum theory from which we start is structurally unstable, though this leads to no bsingularitiesc. 2. The algebra of Fermi statistics is a Cliﬀord algebra.

3. Cliﬀord statistics accounts for Bose statistics as well, through bPalevc statistics. 4. The fundamental representations of the classical groups are Fermi catenations of the “atomic” representations at the terminals of their bDynkin diagramsc.

5. bFermi statisticsc is applied to produce spinor fields, and accords with the empirical bspin-statisticsc correlation (§3.4.3). 6. Random finite set theory, the prototype of exponential logics, can be constructed by iterating a real bClifford statisticsc and restricting vectors to the bstandard basisc of monomials.

7. Cliﬀord statistics is a generic variant of bFermi statisticsc and therefore structurally stable.

8. The Clifford ring of classical bgravityc is a singular limit of a Clifford algebra. Therefore attention is turned first to basic quanta with Clifford statistics (Assumption ??). 152 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

For any vector space V , I’ V := {Iv : v ∈ V } (3.13) designates the space of braced V vectors.

Deﬁnition 1 (Stratum L) The vector space of stratum L is

L S[L] := (PI’) R, (3.14) for both individuals and pluralities on that stratum, where P is the power space operation deﬁned in (??).

The vector space S[L] for any stratum L has 2L dimensions. The self-Clifford algebra S is the Clifford product of its ranks S(r): G S = S(r) (3.15) r in which the first-grade vectors of each Clifford algebra in the product anticommute with those of any other. Klein and bDeWittc assumed that Kaluza’s extra dimensions were compact, closed on themselves. This created a bcompactification problemc: What compactifies these classical dimensions? In ϑo both the internal and external groups are orthogonal groups of subspaces of the event vector space, and they have enough finite-dimensional representations whether they are compact or not. The charge spectra like the coordinate spectra are discrete and bounded merely because simple Lie algebras have finite-dimensional representations; compactness is not required. This replaces the compactification problem by the bgrowth problemc: What causes the queue to grow to macroscopic sizes in four dimensions while all other dimensions remain microscopic? As with snowflakes, graphenes, and soap bubbles, this enormous anisotropy may be a consequence of the structure and dynamical interaction of the structural elements. Per- haps — speculating freely — the four-dimensional structure of stratum 3 triggers a four- dimensional crystallization on stratum E. At first the extension problem is approached semi-empirically. Hopefully it will then become possible to approach it deductively. Simple bquantizationc eliminates non-generic bsingularitiesc like the Wronskian bsingularitiesc of bgaugec theories and the singularities of propagators on the light cone. Instead of infinite renormalization constants, simple bquantizationc has finite quantum constants. bSegal- Vilela-Mendes spacec has three new homotopy parameters and quantum constants: in the present symbols, a space-time quantum X with time units, an energy quantum E with energy units, and a large quantum number N with no units. The usual quantum of action and angular momentum is ~ = NEX. Since a non-relativistic limit is not considered it is convenient to set light-speed c = 1. 3.3. QUANTIFICATION 153

The scalar meson in Minkowski space-time and general relativity both have inﬁnite- dimensional Lie algebras whose elements depend on arbitrary functions, for example functions of time. Full quantization shrinks these Lie algebras to high but ﬁnite dimensionality. 2 Let A = Alg NR be the algebra of N × N real matrices, with dimensionality N . The bmean-square formc iHh of A has signature N. For a familiar example, it is a Minkowskian form of signature 2 on the 2 × 2 matrices, which represent the monadic vectors of S[3]. As N → ∞, therefore, the signature N ∼ o(Dim A), and in some weak or statistical sense the form approaches neutrality, signature 0, as for a Fermi algebra. Thus the normed algebras A = Alg NR interpolate between, and unify, the algebras of Dirac spin for stratum L = 3 and bFermi statisticsc for L → ∞, where with high precision ∞ ≈ 6.

3.3.2 The cumulator

Operations on one stratum have consequences on all higher strata. A one-quantum operator q of one stratum deﬁnes a cumulative many-quantum operator Σ q of the next stratum. In canonical theories, one writes the many-quantum operator as ψ†qψ, where ψ is a vector of annihilators. This must be slightly modiﬁed here to incorporate the stratum-raising operator I. Suppose that sl are generating monadics for S[L], so that a typical polyad of S[L + 1] is a polynomial s = Poly(I sl) ∈ S[L + 1] ∈ S[L + 1]. Then Σq acts on s by applying q to each sl in s in turn, and summing over l:

X d d Σq = ◦ (qsl) =: ◦ (qsl) dsl dsl l by the summation convention. This can be accomplished by replacing each of the outermost braces I in s in turn by Iq and summing all the results. This is a well-deﬁned polarization process and will be written as D Σq = (I q) ◦ I . (3.16)

D Here I is a Frechˆetdiﬀerentiator with respect to the outermost operator I in its D operand. In general x ◦ I replaces each of the outermost Is in turn by the operator x and sums. In particular Σ1 = Grade . (3.17)

The one-quantum operator q can be expressed with basis vectors em ∈ V and dual n D n n basis vectors e ∈ V , e ◦ em = δm:

m n q = em ⊗ q ne (3.18) 154 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

V V For Fermi statistics, a creator ebm : 2 → 2 is defined as the operation of left-multiplication by the basis vector em: ebm := Lem. n Dually, a Fermi annihilator eb is defined with a bGrassmann differentiatorc with respect to en and written as n n D d eb := e L = , den so that n n [ebm, eb ]− = δm (an anti-commutator). Then to form Σ q, one replaces dual basis vectors en in q by n annihilators eb and basis vectors em in q by creators ebm:

m n D Σ q = ebm ⊗ q neb = L q L . (3.19) This way of writing Σ emphasizes its basis-independence. Σ : so[L] → so[L + 1] is a Lie-algebra homomorphism:

Σ [a, b] = [Σa, Σb].

It constructs the action of a one-quantum operation on a many-quantum collection, and converts one-quantum coordinates into cumulative many-body coordinates. 2 † Σ sends each predicate projector q = q = q ∈ Endo V into a bnumber operatorc Nq := Sq ∈ Endo 2V giving the number of individuals with the predicate q in any catenation. In this way it serves as a bquantifierc; not one of the traditional four A, E, I, O, but a numerical quantifier. P and S in turn define operators Π and Σ that convert finite or infinitesimal transformations Ω, ω : S[L] → S[L] of a parent stratum to induced transformations ΠΩ, Σω : S[L + 1] → S[L + 1] of the next stratum:

H H ΠΩ = I(PΩ)I , Σω = I(SΩ)I . (3.20)

H † First the operator I = I unbraces monads, stripping them to their internal polyads. Then Ω acts on the internal polyad within the monad by the extension PΩ. Finally I rebraces the resulting polyad. Similarly for Σ. There are now two natural Lie homomorphisms from the commutator Lie algebra Grade2 S[L] of the second-grade elements sβα = [sβ, sα] ∈ S[L] to that of Grade2 S[L + 1] that must be distinguished: (1) The left-multiplications Lsβα := γβα define infinitesimal operators γβα : S[L] → S[L]. These induce transformations Σγβα : S[L + 1] → S[L + 1]; here Σ is the cumulator already defined. 3.3. QUANTIFICATION 155

(2) Another Lie homomorphism Σ2 is deﬁned by

Σ2[γβ, γα] := [LIsβ, LIsα] =: γβ α. (3.21) Evidently

∀ω1, ω2 ∈ S[L]2 : Σ2[ω1, ω2] = [Σ2ω1, Σ2ω2]. (3.22) Familiar universal, existential, and numerical quantifiers ∀, ∃, N are easily expressed in terms of Π and Σ. Np is a queue variable whose value is the number of monads with the property p in the queue. For any monadic projector p, with complement 1 − p, one defines Np := Σp; (3.23) . ∀p := P [N(1 − p) = 0], (3.24) . ∃p := 1 − P [Np = 0], (3.25) where P [...] is the projector on the subspace defined by the condition [...]. Cumulators can be defined for Fermi, Bose, and Maxwell statistics as well; it was a concept of classical physics before quantum.

3.3.3 Algebra unification Quantum mechanics has one product xy where classical mechanics had two, a commutative algebra product xy and a Poisson Bracket [x, y]P, as was emphasized by bGrginc and bPetersenc [46]. Thus canonical quantization unifies two algebras of the classical theory into one of the quantum theory. There is material for further balgebra unificationc. Space- time vector fields can be multiplied either as Clifford elements or as differential operators. The non-associative inner product v · w and Lie Bracket [v, w]L of space-time vector fields derive from these respective associative products. The full quantization performed here unifies these two algebras into one Clifford algebra on the vector space of the quantum bgravitational fieldc. A Grassmann algebra and a regular symmetric bilinear form on the vectors (first-grade elements) of the Grassmann algebra define a Clifford algebra. A central i is structurally unstable, so we begin with the real quantum theory. Then quantum theory provides a real regular bilinear form, the bprobability formc. Therefore ordinary bFermi statisticsc defines a real Clifford statistics, whose Clifford product unifies the Grassmann product of the bFermi statisticsc with the inner product of the quantum theory. This bproduct unificationc is the analogue for graded canonical quantization of the product unification already mentioned for ordinary canonical quantization.

3.3.4 Spinor spaces Cliﬀord elements of C = 2W can be isomorphically represented as linear operators on an associated vector space, a spinor space Spinor W , deﬁned up to isomorphism. That is, C =∼ (Spinor W ) ⊗ (Spinor W )D. (3.26) 156 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

Its elements, spinors, are represented by the columns of a faithful irreducible matrix representation of C. Aphoristically put, the spinor space of a bilinear space is the square root of its Clifford algebra. It is not the square root of the bilinear space itelf, however, as is sometimes said. To be sure, vectors are (represented by) bilinear forms in spinors, but so are forms of every grade: tensors, axial vectors, and all are bilinear forms in spinors. The Clifford algebra contains them all. It follows that for the duplex bilinear space W = V ⊕ V D, whose Clifford algebra is 2W = 2V ⊗ 2V D , Spinor W = Spinor(V ⊕ V D) = Grass V (3.27) up to isomorphism. Relative to quanta with vector space W = V ⊕ V D, a quantum with vector space V or D V is called a bsemiquantumc, and its vectors are called semivectors. Therefore:

Spinors are the vectors of a Fermi assembly of semiquanta.

The orthogonal group of a vector space V is generated by the grade-two elements sβα V of the Cliﬀord algebra C = 2 . When the sβα act on C by left multiplications Lsβα, C transforms as a direct sum of spinor spaces of V . Under inner automorphisms 4sβα, C transforms as a direct sum of the spaces of multivectors of every grade over V . Cliﬀ V can then be realized as an algebra of matrices whose columns are spinors of Ψ and whose rows are dual spinors of ΨD.

3.4 Fermi algebra

It is particularly easy to factor a Clifford algebra into a spinor space and dual spinor space for the special case of a bFermi algebrac, constructed as follows. There is a familiar process of bBose quantificationc, based on Bose statistics: 1. A vector space V and its dual space V D define

D 2. a canonical Lie algebra a = V ⊕ C ⊕ V which generates 3. a canonical algebra A that can be written as Alg V where

4. V = Poly(V ) is a higher-stratum vector space.

This process is ripe for iteration V → V → V → ..., but singular. There is a closely parallel quantification process, bFermi quantificationc, based on Fermi instead of Bose statistics: 1. A vector space V and its dual space V D define 3.4. FERMI ALGEBRA 157

D 2. a Fermi Lie algebra f = V ⊕ C ⊕ V which generates

3. a Fermi algebra F that can be written as Alg V where

4. V = Poly(V ) is a higher-stratum vector space.

This process is ripe for iteration V → V → V → ..., and is regular. The bFermi graded Lie algebrac f := Fermi(V ) is the graded Lie algebra whose elements constitute the vector D space f := V ⊕ C ⊕ V and obey the graded Lie product relations

∀n, n0 ∈ V D, ∀p, p0 ∈ V : [1, n]g = [1, p]g = 0, 0 0 [n, n ]g = [p, p ]g = 0, [n, p]g = p(n). (3.28)

Vectors in V D are assigned grade −1; those in V are assigned grade +1. The numbers z ∈ Z = C are assigned grade 0. The brackets are graded Lie products, symmetric if one factor has even grade, skewsymmetric if both grades are odd.

Then the bFermi algebrac F := Fermi V is the linear associative algebra of polynomials in the vectors of f modulo the Fermi relations (E:GRADEDFERMI), in which graded Lie products are now read as graded commutators. Grades in f take on only the values ±1, 0. Grades in F range over 2m + 1 values from −m to m. Finally, V = Grass V is the Grassmann algebra Poly(V ) of polynomials in the vectors of V regarded as anti-commuting variables. The dimensions of f, V , and F , respectively, are

2 Dim V + 1 < 2Dim V < 22 Dim V , (3.29) assuming Dim V ≥ 3. The above construction lacks two pieces of information vital for quantum application: There is no mention of the coefficient field of V or V The process works equally well for R, C, or for that matter the binary field 2. There is no mention of the probability form of V and none is provided for V . In both cases, however, if the information is given for V it can naturally generate the information for V . It is only necessary to provide the missing information at the start of the iteration. This suggests that the physical field is R and that the central i is a singular limit of a non-central operator i in the same way that central coordinates q, p of classical mechanics are singular limits of non-central quantum operators q,b pb. This leads to an interesting unification of i and g and is assumed in ϑo. 158 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

3.4.1 The duplex space In standard quantum theories, quantum superposition between input and output vectors never arises. They occupy separate spaces V and V D, linked only loosely by an antilinear adjoint operation H : V → V D. When a vector ψ ∈ V is multiplied by i, a dual vector D D φ ∈ V is multiplied by −i. In the bduplex spacec W := V ⊕ V , the addition ψ ⊕ φ is possible, but this sum is not projectively invariant under the usual action of i; duplex addition is not invariant under i. In ordinary experiments the suspension of quantum superposition between input and output vectors is physically reasonable even if they both belong to some larger vector space like W . The input and output processes are widely separated and generally involve interactions with different macroscopic systems of uncontrolled relative quantum phase, like a polarizer and an analyzer, so quantum superposition of ψ and φ is unfeasible. This incoherence, however, is a typical result of large numbers. For experiments of a sufficiently small scale, coherent phase relations between input and out processes might conceivably occur, of the kind described by vectors in the self-dual duplex space W = V ⊕ V D. Since the Fermi algebra over V is a real Clifford algebra over W , it loses nothing to suspend the assumption that the complex number i is in the center of the operator algebras we consider. In the following the basic quantum system, the queue, has real coefficents, Fermi statistics, no probability form and no central imaginary unit. This creates an obligation: To account not only for the apparent centrality of i in the canonical limit, but also for its global universality and its anticommutation with bWignerc −1 btime reversalc T, which maps i 7→ TiT = −1 (Section §6.1.6). Clearly a Fermi algebra over N is a bClifford algebrac, over the real neutral bilinear space

V = N ⊕ P, P = N D, ∀n ∈ N, ∀p ∈ P = N D : kn + pk := p(n). (3.30) called the bduplex spacec of N. That is, the anticommutationrelations of Fermi N are just those of the real Cliﬀord algebra Cliﬀ V :

Fermi N = Cliﬀ V = 2V . (3.31)

The bduplex spacec W , as the direct sum of complex vector spaces, inherits i from them. On the other hand the operator i : N → N, n 7→ in induces the operator i† : P → P, p 7→ −ip, resulting in a combined operator

j : W 7→ W, n 7→ in, p 7→ −ip. (3.32)

The operators i and j are both square roots of −1 and commute. Physical operators including observables are required to commute with both i and j. Thus the physical 3.4. FERMI ALGEBRA 159 operators do not form a simple algebra, because the vectors n and the dual vectors p are not supposed to have quantum superpositions. In the present work all such restrictions of quantum superposition are dropped, hop- ing that they are low-energy artifacts. Assuming that multiplication of V by i induces multiplication of V D by −i as usual, the duplex space of a complex space is a real space provided with a non-central i. The complex Fermi algebra then becomes a real Clifford algebra over a neutral space. The algebra F = Fermi N does not determine the vector space N from which it stems. In particular F is invariant under the exchange of N and P . A one-dimensional projector Ω ∈ F exists with the property that nΩ = 0 = Ωp for all n ∈ N, p ∈ P , called the vacuum projector for N. The vacuum suffices to fix the split into positive and negative vectors, creators and annihilators. Being a Clifford algebra according to (3.31), F is isomorphic to a full matrix algebra over a spinor space Ψ related to the duplex space W , though not naturally isomorphic. The Clifford algebra defines a Grassmann graded Lie algebra on the same elements, whose graded-commutative product is called the bWick productc of the Fermi algebra. The generators of Fermi N obey the anticommutation relations of the real Clifford algebra Cliff V , Fermi N = Cliff V = 2V , (3.33) where V is the real neutral bilinear space

V = N ⊕ P, P = N D, ∀n ∈ N, ∀p ∈ P = N D : kn + pk := p(n). (3.34)

Using the anticommutation relations, every Fermi product can be expressed uniquely as a bnormally ordered productc, in which all the vectors of P stand on the left of all the vectors of N = P D. Therefore

Fermi P = Grass P ⊗ Grass P D. (3.35)

It follows from (3.35) that up to isomorphism the spinors of V are the many-quantum vectors of the Fermi catenation over P :

Spinor V = Grass P. (3.36)

Brieﬂy put:

3.4.2 A spin is a queue of semiquanta. This is emphasized, for example, by Wilczek and Zee [86]. They consider the possibility that the three generations of leptons, and also of quarks, are physical elements of the Fermi assembly represented by a higher-dimensional spinor. 160 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

Here the possibility is also considered that the leptoquark generations belong to different strata of a queue. The Fermi quantification process is a general way to construct a spinor space Spinor V over any quadratic space V , and so for any simple quantum system I[V ] [16]. This is a rare example of stratum structure in present quantum theory, and so merits attention. bBrauerc and bWeylc liken their spinor construction to “bsuperquantificationc”, by which they mean what is usually called“bsecond quantizationc” and is here called bFermi quantificationc. They do not propose a physical interpretation, which would raise hard questions about the bspin-statisticsc correlation and conservation of angular momentum, but limit themselves to the mathematical problem of explicitly constructing a spinor representation. bBrauerc and bWeylc, bCartanc, and bChevalleycconstruct a spinor space in three steps [16, 18, 19]:

Neutralize. Cartan and Chevalley assume that the quadratic form Q of V is bneutralc. One may bring this about if necessary by extending from real to complex coordinates, or by replacing V by Dup V with the duplex norm k ... kDup. Split. V can then be decomposed into two isomorphic maximal null subspaces P,N ⊂ V of half the dimension of V , such that V = P ⊕ N and

∀p ∈ P, ∀n ∈ N : kn + pk = p · n. (3.37)

We designate one such arbitrary choice (improperly) by

P = Semi+ V,N = Semi− V (3.38) and call a vector in either of them a bsemivectorc. The hypothetical quanta I[P ] and I[N] are called bsemiquantac of I[V ] in this context. Possibly Cartan and Chevalley chose the letters P and N in this construction to recall the positive-energy and negative-energy spaces of Dirac electron theory.) Power. Then identify V , which we have already factored into creators and annihilators, with the V of (3.34) and form the power space of P ⊂ V , the Fermi algebra

Cliﬀ V = Fermi P = Grass P ⊗ Grass P D. (3.39)

Therefore Grass P is a spinor space of V . And Grass N is another.

P = Semi+ V,N = Semi− V, Spinor V := Grass Semi+ V. (3.40)

Abspinorc space is a vector space supporting a faithful irreducible representation of the Clifford algebra. It is also by definition the vector space of a Clifford assembly. Therefore one may just as well say that a bspin 1/2c is a Clifford assembly of semiquanta. 3.4. FERMI ALGEBRA 161

3.4.3 Spin-statistics correlation

Let h] be a probability amplitude vector in the dynamical stratum F. h] is blocalizedc if for all group elements g ∈ SUF(C) outside a small neighborhood n ⊂ SUF(C) of the identity, [hgh] = 0. (3.41) The exchange parity of is the operator X = X() representing a homotopy g(θ) ∈ SUF(C) (0 ≤ θ ≤ 1) that exchanges h] and hgh] where they occur as factors:

hg(θ)h] ∨ hg(θ)hgh] = h] ∨ hgh] for θ = 0, = [hgh] ∨ h] for θ = 1. (3.42) . X2 = 1, so X = ±1. The spin parity of , is the operator W = W () representing a continuous rotation b c . of one through 2π. W () = ±1is +1 if the spin of epsilon is even in units of ~/2 and −1 if odd. The observed bspin-statisticsc correlation is

W () = X() (3.43) for all quanta of the dynamical stratum. Since we violate this equality we require separate names for its terms. Cartan’s mathematical definition of spinors in terms of Clifford algebra, and the Brauer Weyl construction of spinors by Fermi quantification, both require further physical interpretation before they can be taken seriously as a physical model of the phenomenon of spin 1/2. The process bSemic is unnatural: there are an infinity of minimal left ideals, and Semi selects one of them. Something in nature must make this selection. Choosing one ideal out of them all breaks a symmetry. This suggests an organization on a stratum higher than that of the system. If V is the vector space of a quantum V; in the case of greatest present interest, V = S[L]. Then a Fermi assembly of V’s has vector space W = 2V . Suppose that V is half full, in the sense that a subspace N = Semi− V is full and an isomorphic subspace P = Semi+ V is empty. Then V can be represented as a duplex space V = Dup N. This defines a duplex norm HN on V , in which all the vectors of N and P are null vectors. Let C be the Clifford algebra over V with respect to this duplex norm. With respect to left multiplication, the elements of N act on C as creation operators and those of P act as annihilation operators. Then the subspace N generates a Clifford sub algebra 2N that is a left ideal of C. For the elements of N adjoin factors in N, and the elements of P remove factors in N, N V 1 and both map N → N. Define a “vacuum” vector ΠN ∈ 2 ⊂ 2 , of grade 2 Dim V , as 162 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC a top vector of N.ΠN is an arbitrarily ordered Grassmann product of all the vectors in a minimal basis for N ⊂ V , analogous to the negative-energy Dirac sea. Let ψ ∈ P and let ψ be the corresponding first-grade vector in 2N . Then any swap ω ∈ so(V ) acts on ψ as the inner automorphism

Σω ψ = [γω, ψ]. (3.44)

V induced by spin operator γω ∈ Grade2 2 .

Assertion 1 The many-quantum vector ψ ΠN is a spinor of SO(N).

Argument: 2N is a Grassmann algebra, and Πn is its top element. Therefore, ∀n ∈ 2N : nψ = 0. Therefore nγω = 0. Therefore the inner automorphism generated by γω reduces to left-multiplication by γω, a spinor transformation. The many-quantum vector ψ ∨ ΠN represents one spin in the vacuum. This makes spin 1/2 anomalous in the same sense that spin 1/3 is anomalous in anyon theory. There is a larger quantum system whose swaps are whole numbers, having the bClifford algebrac Cliff V as vector space. This vector space factors into two dual subsystems as Cliff V = Cliff N ⊗Cliff P . When we observe a spin 1/2, we are observing one of these subsystems and omitting a vaccum-full of the other because it belongs to the metasystem. bmetasystemc. If W = Dup V is a Minkowskian space then a bspin-statisticsc correlation would seem to imply that a hypothetical quantum entity b with vector space V ⊂ W would be a boson; yet to make spinors the cited authors forn the Fermi algebra Fermi V of integer-spin vectors of the boson b, seemingly violating the bspin-statisticsc relation. Similarly, the components of the Dirac γµ have negative exchange parity and positive spin parity. Evidently the bspin-statisticsc correlation for quanta does not apply to lower strata; and indeed Dirac spin matrices are not particle vectors. If this famous spinor construction has physical meaning — and let us suppose that it does if only for the sake of a reductio — then it omits some important physical element that absorbs the right action of a Clifford element S(L) associated with a Lorentz transformation L, leaving only the left action. As mathematicians we may evoke minimal left ideals at will; for the physical interpretation as vector spaces, we need a physical agent to conserve angular momentum or swap-invariance. The vacuum, the ambient organized stratum-F queue, often serves as this agent. Sup- pose that a one-dimensional vacuum projector Ω = Ω2 = Ω† is invariant under all the antisymmetric generators S(L) = −S†(L):

[S(L), Ω] = 0. (3.45)

Then S(L)Ω = ΩS(L) = ΩS(L)Ω = −ΩS(L)Ω = 0 (3.46) 3.4. FERMI ALGEBRA 163 and δLΨΩ = S(L)ΨΩ. (3.47)

That is, the natural spinors are not the multivectors of PoV themselves but the elements of the orbit PoV’ Ω of Ω under these multivectors. Now each part of the spinor construction can have a physical interpretation.

1. L is a Lorentz transformation that relates two experimenters in relative motion (say) assigning vectors to some quantum entity b of bstratumc 2 with vector in Vb = V ⊂ W . 2. S(L) is the induced action of L on the Cliﬀord catenation of b’s, with vector space Ψ = PV generated by creation operators.

3. ΨΩ is the orbit of a suitable isotropic vacuum Ω under these creation operations.

The real spinor space S = 4R of a Minkowski√ tangent space M is a square root of the Clifford algebra C = PM = S ⊗ SD, 4 = 16; not merely a square root of space-time as one sometimes hears. The vacuum takes that square root in step 3, where the dimension falls from 16 to 4. In the classic constructions cited, especially Chevalley’s, Ω represents a full Dirac sea. If this standard theory of spinors is to be taken seriously, as I attempt here, then electron spin 1/2 is anomalous like anyon spin 1/3, dependent on an ambient organization. Spin is a relative angular momentum of a Fermi catenation of several even entities relative to a coherently organized background Ω of many such entities, in an infinite limit. In the present project, this means that although we build with Fermi catenation, we should expect to encounter no spinors or spin 1/2 entities below the dynamical stratum, where vacuum organization can occur. And the spinors arising there will be Fermi catenations of simpler quantum entities describing “subspins.” For Minkowski space-time, spinors of 4 components describe Fermi catenations of 0, 1, or 2 identical subspins: 4 = 1 + 2 + 1. In a space-time of 6 dimensions, a spinor of 8 components describes 0, 1, 2, or 3 subspins: 8 = 1 + 3 + 3 + 1. Schematically √ √ Spinor V = Cliff V = 2V . (3.48)

bClifford algebracs can be represented as matrix algebras, and then the vectors they act on are called spinors: A bspinorc space of a bClifford algebrac C is a module σ provided with an irreducible isomorphic representation of C in Endo σ. The Clifford algebra associated with the quadratic form Q of full signature (n+, n−) is isomorphic to a matrix algebra with matrix elements in a ring R(σ) depending only on the signature σ := n+ − n− mod 8 according to the bSpinorial Clockc [17] of Figure 3.1. The matrices have size 2D ⊗ 2D/d, where D = Dim V and d = Dim R(σ) are real dimensions. Also, when the dimension of V increments by 1 the real dimension of the bClifford algebrac 164 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

R C 0 2R 7 6 1 '$ H 6 2 R 5 3 2 4 H&%C H

Figure 3.1: The Spinorial Clock [17] Find the signature σ mod 8 of the quadratic form Q of a vector space V on the inner circle. Next to it on the outer circle, read the coeﬃcient ring R(σ) of the spinors of Q.

V 2 doubles, starting from V dimension 0 and spinorial dimension 1 at σ = n+ = n− = 0. This leads to the tabulation of the spinor spaces of the real Cliﬀord algebras in Table 3.1, based on the Spinorial Chessboard of Budinich and Trautman [17].

0 1 2 3 4 5 6 7 ... (n+) 0 RR2 2R 2C 2H 2H2 4H 8C ... 1 C 2R 2R2 4R 4C 4H 4H2 8H ... 2 HC2 4R 4R2 8R 8C 8H 8H2 ... 3 H2 2H 4C 8R 8R2 16R 16C 16H ... 4 2H 2H2 4H 8C 16R 16R2 32R 32C ... 5 4C 4H 4H2 8H 16C 32R 32R2 64R ... 6 8R 8C 8H 8H2 16H 32C 64R 64R2 ... 7 8R2 16R 16C 16H 16H2 32H 64C 128R ...... (n−)

Table 3.1: PERIODIC TABLE OF THE SPINORS. The dimension and coeﬃcient ring of the spinor space for the real bCliﬀord algebrac over (n+R n−R). R2 and H2 are the rings of real and quaternionic diagonal 2 × 2 matrices.

3.5 Cliﬀord statistics

An individual with bvectorc space V is said to have bClifford statisticsc if its quantification produces a system with vector space Cliff V = 2V . Such individuals may be called 3.5. CLIFFORD STATISTICS 165

bcliffordonsc but they are also fermions and obey an exclusion principle. Clifford statistics has the Wilczek-Zee bfamily propertyc [86]: When the orthogonal group SO(W ) is reduced to a subgroup SO(V ), V ⊂ W , with dimensional difference δ := Dim W −Dim V , then the Clifford representation of SO(W ) reduces to a family of 2δ copies of the Clifford representation of SO(V ). If these copies are the families of quarks or leptons, then since 3 is not a power of 2, this suggests at least a fourth family. In the present work the conjecture is natural that the generations of leptoquarks are generated by I acting on some stratum below stratum F. This is not inconsistent with the Wilczek-Zee hypothesis, and might be a model for it.

3.5.1 Spin-statistics anomaly

Iterating bClifford statisticsc conspicuously violates statististics conservation. The vectors of each bstratumc are polynomials in the bracings of the vectors of the the previous stratum, subject to the Clifford Clause. An element of even grade has a bracing of odd grade 1, as though an even number of fermions could form a fermion. On the other hand, Clifford quantification conserves angular momentum. This flagrantly violates conservation of statistics and any bspin-statisticsc correlation across strata. Moreover a similarly unnatural re-grading happens in Dirac’s theory of bspinc, which works well, and which it is supposed here is an earlier iteration of the same Fermi statistics. Dirac formed a bClifford algebrac over the Minkowski space-time tangent space, thereby assigning anticommutation (Fermi) relations to the four tangent vectors γµ which have spins 0 and 1. The conclusion is that the bspin-statisticsc correlation that works on stratum E does not apply to stratum C. This suggests that the empirical spin-statistics relation results from organization. The strong dependence of the spin-statistics correlation on space-time continuity also suggests this. For skyrmions, or topological solitons in a tensor field subject to non-linear constraints, spin and statistics are correlated because a rotation and a pair exchange are connected by a homotopy. No such construction is possible on lower strata. It is a concept of the singular limit. µ Moreover the Dirac spin operators γ transform dually to momentum generators pµ. In a fully quantized theory like ϑo the event coordinates and momenta pµ become singular ν6 limits of swaps like ω(ν5, ω of the so(dC) of a deeper stratum C. C = 3 is the this most economical possibility and is provisionally chosen here. As the Dirac Clifford algebra shows, the usual bspin-statisticsc connection deos not apply to deeper strata like C, since it concerns physical quanta, and these tangent vectors are not vectors of quanta. The standard theory keeps space-time and quanta in separate strata so that these spin-statistics anomalies are harmless. Here I bridges strata and such anomalies become the norm. The spin-statistics correlation at the many-particle stratum F now seems anomalous and requires special handling. The usual arguments for the spin-statistics correlation based on continuity or analyticity work in a singular limit. They must have correspondents in a fully quantum theory or the 166 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC theory must be corrected or abandoned.

In the queue theory ϑo spin 1/2 and the bspin-statisticsc correlation are the true anomalies, existing only at stratum F and only as a result of the special organization of the vacuum. This returns to early days of quantum theory when spin 1/2 was still considered anomalous, but today there is also the example of anomalous anyonic spin 1/3 [87].

3.6 Measurement

Quantum theories swallow their parent classical theories alive (as Teller put it). Classical theories continue to work well for some experiments in the quantum universe. To see this, since experiments begin and end with measurements, it is important to recognize that the physical processes called measurements in classical physics are exceedingly different from those called measurements in quantum physics; and yet the quantum theory, if it to be comprehensive, must be able to represent both processes. When quantum theory examines a measurement closely it sees an interaction. The analysis had to wait until now because the linear logic could not analyze an interaction between two systems, but the polynomial logic can. The correspondence principle deals with the limit ~ → 0. If the classical measurement process is to be represented physically within a quantum theory, however, then the physical value of ~ must be used, not the singular limit. One difference concerns degree of isolation. Typically the macroscopic system under study is interacting and exchanging quanta with the extrasystem during the entire experiment. The actual elements at any one time are uncertain. One may describe such a process by a statistical operator on a many-quantum vector space of the kind used in quantum field theory, which allows the number of elements to be a variable. A second difference concerns the strength of the measurement interaction. An actual macroscopic measuring process generally has a much smaller effect on the system than the input-output processes represented by vectors in quantum theory, one that is usually neglected altogether in a classical theory. A visual inspection, for example, couples directly only to the tiny fraction of atoms near the surface of the macroscopic system, its skin so to speak. Such a gentle process may reasonably be called an observation, in the sense that the observer does not intervene significantly in the development of the system observed. Then no distinction need be made between input and output classical measurement processes, since what is going on may be a continuing process. In contrast N input process represented by a vector erases the prehistory of the system and reduces the system entropy to 0. One may represent an observation process by one member of a parametric family of probability operators ρ(y), where y = {yk} designates a family of parameters, much smaller than the family of all possible probability operators. One then uses the observation results to fix the parameter set y. It is supposed here that the parameters y are expectation values 3.6. MEASUREMENT 167 of chosen macroscopic variables collectively designated by Y :

yk = Tr ρ(y)Y k. (3.49)

The method of coherent states [49] is one way to describe classical behavior of quantum systems. Coherent states are called coherent vectors here for the sake of consistency. They form a parametric family of vectors hy with a set y of real parameters. For coherent vectors, hy is bcompletec: X hααh = h1h, (3.50) α but not orthonormal: βhalpha 6= δαβ, (3.51) and bstrongly continuousc in y:

∀y0 : y → y0 ⊂ khy − hy0k → 0. (3.52)

The coherent probability operator hyyh (for any value of y) is then a one-dimensional projector, of entropy 0; here this is replaced by the statistical operator ρ(y) of greater entropy. One may assume that the classical observation operators too are a spectral family of projections X ρ(y) = h1h, ρ(y)2 = ρ(y) = ρ†(y), (3.53) y and strongly continuous in y, but not orthogonal:

0 ρ(y)ρ(y ) 6= ρ(y)δyy0 . (3.54)

In applying coherent vectors, the approximation is made that the statistical development carries any coherent probability operator hyyh into another operator hy0 y0h of the same family, with a possibly diﬀerent parameter value y0. An observation then selects one of the probability operators hy yh. The dynamics is then approximated by a time-development of the parameter y = y(t). These vectors are coherent in the sense that the wave packet hy coheres — sticks together — during the development. This classical use of probability operators diﬀers from the quantum use. The coherent- vector probability operator hααh, since it expresses maximal information and 0 entropy, does not accurately describe a physical macroscopic input-output process, and cannot be determined by inspecting the input-output equipment, as in the usual quantum kinematics. It describes the result of statistical observation, as in classical statistical mechanics. A more general macroscopic probability operator ρ(y), however, may be a microscopic physical description of a macroscopic input-output process. [Do080318] To be continued 168 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

Coherent vectors are often used to approximate a canonical quantum theory by a canonical classical theory. Coherent spin vectors can similarly be used to approximate a spin quantum theory by a canonical quantum theory. [Do: Extend the method of coherent vectors [49] to approximate the canonical quantum Lie groups of canonical quantum physics within the orthogonal group of S.] Chapter 4

Exponential quantum logics which concerns iterated quantiﬁcation.

The maximal descriptions of an isolated physical system are supposed to correspond to rays in a vector space S associated with the system. For an assembly, S has a product representing the union of assemblies. Just as iterated addition is multiplication, iterated multiplication is exponentiation. To form polyads of polyads, however, one ﬁrst converts a polyad into a monad. The bracing operator I does this. An old example of this process is considered next before taking up the general case.

4.1 Spinors

There are already traces of iterated bFermi statisticsc in present physics that are worth exploring further. The bstandard modelc applies bFermi statisticsc only to spinorial fields, of spin 1/2, not to fields of integer spin. But spinorial fields are themselves Fermi algebras over deeper spaces [16, 86]. Suppose that the one-particle vector space is

Ψ1 = Spinor V0 = Grass Semi V0 (4.1) for some quadratic vector space V0. Then the many-particle Fermi vector space is

Ψ2 := Grass Grass Semi V0 ←Cliff(Cliff Semi V0), (4.2) a singular limit of iterated Clifford statistics, which we may assume is the more accurate theory. We may take Semi V0 to be a Minkowski tangent space M, disregarding functional dependence on space-time coordinates. But M is itself presented as a thrice-Clifford algebra M = Cliff3{0} in (4.22). Combined with the previous result (4.2) this gives a five-fold iteration Cliff5. We suggest physical interpretations for these lower strata in (4.33).

169 170 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

A quantum set or bqueuec in a broad sense is a quantum system whose vector space is a Clifford algebra with specified interpretations for the algebra operations. Where the quantum element I[V ] has bvectorc space V , the queue of I[V ]’s has the vector space 2V = Fermi V. (4.3) This queue theory includes classical set theory as a commutative reduction (§2.2.3). Classical set theory can also be reconstructed as a stratified Clifford statistics with the binary coefficient field Z2 = {0, 1} instead of R but the binary Clifford algebra does not handle classical probability theory as efficiently as the real Clifford algebra and will not be used.

Assertion 2 When V = S the classical power set 2S is a commutative reduction of the quantum power set with vector space 2V .

Argument: Let {hn : n = 1,...,N} be a basis for V . The operators h1h, h2h, ..., hNh form a maximal commuting set of independent operators in 2V and generate a Cartan (maximal commutative) subalgebra of 2V , isomorphic to the set algebra generated by the classical objects {1},..., {N}. To ease the conceptual transition from classical sets to quantum sets let us begin with a back-formation expressing classical set theory as a commutative reduction of Cliﬀord statistics.

4.1.1 Baugh numbers The sets used here are finite and form an infinite set S generated by forming finite products of what has already been formed, and by bracing I, which turns each monomial into a first- grade monomial. One designates the monomial with no factors by 1. It is the multiplicative identity. s Recall that if s is any set, 2 , the bpower setc of s, is the set of all the subsets of s. and 2I[s] is the random object whose values are the subsets of s. If we peel the outermost braces from a set we expose the elements of the set. If we repeat the process indefinitely, the sets thus exposed are called the bancestorsc of the set. It is helpful to think of a finite set in the way introduced by bBaughc, as a (natural) number in a positional notation with an expanding base [7]. The usual fixed-base positional expansion for a number N s a weighted sum with coefficients cn defined by the number N and weights bn defined by the base alone: ν X bn+1 N = (N N ...N ) = N b , 0 ≤ N < . (4.4) 1 2 ν b n n n b n=0 n

In the bb-ary expansionc of the numbers,

b0 = 1, bn+1 = bbn, 0 ≤ cn < b. (4.5) 4.1. SPINORS 171

In the bBaugh expansionc, the coeﬃcient Nn in any place n = 0, 1,... ranges over the numbers that can be expressed with the previous places alone, and the weight bn is the smallest number that cannot be so expressed:

bn b0 = 1, bn+1 = 2 , 0 ≤ cn < bn. (4.6)

The Baugh number B(s) of a set s is then determined by the rules that for all disjoint sets u, v:

B(u ∨ v) = B(u) + B(v), B B(Iu) = 2 (u). (4.7)

The monads of Table 1.1 generate the polyads in the vector space S. Let sb designate the polyad of Baugh number b. Then s0 = 1; s1, s2, s3, and s4 can be identiﬁed with the Dirac gamma matrices γ1, γ2, γ3, and γ4, with γ4 being timelike; and what is usually designated by γ5 is

γ5 := γ4γ3γ2γ1 = s4s3s2s1 = s4+3+2+1 = s10. (4.8)

The set S has no symmetries. No two sets are the same. This is useful, because nature has no symmetries; an exact symmetry would be unobservable, as Wigner pointed out [84]. The moon disturbs the roundness of the Earth measurably; it disturbs the roundness of a terrestrial hydrogen atom in its ground energy stratum too, though immeasurably less. Symmetries in physics result from ignoring some asymmetries, especially of the metasystem. They will be represented by corresponding symmetries in queue theory resulting from ignoring some remote ancestors of the queue, terms at the tail of the Baugh expansion.

4.1.2 Random sets

A brandom setc I[S] is a random object with state space S sampled in Table 1.1. Then its probability distributions form the simplex in S whose edges are the coordinate vectors sb ∈ S. Thus the state space S of a random set is a set of sets. This is a commutative reduction of a quantum concept: A b queuec is a quantum entity whose vector space is a (ﬁnite) stratum S[L] ⊂ S.

4.1.3 Quantum cells The general element of 2V is also an element of the Grassmann algebra over V . In any frame for V , an element of 2V expanded in the induced frame for 2V has the form given by Chevalley for a chain associated with a simplicial complex [20]. The n basis elements are the vertices, and the monomials are the simplices. In the Chevalley theory, however, the vertices are absolute, while in queue theory the vertices are relative to a basis, subject 172 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS to quantum superposition, and can be transformed into other vertices by arbitrary linear transformations in GL(n). Such a chain can be regarded as a vector of a bqueuec, a possible input process for it. The monomials are basic vectors for the queue. The Grassmann product abc . . . d of any sequence of vectors a, b, c, . . . d ∈ V cam also be regarded as a volume element with edges a, b, c, . . . d. This is the object that Grassmann and Cliﬀord called an element of extension. Its measure is the square root of the Cliﬀord norm of the Grassmann product, µ(abc...d) := [Q(a ∨ b ∨ c ∨ ... ∨ d)]1/2. (4.9)

It is convenient sometimes to speak of the bClifford algebrac elements as vectors of quantum cells rather than queues because µ(abc...d) is the squared measure of the cell, not the set, defined by the edges a, b, c, ..., d. Subsets of {a, b, c, . . . d} define subcells or faces of the cell abc . . . d. To recover the classical set construct from the q, one fixes an orthonormal basis in V , forms the commutative set S of the projections on the basis vectors, and takes the states of I [S] to be the elements of S. To iterate Cliff we require an operation that associates a high-grade element of a bClifford algebrac into a first-grade element of the next-stratum Clifford algebra. This is the bbracec I of the next section.

4.1.4 Bracing

The bbracec I : x 7→ {x} = x (4.10) forms monads and is used to express the idea of association, or of one thing being “at” another. In bﬁeldc physics today, one puts a ﬁeld variable f, typically an operator, at a point x by forming the pair IA ∪ Ix. In a physics of atoms, one might describe an atom A at a space-time event x by forming the pair

{A, x} := {A} ∪ {x} =: IA ∪ Ix. (4.11) Every physical theory uses I, usually tacitly. Often it is used to couple metasystemic and systemic operations. Time, for example, is measured in a clock of the bmetasystemc and is then used to control measurements on the system, and q(t) associates the metasystemic variables t and q into the system variable q(t). If the strata of physics are to communicate, some such operation must be possible. Statistors for a queue with Fermi statistics belong to a self-Grassmann algebra S defined constructively thus: First one defines a bracing operator in general. Let V be any normed vector space. Let W be the set of all unit classes {v} for v ∈ V . The bbracec operator I : V ↔ W is the unit-set forming operation Ix = {x} = x (4.12) 4.1. SPINORS 173 modulo linearity, the identification

(∀x, y ∈ V )(∀a, b ∈ R): I(ax + by) ≡ aI(x) + bI(y), (4.13)

This makes the set W = I’ V of braced vectors a linear space. I is also designated by a bvinculumc as in I(abc) =: abc (4.14) to emphasize that it is an associator. In the quantum theory it is also necessary to deﬁne the probability form H for W . H D This can be done by deﬁning the adjoint operator I := I . Two physically equivalent possibilities have been considered: H † (1) I is an isometry S → S. Then I := (I) is the left inverse of I,

H H 2 I I = 1, II = Grade1 = (Grade1) . (4.15)

Grade1 is the projector on the ﬁrst-grade subspace S1 ⊂ S, the range of I. D (2) I and I are related as generalized canonical boson creator and annihilator:

D D I I − II = 1. (4.16) √ 0 If I satisfies (1) then I := I Γ satisfies (2). The matrix elements for I differ between (1) and (2) only by factors depending on the strata. The choice is merely a choice of notation. Nevertheless such differences have practical consequences when theories are being constructed; a coupling may look natural with one formulation and unnatural with another. In quantum theory it seems more natural to assume canonical commutation relations D as in (2) between I and I than to assume that I is an isometry as in (1). The canonical H D brace relations (E:CANONICALBRACE) are adopted here. Then I := I has the formal properties of a differentiator with respect to I. I adds a bar to the stack, I multiplies by the number of bars on top of the stack and erases one of them. It annuls an amonadic operand, since the number of top bars is then 0. The analogue of the boson number operator is

D h∆Rankh := II , (4.17) called the brank jumpc, which counts consecutive bars in its operand, downward from the top to the ﬁrst non-monad. 0 Statistors hp ∈ S of the form hp = hIhp have grade 1 and are called bmonadicc. Statistors orthogonal to all monadic vectors are called bamonadicc and form a vector space designated by S6=1. They have the deﬁning property

H I hu = 0. (4.18)

H I is 0 on S6=1. ∆Rank counts ranks above the amonadic, in the sense that: 174 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

n Assertion 3 If a is amonadic and hb = hI ha, then h∆Rankhb = hb · n, and hb has norm

bhb = n! aha. (4.19)

Argument: Omitted. The canonical relations for creators and annihilators lead to a linear ladder of vectors. The canonical relations for bracing and de-bracing lead to a fractal network of vectors, with a replica of the entire network growing from every node in the network. About notation: The most familiar Clifford algebra in physics is that of the γµ of the Dirac equation. These operate on the spinorial vectors of the hypothetical single electron. The elements of the Clifford algebra S are used here as vectors of a quantum system, not as operators on vectors. The two kinds of algebra can be isomorphic and can be related but they cannot be identified. To reduce confusion, generic vectors in S will usually have the root symbol s or ψ, while generic operators on S will usually have root symbols γ and ω. S is partitioned into nesting strata ... ⊂ S[L] ⊂ S[L + 1] ⊂ ... (L ∈ N) defined by

L S[L] = [PI’] R, (4.20)

Here R serves as a trivial singlet Clifford algebra S[0] of stratum 0 to begin the construction. A more trivial Clifford algebra S[−1] of one element 0 is stratum −1. The lower strata of S are shown in Table 1.1. To show the close connection between Clifford algebra and set theory one may express the usual concept of a finite set as a commutative reduction Sc of the bstratified Clifford algebracS: Let the classical state space Sc ⊂ S be the set of vectors generated from R ⊂ S by I and × alone. The brandom setc (understood to be finite) is I [S], where S is the stratified self-power set. Sometimes it is useful to extend the term quantum to systems whose vectors do not fill out a single vector space but a union of several vector spaces having only their origins in common. This is a way of partially suspending coherent superposition but still allowing bincoherentc or classical superposition, which is mixing. ∼ Since the set of rays in Sc is isomorphic to the self-power set S,I[S] = I[Sc] is also, in this extended use of the term quantum, the quantum object whose vector space is Sc. The predicates of I [S] are those of S that are diagonal in any basis B ⊂ Sc. The transformations of I [S] are in 1-1 correspondence with the operators on S represented in one such basis by matrices with elements 0 and 1 and with only one non-zero matrix element in each column.

4.1.5 Critique of the brace operation The quantum brace operation resembles the pre-quantum one that is still in general use, in keeping with the principle of least change. But the pre-quantum brace is loaded with assumptions that contradict the core ideas of quantum physics. 4.1. SPINORS 175

In the first place, set theory is dedicated to a creation from nothing; I permits all its sets to be constructed from the empty set. This seems to be as much a reenactment of a certain theological scenario as a representation of known physical processes. In quantum physics the creation operators represent multiplication by vectors, which in turn represent input from the metasystem, not from nothing. In particle experiments, every particle creation is balanced by annihilations so that what happens is transmutation, not creation from nothing. Processes that resemble a creation from nothing in physics might conceivably go on in the regions represented as singularities of the gravitational field in present cosmology, not because there is any evidence for them, but because we have too little evidence to exclude them. It is absurd to take a process that has never been seen as the foundation for all physics, unless there is no alternative. It seems more likely that all the processes described by bracing are actually results of binding, than the converse. In the second place, the bracing operation of set theory accepts sets of arbitrarily high grade and delivers a set of grade 1. It is therefore highly inhomogeneous, and reduces to operations that accept sets of a given arbitrarily high grade and delivers a set of grade 1. The quantum brace I, correspondingly, accepts vectors of arbitrarily high grade and produces one of grade 1. This reduces into an infinity of irreducible operators that each transforms only vectors of one grade, annulling all other grades. There is no evidence at the quantum level of resolution either for such highly reducible operations, or for irreducible operations producing such high grade change. Finally (for now), mathematical set theory does not undertake to represent the metasystem, including the mathematician, as a set. Perhaps cognitive science attempts to model processes of the mathematician; but cognitive science is not considered to be mathematics. The quantum field physicist, however, ordinarily seeks a physical model that works for any system under study, including selected portions of what served as metasystem for a previous study. The processes of mathematics need not include the life-processes of the mathematician, but the processes of physics must include the life-processes of the physicist. There is no reason to doubt that the standard model has this kind of self-consistency; its regularization can also have it, therefore. If one were to give up the model of classical set theory entirely and replace all bracing by binding, the resulting theory would use only one stratum. Then to account for a dynamics vector space with 2N dimensions, an event vector space would need N dimensions, and N would also be about the number of events in the history of the cosmos. The idea that such a large number is a datum rather than a result of internal combinatorial structure is uncomfortable for someone who believes in a fundamental theory without arbitrary constants, or in design by an intelligence that cannot remember many digits; but it is in keeping with the empirical principle. Matter indeed exhibits a modular structure of several strata: the electron, the atom, the molecule, and so on up to the cosmos, with the human being near the middle. But not long ago a physicist as advanced as Mach regarded such structure as in principle unverifiable, and insisted on a continuum model of matter. Perhaps we are in a similar 176 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS position in regard to space-time today. If the analogy between space-time and matter is at all valid, then it is significant that none of the material strata is constructed from lower ones by bracing today. The stratified structure of matter, or of nature above the space-time stratum, results from binding with forces of various ranges, resulting from the exchange of various quanta, not from bracing. Two or three strata seem sufficient for this. On the other hand, the theory of binding rests on the lower stratum of space-time, which is assembled by set theoretic methods. If space-time itself is crystalline, its structure cannot result from interactions like binding that require a space-time substrate. To account for so many interrelated events seems to call for combinatory processes as well as dynamical ones. So does the transition from a one-quantum theory to a many-quantum theory. In sum, it seems impractical to entirely eliminate combinatory processes in favor of dynamical ones. The strategy adopted here is the opposite, combined with structural stabilization and stratification.

4.1.6 Queues

The (finite) queues form a stratified structure in which higher strata are formed from lower by bIc, b∨c (union) and quantum superposition b+c. The rank of a set in this hierarchy is the number of nested bracings in its construction. Stratum L is the direct sum of all ranks r ≤ L. For example, two elements a, b ∈ S[L], the Clifford algebra of stratum L, may have grade 2 in S[L], and the generators Ia, I ∈ S[L + 1] have grade 1. The bqueuec is the quantum system I[S]. The first few strata of the queue are shown in Table 1.1. The Clifford algebra S[0] of stratum 0 is R as a full matrix algebra of 1×1 real matrices. The Clifford algebra S[1] is a full matrix algebra of 2 × 2 real matrices. The Clifford algebra S[2] is a full matrix algebra of 4 × 4 matrices over R. The Clifford algebra of every stratum L > 3 is a full matrix algebra over R of 2L-square matrices, generated by the 2L anticommuting elements proper to stratum L and all lower bstratumcs. Thus S[4] has 16 generators, all shown. S[5] has 65536˜ generators, 30 shown. The SO[L] form a nested sequence of broken groups:

SO(1) ⊂ SO(2) ⊂ SO(3, 1) ⊂ SO(10, 6) ⊂ .... (4.21)

This vector space S is infinite-dimensional. Free use of S and operators on S would allow infinities into the theory. We therefore restrict ourselves to some finite stratum S[L]. L = 6 serves for the present initial study. For any quantum system I[V ], the queue on I[V ], written as 2I[V ], is the quantum system with vector space 2V := Cliff V . 4.2. SIMPLIFYING QUANTIZATION 177

When we wish to emphasize the Cliﬀord algebraic structure of these sets, we call them cells, with vertices of I[V ]. The Cliﬀord norm gives squared Euclidean measures for cells (rectangular parallelepipeds).

We must ﬁnd bclassical logicsc within the quantum logics, since they work sometimes. Classical sets can be imbedded within queues using the following concept of classical frame: A classical frame C ⊂ S is a basis of S that includes 1 and is closed up to sign under × and I. C is unique up to the signs of its elements. To reconstruct the classical set we introduce a superselection law: The classical set variables are those that are represented by diagonal operators in a classical frame. The classical random set is the random variable object having the classical variables as its variables. Then the classical random set is I [C]. The quantum theory of probability for the queue reduces to the classical theory of probability for the random classical set.

The queue produced by L recursions is called the queue I[S[L]] of bstratumc L. We will use only low strata, L ≤ 6, in the applications that follow. We define the 0-stratum queue by its vector space, the one-element Clifford algebra 0 S = {0}. We then iterate the functor Cliff I. 1 0 Rank 1 of queue algebra has vector space S := PS = R, since the product of no factors is 1. The Clifford norm of 1 is 1. 2 Rank 2 has as vector space the two-dimensional Clifford algebra S = 2R with positive definite Clifford norm. Rank 3 is four-dimensional and has a Minkowskian signature, since the product of two anti-commuting monadics of positive square is a dyadic of negative square. As a bilinear space, Cliff3{0} = M, (4.22) the Minkowskian bilinear space. All higher stratum s also have indefinite norms. b c √ The signature is the square root of the√ dimension, σ = D, for every stratum L ≥ 3. This follows from the fact that there are D more symmetric basis matrices than skew- symmetric, namely the diagonal ones. Thus stratum 4 has dimension 4 and signature 2, like Minkowski space-time, and stratum L as L → ∞ has dimension approaching ∞ and signature negligible compare to the dimension, σ/D → 0. For the fermionic algebra of annihilators and creators the signature is 0; in this sense Clifford becomes Fermi in the limit of infinity dimensions.

4.2 Simplifying quantization

The heuristic process of simpliﬁcation by quantization might consist of the following steps: 178 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

4.2.1 Choose a simple Lie algebra adjacent to each bcompoundc Lie algebra of the singular theory. This generally requires appending extra variables: variables appended to the generators of a kinematical Lie algebra so that it has a simple Lie algebra in its inﬁnitesimal neighborhood. A Lie algebra has no simple one nearby if its dimension is not that of a simple Lie algebra. In such cases one can adjust the dimension by adding variables, but must then posit organizations to freeze out these variables in present experiments. One could stay closer to present experiment by eschewing such variables and accepting a nearby semisimple Lie algebra. This amounts to suspending quantum superposition at some low bstratumc of catenation, indicating a residual frozen organization. Requiring simplicity forces a resolution of such organizations into quantum elements.

4.2.2 Choose a vacuum organization to freeze any extra variables to constants, like aligned spins in a ferromagnet. One may be able to correlate some structural instabilities of present-day physical theory with vacuum organizations already posited for the bstandard modelc and inﬂationary cosmology.

4.2.3 Choose a faithful irreducible representation (FIR) of the semisimple Lie algebra. Generally there are infinitely many. After these choices the simplified theory is finite, since all divergent integrals have been replaced by finite traces. Canonical quantization, based on the Poisson Bracket, quantizes only the dynamical variables of one stratum (E or F), leaving the infrastructure unanalyzed, classical, and structurally unstable. Canonical quantization thus disrupts some inter-stratum cell relations. To preserve relations between two strata, one must quantize both or neither. Full quantization completes quantization on the dynamical stratum and extends it to the space-time stratum. It can therefore preserve inter-stratum structural relations. The bBohrc correspondence principle gives the action quantum ~ a central role: It requires quantum predictions to approach classical as ~ → 0; which means, more explicitly put, for a sequence of experiments with action scales A/~ → ∞ . Full quantization is easier to rationalize than canonical quantization. Since measurement always has margins of error, it is reasonable to require a theory supposedly founded on measurement to be insensitive to small errors; to be structurally stable. A structurally unstable theory expresses a faith in events of probability 0 that goes beyond experiment. In addition it leads to dynamical instability and infinities. The simple algebras of fully quantum theories, on the contrary, are structurally stable and have complete sets of finite-dimensional representations, which give finite predictions for all observables. We tentatively restrict ourselves to subsets of S to insure finiteness and reduce the number of possibilities. 4.2. SIMPLIFYING QUANTIZATION 179

We sometimes use the tilde xe and the circumflex xb to designate a skew-symmetric and a symmetric operator respectively. If Xp is a simplification path in the space of structural tensors, the Xp for different values of the parameter p are isomorphic to each other and not to X0. One often speaks elliptically of a path of Lie algebras ap instead of products Xp. The historic simplification parameter is lightspeed c, the parameter for the path from the simple Lorentz Lie algebra to its singular limit, the bcompoundc Galileo Lie algebra [47]. Simplifying quantization is an inverse of a contraction from a simple Lie algebra to a singular limit. It generalizes and extends canonical quantization to strata below the dynamical, and replaces the canonical Lie product X0 by some other Lie product Xp with

d X0 = Xp . (4.23) dp p→0

Flato (1977) formulated a concept of bdeformation quantizationc that does not take structural stability and stratification into account but retains a structurally unstable classical manifold with central coordinates [9], [41]. A singular limit is usually called a “contraction” of the regular Lie algebra [47] although it is infinite where the regular one is finite. The term originates in the fact that the singular and generic commutation relations agree well only in a neighborhood of the identity; but each approximates the other and so could be considered a contraction of the other if that were the intended sense. Implicit in the name “contraction”, therefore, is a priority of the singular over the regular. For example, a contraction may convert a compact group to a non-compact, and the term “singular limit” seem more descriptive for the present use. Simplification is a special case of what is usually called “deformation” or “jump deformation”, pejoratives expressing a preference for the singular. We simplify in order to reform the theory, not deform it. It is the classical theory that is deformed. The bcompoundc Lie algebras of physics often contain an element 1 that is represented by a scalar matrix in the algebra of the theory. In the simplified variant it becomes an extra variable. These variables must have hidden from past experimenters. To account for this one may also adduce some ambient spontaneous symmetry breaking, degenerate vacuum, or, in the usage of Laughlin [53], organization, analogous to ferromagnetism, of the kind already suggested by the Higgs mechanism and Guth inflation. The organization is required to freeze the extra variables. Disorganization can thaw them and make them observable. There are semisimple Lie algebras of every dimension, for one can make a Lie algebra generic, whatever its dimension, by a generic homotopy, and if a Lie algebra is generic, its bKilling formc is regular, implying that the Lie algebra is semisimple. Simple quantization has problems. First, it leaves open too many possibilities. The simple Lie algebras near a given Lie algebra are rather few in number, but each stratum 180 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS of a theory has such choices, and the number of possibilities multiply. If the choices are independent, the probability of making them all correctly is rather low. Second, after choosing Lie algebras must one choose their matrix representations. A representation is defined by several quantum numbers, depending on the Lie algebra, and the number of possible values they can have is in principle infinite. Third, the most promising statistics for the elements of the queue, the Fermi-Dirac anticommutation relations, define a Clifford algebra over a neutral bilinear space. This is not a Lie algebra but a graded Lie algebra. A Clifford algebra of finite dimensions has a unique faithful matrix representation that defines a finite quantum theory generalizing the theory of spin. If we consider structural stability within the domain of Clifford algebras, those Clifford algebras with a regular Clifford norm on their vectors are the structurally stable ones. The Clifford algebras have unique preferred representations just like the bHeisenbergc algebras, eliminating the infinite choice left open by the simple Lie algebras in general, and finite-dimensional. And finally each Clifford algebra C has a useful orthogonal Lie algebra in its second-grade subspace C2, defining a bPalev statisticsc, and another in its Lie algebra of isometries C → C. This points rather clearly to a much stronger quantization process which we take up next.

4.3 Fermi full quantization

In bFermi full quantizationc one greatly constrains the choice 4.2.1 by requiring the simplified theory to have a finite-dimensional subspace C ⊂ S of the self-Grassmann algebra S for a vector space. This eliminates the choice 4.2.3 altogether, since a Clifford algebra uniquely determines a faithful irreducible representation. The physical interpretation of such a theory has already been sketched: The addition in C is quantum superposition, already discussed. The scalar element 1 (projectively) represents the empty set of systems. A top element > ∈ C represents the full set. The product operation combines systems into a plurality. The Clifford law ψψ = kψk produces a scalar, a multiple of the vector 1 for the empty set. This gives rise to the exclusion principle. It is natural to hope that C is some entire stratum S[F] ⊂ S. One must still choose the stratum F, but the dimension grows so explosively with stratum that if we take the least usable stratum, even a rough count of dimensions will fix this choice. In the present models stratum 6 suffices for the largest system that we can measure at the quantum stratum of resolution. Bose statistics is defined by a canonical Lie algebra and therefore fails the Segal structural stability criterion. bFermi statisticsc is not defined by a Lie algebra in the original sense, however, but by a Grassmann algebra, a graded Lie algebra and requires separate consideration: The Grassmann Relation v2 = 0 for v ∈ V is structurally unstable in the manifold of 4.4. FULLY QUANTUM VACUUM ORGANIZATION 181

Clifford algebras on the vector space V , for which v2 = kvk can be any quadratic form. The regular Clifford algebras are those with regular quadratic form. Grassmann algebra is a singular limit of the regular Clifford algebra with x2 = Ckxk as C → 0. The Fermi operator algebra on Grass V , however, is stable against these variations, and is used for all physical predictions. Therefore bFermi statisticsc is singular for present purposes. While full quantization changes Bose statistics into a Palev statistics, it respects Fermi statistics. . The two main Clifford algebras of the standard model, the Dirac spin 1/2 Clifford algebra defined by the time form and the Fermi statistics Clifford algebra defined by the probability form, are both subalgebras of Alg S, one on stratum 4 and the other on stratum L → ∞. Each epoch requires its own stability construct. For example, Segal stabilized theories only against variations within the manifold of Lie products, preserving the Jacobi identity and co-commutativity. Since these idealizations cause no infinities in the present theory, I retain them. Perhaps a closer critique of the operational basis of physics will eventually justify variations that convert Lie algebras into Hopf algebras, and groups into quantum groups, as in the theory of harmonic analysis on non-abelian groups [67]. Any bgauge theoryc has singular constraints that arise from the vanishing of a bHessian determinantc, clearly a structurally unstable condition. A bfull quantizationc removes these bsingularitiesc too. Because the regular algebra can be arbitrarily close to the singular one, bfull quantizationc can preserve the experimental meanings and agreements of the quantum theory in the present experimental domain, but the regular and singular theories eventually diverge greatly. To be sure, the internal Lie algebra

aSM = su(1) ⊕ su(2) ⊕ so(3) (4.24) of the bstandard modelc is only semisimple. While this is suﬃcient for structural stability, it is natural to consider whether this semisimple Lie algebra results from a simple one through centralization — “superselection” — relations resulting from organization; queue theory is supposed to describe the disorganized vacuum as well.

4.4 Fully quantum vacuum organization

There is evidence of spontaneous vacuum organization that results in a vacuum multiplet projector hΩh of multiplicity Dim hΩh = Tr hΩh > 1, (4.25) at least in the canonical limit. To fully quantize the concept of a self-organized vacuum, first the canonical theory is reviewed. In a canonical field theory, a vacuum is defined by an instant vector ψ0 that is an eigenvector of the system Hamiltonian with minimal eigenvalue. 080522 ...... 182 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

One vacuum organization that we require is a non-vanishing vacuum expectation value for i2: F 2 2 Ωh(ω56) hΩ =: −N 6= 0. (4.26)

F F This implies that the vacuum vector is invariant under neither ωµ6 nor ωµ5:

F F hωµ6hΩ 6= 0 6= hωµ5hΩ. (4.27)

It is natural to assume that the dynamics hP si has these symmetries.

4.4.1 Stratum assignments

Some properties found on one stratum derive from lower strata and some from higher. For an electron in a ferromagnet, an SO(3) symmetry of stratum E is induced from stratum D and broken on stratum F. Gravitation is assumed in ϑo to resemble magnetism in this respect: It has symmetries on stratum E that are induced from stratum D and broken on stratum F. The question then arises for each construct employed on a given stratum whether it is to be constructed from a lower stratum by bracing and catenation, or is proper to the stratum itself, or is a reduction from a still higher stratum. It is possible for some construct A of a stratum L to ﬁgure twice in the next stratum L + 1, once in an binducedc version 0 A = ΣA from stratum L, using the bcumulatorc Σ of (3.20); and once in a breducedc version A000 = Av ΣΣA of a construct A00 = ΣΣA of the higher stratum L + 2:

Rank L+2 A00 = A00 Σ ↑ ↓ Av Rank L+1 A0 A000 (4.28) Σ ↑ Rank L A

A theory keeping only stratum L + 1 loses the physical connection between A0 and A000. Then if A0 and A000 are too diﬀerent, they may seem to be unrelated, and if they are too close, the diﬀerence between them may be overlooked. This is not an academic possibility but happened for coordinate time and proper time, which agree for two experimenter events but not for two system events.

A fully quantum correspondent of a canonical bﬁeldc theory must have a vector space of dimension great enough to cover the same experimental domain as the canonical theory. We cannot measure the cosmic queue at the same level of resolution as a small region. All but a logarithmically small part of the cosmos is the metasystem, including the experimenter, and must be described at low resolution to protect it from destruction. In the present work only small regions are described with maximal quantum resolution. 4.4. FULLY QUANTUM VACUUM ORGANIZATION 183

Each stratum S[L] has a multiplicity (number of dimensions in its vector space) ML and a maximum grade ML−1, attained by its top polyadic

s> := sML sML−1 . . . s1 (4.29)

Call the expectation value of the number of events in the vacuum set history the bevent numberc: E h Hα i N := Tr hΩhs hsαh . (4.30) Since the Planck time is about 10−35 s and the Cosmological Time is at least 1021 s, it is conservative to say that N E > [1046]4 ∼ 10184 ∼ 2552; (4.31) this leaves out the field variables at each event. Then there must be at least 2552 kinds of events for so many to exist in one set. The lowest stratum in Table 1.1 with that many different possible events is stratum 64K E = 6, of dimension 2 . The event vector space Grade1 S[6] is tentatively adopted for the events in the queue. The differential stratum can then be taken to be stratum D = 5. The Lie algebra sl(16) of stratum C = 4 accommodates both the Segal-Vilela-Mendes so(6; σ), including a quantized imaginary [C] [C] i = ω65 /l, (4.32) and an so(10), hopefully adequate for GUT, leaving the signatures to be determined. We take stratum C = 4 for the sake of the sl(16) invariance of its monadics. The example bSegal-Vilela-Mendes spacec shows that canonical space-time coordinates, momenta, and i can be represented as singular approximations to high-dimensional representations of 9 of the 240 generators of sl(16), the group acting as its defining representation on the monadic vector space Grade1 S[4], and that no lower stratum suffices. In this model the differential stratum D=4 is the highest microscopic stratum and stratum E=5 is the lowest macroscopic stratum. It is harmless to use a stratum A=1 with vector space S[1] = 2R as the seed of the complex plane C for the amplitudes of complex quantum mechanics. The generator of so[A] can serve as the seed for the imaginary i of canonical quantum dynamics. The vector spaces S[L] are tentatively used in ϑo to support the following significant groups and vector spaces, with the given classical correspondents: L S[L] so[L] [A] A = 2 2R SO = U(1) = complex phase group [B] B = 3 4R so = Lorentz group [C] charge C = 4 16R so ⇒ Poincaré,C-gauge groups 16 [D] (4.33) differential D = 5 2 R so ⇒ ? [E] event E = 6 26R so ⇒ udiff, E-gauge groups [F] field F = 7 27R so ⇒ canonical, unitary groups metasystem G = 8 184 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

The canonical group referred to for stratum F is the canonical group of classical gravity and the unitary group of the standard model field system. Rank G is assigned to the ]ixmetasystem, discussed in §1.7.5. This assignment of Lie algebras and therefore of groups has a conspicuous incongruity. The strata of the fully quantum theory nest, and therefore so do their groups and Lie algebras. The strata of canonical physics do not always nest: A coordinate differential is not an event coordinate, an event coordinate is not a dynamical field. But the Lie algebra nestings B ⊂ C ⊂ E ⊂ F agree with the canonical theory, and the others seem harmless. The Einstein group Diff(4) is too large to be well approximated on stratum 4, which has dimension only 216 = 64K . Therefore stratum 4 is used as a filler between strata C and E. It seems harmless to associate it with the differential stratum D of classical gravity. 16 Then Diff(4) can be an approximate singular limit of SO[E] =∼ SO(22 ; 64) acting on the 2216 -dimensional event vector space. A field is usually represented as a set of pairs of events and field values. Following the bKaluza strategyc the field variable is another event coordinate. It is natural to assign the queue corresponding to the field to stratum F = 6, the successor of rank E in (4.33). The canonical group of gravity theory is generated by gravitational potentials and their canonical conjugate momenta modulo constraints. It can then be approximated, one expects, by a singular limit of SO[S[6]]. Chapter 5

Quantum space-times which examines early quantum space-times and formulates a fully quantum one.

Theoretical physics today is a curious sandwich, with a thick upper slice of classical bread, the macroscopic world of metersticks and stars; a thin slice of quantum cheese, the microscopic world of quarks and bgaugec quanta; and a thick lower slice of classical bread, the submicroscopic worlds of space-time events and their differentials, classical all the way down. The upper disconformity, between the quantum microscopic stratum and the classical macroscopic stratum, is understandable. It happens when we pass from fine quantum resolution to coarse macroscopic resolution, and can be shifted with some effort by improving our experimental resolution of the part of the bmetasystemc just above the disconformity. A maximal observer of quanta cannot be maximally observed. To transform changes of one quantum into macroscopic changes, there are necessarily unstable elements in any observer of quanta, like the visual purple in the eye. Maximally observing the observer would cause some of these elements to discharge, disturbing the observation.

Since the bclassical logicsc are reductions of the quantum logics, one can claim to be using a quantum logic even when one does classical logic, attributing the approximate commutativity of the macroscopic stratum to the bLaw of Large Numbersc, as one attributes the approximate absoluteness of time in daily life to our low celerity (cosh−1(v/c)). It is supposed here that the lower disconformity also results from a lack of experimental resolution, now of space-time events, but this one is practically immovable at present. We can supply reasonable quantum models of classical laboratory instruments, but not of classical space-time events. This is probably a failure of imagination. If there are space-time atoms they are surely very small. There is surely no physical reason for smaller things to be less quantum than

185 186 CHAPTER 5. QUANTUM SPACE-TIMES larger. Classical geometry, whether Euclidean, Minkowskian, or Riemannian, is therefore dubious for microphysics just because it is an exercise in a classical logic, which is a macrophysical logic. One way to repair the lower disconformity is by a bfull quantizationc, replacing classical logical elements of space-time geometry by quantum correspondents. One might consider performing a full quantization by expressing a canonical theory in formulas of classical logics and reading them in quantum logic. This does not work at all. Much of the logical structure that is used in a systematic set-theoretic model of classical geometry lies far below the stratum of observation of classical physics and has almost nothing to do with experiment. It is invented merely to build up a continuum from the discrete binary elements natural to classical logic. This infinite infrastructure can be set up in an infinity of ways that differ mathematically in ways that make no physical difference, since there is no trace of any such classical set-theoretic infrastructure in nature. From the viewpoint of physics most of the lower strata of the present physical theory are junk theory, and provide no guide to lower strata of nature. Conversely, there is no trace of the continuous symmetry of space-time such as Lorentz invariance in its axiomatic infrastructure, classical set theory. Such invariances first appear in the infinite classical construct. Thus all this infinite classical infrastructure is inappropriate in a fully quantum formulation, which incorporates continuity into its probability amplitudes from the start. Most of the classical logical infrastructure can therefore be left behind. One point of a full quantization is to replace this imaginary infinite classical infrastructure by a finite quantum one that has at least the potentiality of representing natural processes, which are quantum. The model ϑo replaces an infinity of classical strata to the five quantum strata B – F. On the other hand, the key classical concept of equality a = b as a relation between two set variables has no invariant quantum correspondent as a relation between two queues. To be sure, the relation of symmetry, expressed by a symmetrizing projector P+, is usually regarded as expressing identity in bosonic statistics but it lacks the important property of transitivity, and does not imply that two quanta in this relation have the same values for their variables. One needs a more knowing guide to the microcosm than classical logic. Historically, group theory has been used from the start to move the house of physics from its classical foundations to stabler quantum ones. Groups entered physics silently decades before physicists began to speak of the “group plague”. Canonical quantization itself is group-directed. It is set up to preserve isomorphically the 7-dimensional Lie group k generated by the three coordinates q and three momenta pk of each electron, simply replacing the classical Poisson Bracket by the quantum commutator times i/~. It further reconstructs the larger canonical Lie group, which transforms functions of the classical coordinates and momenta, as a singular limit of the unitary group of a Hilbert space. On the other hand, set theory infiltrated quantum physics almost as early as group theory. Quantization was designed to preserve certain membership or product relations among sets, simply replacing Cartesian products by Grassmann products. The classical 187

Rutherford atom A is the set whose elements are the nucleus N and the electrons en, as far as its phase space is concerned:

A = {N, e1, . . . , en} = {N} ∪ {e1} ∪ ... ∪ {en}. (5.1)

It was still modeled as a composite of its nucleus and its electrons in the early days of quantum theory, with Grassmann products replacing Cartesian products. When we form Grassmann products to make multi-electron orbitals we are still tacitly representing classical set structure within the fully quantum theory. Every simple Lie group mathematically defines both a quantum praxis and a quantum statistics, though most of these have had no physical application. The predicates of the first-order praxis are the projection operators of the bilinear space on which the Lie group acts in its defining representation. The statistics is defined by the same commutation relations that define the Lie algebra. This is the class of bPalev statisticsc recapitulated in §5.2.7. To be completely constructive, fix attention on the defining representation of each of these simple Lie algebras as the isometry Lie algebra of a quadratic space. The quantum predicates of the praxis are then represented by idempotent matrices in the same matrix algebra V ⊗V D as the defining representation of the Lie group. The set-theoretic structure and the symmetry group structure are both traces of the same quarry. Segal too explicitly used group structure as guide to the quantum underworld. In bsimplification by quantizationc, one replaces every bcompoundc (= non-semisimple) Lie algebra in the formulation of a theory by a simple one of which it is a singular limit, as did Vilela-Mendez [75]. We describe further examples of bsimplification by quantizationc below (§5) and then specialize to full quantization. In these instances of simplification by quantization, unfortunately, the physicist must not only choose a nearby Lie algebra, sometimes of rather small dimension, but one must also choose a representation of that Lie algebra, typically of huge dimension. The choice of representation is then embarrassingly rich. Full Clifford quantization narrows the choice to a line of Lie algebras, those of the real special orthogonal groups, especially those resulting from Clifford bquantificationc. Here the possibility is explored that all quantum kinematics is derived from quantum statistics; that the high-dimensional representations of low-dimensional Lie groups that occur in particle and space-time physics encode structural information as in atomic physics. A catenation of many identical systems having the defining representation of the same Lie group as their bkinematical groupcs has a high-dimensional representation algebra of this group as its bkinematical algebrac. Indeed, once one fixes on a family of statistics it can be iterated, for example to form a Fermi catenation of Fermi catenations. This generates a line of representations of exploding dimensionality, all in the same Cartan family, which may be adequate for present-day physics, granted singular limits and organizations. Choosing a representation from this line is easier than choosing it from the jungle of representations that bsimplification by quantizationc leads us into. 188 CHAPTER 5. QUANTUM SPACE-TIMES

bCliﬀord statisticsc (§3.5) seems especially appropriate for such iteration, since it is a structurally stable variant of bFermi statisticsc, contains both Fermi and Bose statistics as singular limits, and includes the Dirac spin Cliﬀord algebra as a low stratum S[4] in the sequence it generates.

5.1 Problems of classical space-time

5.1.1 Structural instability of time

The bGalileoc Lie algebra of rotations and boosts is the limit (as c → ∞) of a sequence of Lie algebras all different from itself as Lie algebras. The commutative algebra of classical mechanics is also the limit (as ~ → 0) of a sequence of isomorphic Lie algebras different from itself. So is the canonical Lie algebra. Such Lie algebras are said to be bstructurally unstablec. Variables that commute with all other variables in the algebra form the bcenterc of the algebra and are called bcentralc. In standard field theory, space-time coordinates are central. Structural stability requires non-central space-time variables. Central space-time coordinates are unnatural in the canonical theory of bgravityc too, for what seem to be similar reasons of structural instability [11, 10]. The structural instability of the Galilean Lie algebra could have told us that it was almost certainly an approximation to a more physical stable Lie algebra long before this was discovered by more experimental arguments. Now the structural instability of the bHeisenbergc commutation relations tells us that canonical quantum theory and Bose statistics are almost certainly approximations, not the end of the quantization trail.

5.1.2 Dynamical instability of time

Structural stability correlates with dynamical stability. If the Lie algebra elements are to be observables, some compound groups of present physics force us to infinite-dimensional matrices, most of which are undefined on most vectors. In such theories an energy spectrum can be unbounded below, like those of the classical hydrogen atom and the Dirac one-electron quantum theory. Thus structural instability permits the dynamical instability of unending radiative decay. Any nearby semisimple group has finite-dimensional representations in which all observables have bounded and finite spectra. In such theories dynamical instabilities are impossible. The work at hand is to present the standard physical theories as singular limits of such a stable theory, and of one that is also more physical. Almost all quadratic forms are regular. Almost all matrices have inverses. Regularity is normal, singularity is singular. Experiments, however, have error bars; experiment is generic. Therefore a singular theory cannot be based entirely on experiment, which is always generic, but must also postulate some structure of probability 0, often an idol in 5.1. PROBLEMS OF CLASSICAL SPACE-TIME 189 the Baconian sense rendered invisible by habituation. This idolization facilitates some calculations but also corrupts them; infinity in, infinity out. Present physical spaces have infinitely many infinite tangent planes and produce infinities. Simple quantum spaces are built from finitely many finite quantum elements like spins and produce finite answers.

Present-day quantum bfieldc theories assume space-time coordinates of infinite precision and range, with no complementary variables. They assume an empty background universe with a high symmetry, but a symmetry that is broken by the contents of the universe. These assumptions are vestiges of celestial mechanics, where the probe process has negligible effect on the process probed, and the system has small effect on the metrical structure of the space-time [25]. There seems to be need for a post-canonical quantum theory even less commutative and structurally more stable than quantum theory. It is not yet clear which present experimental processes best expose the hypothesized quantum structure of space-time; it is assumed here that all polyadics act on this structure. This gives the physicist license to hypothesize possible experimental operations and fit them into the framework of quantum kinematics, leaving the manner of their experimental implementation for the future; as Dirac did when he invented his theory of electron spin. Quantization gave us much understanding of the Periodic Table of the Elements but we have not had as much success with the Particle Table. One major difference is that one number, the atomic number, fixes the position of a chemical element in the Periodic Table, and determines its main properties in a systematic way, while it takes several numbers to fix the properties of a particle systematically. This seems to be because one group dominates chemistry, the rotation group; one statistics, Fermi; and one mass, the electron. The Lenz group expressing the conservation of the perihelion is an approximate symmetry group, valid for the non-relativistic Coulomb central potential, and also influences the Periodic Table, but less than the rotation group. Several groups and two statistics and several masses enter into the Particle Table, according to the bstandard modelc. GUT provides a Periodic Table of the Particles based on so(10). These approaches do not analyze the space-time continuum but hang additional dimensions onto it, quantum or continuous. The fully quantum event space is built up of simple quantum cells. Particle theories like the standard model and GUT can suggest structures for these cells. Relativity already provides particles with a deeper and still unresolved internal structure. Relativistic mechanics represents an elementary particle as a curve in space-time, its path or history. This can be composed out of a linear sequence of differential elements dx, by integration. These differential elements are its internal structure. A compound particle like the Solar System is represented classically by a braid of such space-time curves. If space-time is quantum then under high resolution, quantum particles resolve not into such continuous braids but into cell networks of complex quantum space-time processes. 190 CHAPTER 5. QUANTUM SPACE-TIMES

5.2 Earlier quantum space-times

In canonical theories, energy i~ d/dt generates a change in time t and conversely, in virtue of the relation [E, t] = i~1, [1,E] = [1, t] = 0, (5.2) although t and E are not observables. This defines the singular bcanonical Lie algebrac h(1) := a(t, d/dt, 1). One structurally stable variant is Segal’s so(3; σ): [t, E] = r,[r, t] = E, etc., with extra variable r much like t and E, and signature σ. This time is a quantum variable, one of the three generators of so(3; σ), and has discrete, bounded, and uniformly spaced spactrum, like any angular momentum. The passage of this bquantum timec is not effected by a one-parameter Lie group. Energy is another such variable, and so is r, the commutator of time and energy, which is supposed frozen to the immediate vicinity of i~ in the vacuum queue. Such a theory can still represent the dynamics of the system through a generic history- amplitude instead of a time-translation generator. A history is typically a queue of a great many spinlike input and output actions, in which the observed system is an excitation. In canonical physics we analyze the bfieldc into couples (x, f) of space-time events and field values, coupled by a compound group. ϑo replaces the field by a queue with still more coordinates, now coupled by a simple group. The queue events have momentum-energy and angular momentum as well as space-time coordinates, and in general each reference frame in the vector space resolves these variables differently into space-time and energy- momentum.

5.2.1 Event energy Simple quantum relativity continues Einstein’s physicalization of geometry. It supplements formerly geometrical event coordinates, like position in space-time, with apparently dynamical event coordinates, like momentum-energy. Quantum events like those of bSnyderc, bVilela-Mendesc, or Baugh have, in each admissible frame, not only space-time coordinates but also momentum-energy and other coordinates, mixed by the simple invariance group. This is counterintuitive, for it relativizes the construct of absolute space-time point that has pervaded physics at least since Aristotle. Events are ordinarily supposed to have space-time coordinates but no momentum-energy coordinates. Hume pointed out that we never experience space or time by themselves. It is reasonable to ask how the notion of an event as a purely spaciotemporal entity, with no other mechanical properties, entered physics and why it has lasted, when we never encounter such an entity in our experience. The root of “line” means linen string and the root of “point” means puncture. These recall that geometry sprouted from annually flooded fields along the Nile, with strings of Egyptian linen as its lines and sharp stakes piercing the ground as its points. A typical Baconian idolization seems to have occurred: Surveyor’s stakes and strings have well- defined momenta, but their momenta are small in the terrestrial reference frame because 5.2. EARLIER QUANTUM SPACE-TIMES 191 they and the observer were both attached to one well-organized condensate, the Earth. These small momenta were first tacitly taken to be exactly zero, and then dropped from thought completely, due to habituation. The idea of non-mechanical point emerged from experience with mechanical ones whose masses and momenta could often be ignored. In space-time too one never encounters a massless event. To arrive at the space-time concept of today from actual space-time happenings, which all carry momentum-energy, we must have approximated the momentum-energy by 0 and then forgotten it. One now corrects this idealized construct of empty space-time by attaching an energy- density tensor to each event. For other media, such tensors are surrogates for an underlying atomism but space-time defies analysis into classical atoms in any relativistically invariant way. Here a quantum analysis is considered that accommodates the classical relativity of space-time within the quantum relativity of vector space. As a first step, the simplicity principle, combined with Lorentz invariance, leads one to restore the lost energy-momentum variables pµ — as Snyder already did [69] — and to posit a symmetry between xµ and pµ at the event stratum —- like Segal [66] — , broken in the singular limit where pµ → 0, perhaps by an organization of events into space-time. An organization of space-time would now be expressed by a vector, fixed by the experimenter, not by the operator algebra, fixed by the system. In canonical quantum physics, the dynamical phase-space of a theory is built on an allegedly deeper space-time, at least on a time axis; geometrical symmetries are regarded as fundamental symmetries of a hypothetical empty universe and are built into the dynamical transformation group as a preferred subgroup; particles are characterized by the representations of the space-time group that they define. In cellular q physics the macroscopically extended quantum space-time is a contingent organization of the system queue. With no absolute space-time, there is no fundamental space-time group. The dynamical group is now simple and prior to the canonical singular geometrical group. The system queue is a combination of event queues. Space-time symmetries are at best phenomenological approximate statements about the ambient queue. General covariance was indispensable for the creation of general relativity. It has a natural correspondent, fully quantum covariance (6.107), which implies covariance under a certain so(n; σ) for each stratum, with large n for the event stratum E. Full quantum covariance is used in building a fully quantum theory of bgravityc in the way that general covariance was used to build general relativity. General covariance is treated here as not a fundamental and exact law but as an approximate phenomenological statement about the current ambient queue, that fails, for example, when the queue melts down. Central space-time coordinates are built into the usual field theory, as part of the strategy of absolute space-time. Consider the theory of bgravityc, for example. In his gravitational action principle Hilbert varied bgravitational fieldc variables gµν(x) without κ varying space-time coordinates x = (x ). In the resulting Poisson Bracket Lie algebra, gµν commutes with xκ, and the xµ are central in the Lie algebra of fields. There are no central time coordinates in actuality, however. Our actual physical 192 CHAPTER 5. QUANTUM SPACE-TIMES coordinates xµ for a remote event are based on signals, usually electromagnetic, that reach us through the intervening space-time and therefore inform us about the intervening bgravitational fieldc as well. The lattice of rods and clocks imagined by Einstein to coordinatize events is not different in principle from optical coordinates since it too is connected by electromagnetic interactions and modified by gravitational fields. Physical coordinates are more relative than general relativity admits, being influenced by the ambient field as well as the choice of reference frame. They are but field variables under another name, and non-local as well. Since fields are non-central, so are coordinates.

5.2.2 Indeﬁnite probability form

Fermi bfull quantizationc with real coefficients and a cut-off at a finite stratum leads to a real finite-dimensional vector space with no statistical metric. If the Lorentz group (for example) is to be represented in this space, it cannot be by a unitary representation. One must learn to do fully quantum theory either with no absolute statistical metric or with an indefinite one. The interpretation of indefinite probability forms is not yet fixed. One interpretation favored by bDiracc is that nature has good credit: its counters can run into the negative and therefore so can probabilities. Another interpretation used in bGupta-Bleulerc electrodynamics, which also has a vector space V of indefinite metric, is that negative-probability transitions do not occur for other reasons. Each experimental frame can use a maximal subspace V+ ⊂ V with positive-definite metric to represent feasible operations on the system. The complementary negative-definite subspace V− is reserved to represent transformations that are unfeasible as experimental operations but permissible as relations with other experimental frames; just as timelike vectors represent feasible translations of a mechanical system while space-like vectors represent unfeasible translations, which are nevertheless necessary to relate some spatially-separated observers. The canonical theory of bgravityc has at least three strata D, E, F. The fully quantum theory of gravity in ϑo imbeds these classical strata, with due regard for their cardinalities, in the strata of a fully quantum theory. Here are some earlier quantum spaces that influenced this work.

5.2.3 Feynman space bFeynmanc (ca. 1941) investigated quantum space-time before he developed his quantum electrodynamics. He hoped that by eliminating arbitrarily small distances and times he could avoid all ultraviolet divergences. The trouble with a classical lattice spacetime is that it destroys the continuous symmetries of spacetime under translation, rotation, and Lorentz transformation, leading to violations of the conservation of momentum and angular momentum. 5.2. EARLIER QUANTUM SPACE-TIMES 193

He considered a form of quantum space-time whose coordinates were sums of many independent commuting replicas of the Dirac operator-vector γµ [33]:

µ µ µ xb = X[γ (1) + ... + γ (N)]. (5.3) X is a fundamental subnuclear scale size to provide the physical dimensions of time. Then the four coordinates of a point do not commute with each other. This permits them all to have discrete spectra even though the theory is Lorentz invariant, forming a kind of quantum lattice; just as all the components of angular momentum have discrete spectrum even though their theory is rotationally symmetric. Nor are the coordinates xµ Hermitian in any definite metric; therefore they are not observable quantities in the strict sense. In the queue models considered below there are many vector-operators γµ(n) but they mutually anticommute, generating a large Clifford algebra, instead of commuting. The bFeynman spacec model shows that quantum space-time, regularity, and Lorentz invariance are compatible if one sacrifices unitarity. Its time coordinate has a discrete bounded uniformly spaced spectrum with a time quantum X. Some propose on dimensional grounds that the time quantum X be identified with the Planck time b c √ TP = ~G. (5.4) Organized media typically have several scales of length and time, so the dimensional argument is weak. Even an electron carries at least three very different empirical scales of length, its Compton wavelength, its classical radius, and the range of its form-factor. We should allow the physical space-time queue as much. Indeed, all the lengths in the particle spectrum are presumably characteristics of the vacuum. Dimensional arguments, moreover, cannot tell us how Newton’s G or Feynman’s X depend on the large number N. Therefore X and G will initially be kept independent, hopefully to be related later. bFeynmanc and bHibbsc formulated a model of a quantum particle moving in a classical discrete space-time similar in spirit to the Feynman quantum space [35]. It is usually called the checkerboard model and has also been studied by Jacobson [48] for example. The checkerboard is the infinite square lattice in which a piece moves like an uncrowned piece in the game of checkers: forward to the right or forward to the left. A rank of cells is interpreted as a space line, a file of cells as a time line, and a diagonal of cells as a light line, The cells accessible to a piece from a given cell form a discrete future light cone with apex at that cell. On the one hand these models have no useful continuous symmetries and are therefore quite unphysical. On the other hand there is a surefire way of giving them the continuous orthogonal group symmetries they lack: One writes them in finite set theory and reads them in queue theory. It is therefore worth examining the Feynman model further. A path of a checkerpiece is represented by a sequence of position vectors t(n) x(n) = (5.5) x(n) 194 CHAPTER 5. QUANTUM SPACE-TIMES

These deﬁne displacement vectors δ(n) := x(n + 1) − x(n). By the rules of checkers, each displacement δ(n) has one of the two constant forms

+1 δ(n) = =: c , (5.6) ±1 ± corresponding to motion at light speed to the right or left. A motion-reversing operator R, with

Rc± := c∓, (5.7) is useful. A Feynman path probability amplitude A is assigned to a path of such a piece by summing contributions δA from each of the vertices (squares) x(n) in the path: A = P n δA(n). The contribution δA(n) of vertex n to the sum is taken to be δA(n) = +1 if there is no reversal during that segment, δA(n) = iM if there is reversal:

δA(n) = 1, for δ(n + 1) = δ(n), = iM, for δ(n + 1) = Rδ(n). (5.8) iM is thus the probability amplitude for motion reversal relative to non-reversal. It follows that a vector for one time can be expressed by an amplitude function of the coordinates x, δ of a square and a displacement vector at that square, giving the location and motion of a piece. This vector will be written as a 2-component column vector

ψ1 hψ = , (5.9) ψ2 the 1- component giving the amplitude for a move to the right, and the 2-component to the left. The amplitude function representing ψ is

ψ(x, δ) =: [x, δhψ]. (5.10) hψ propagates according to

[x(n + 1), δ(n + 1)hψ] = [x(n + 1) − δ(n + 1), δ(n + 1)hψ]+ + [x(n + 1) − Rδ(n + 1), Rδ(n + 1)hψ : iM]. (5.11)

A still more cellular version can give further insight into the Dirac equation. Consider the two displacements c± of the checkerpiece as two abstract objects; call them bmovesc. Then a path on the checkerboard is a classical Maxwell-Boltzmann catenation — or sequence — of moves. From this Maxwell catenation one forms a classical Bose catenation by ignoring order, or symmetrizing over all permutations. This is the resultant displacement of the end of the path from the beginning. The two null coordinates n± of the end of the 5.2. EARLIER QUANTUM SPACE-TIMES 195 path relative to the beginning are occupation numbers of this Bose catenation. The time duration of the path is the total occupation number t = n∗ + n−. The spatial extent of the + path is the diﬀerence x = n − n−. The Feynman model lacks momentum variables and several strata, supplied in what follows.

5.2.4 Snyder space bSnyderc space renounced geometric space-time in that its elementary events have energy- momentum and angular momentum variables as well as time and position. They might therefore be characterized as “particle-events”. bHeisenbergc had suggested that there is a bfundamental lengthc in nature, but attempted to incorporate it into non-linear bfieldc equations without modifying classical MInkowski space-time. Apparently following up an idea of Heisenberg communicated to him by bOppenheimerc, perhaps merely that of a fundamental length, bSnyderc introduced new commutation relations for the space-time position coordinates xµ involving a fundamental length X, which became a quantum of the positional coordinates. This led him to a Poincaré-invariant theory with a discrete spectrum for each spacelike coordinate and a continuous one for each timelike and energy- momentum coordinate [69]. No working bfieldc theory was erected upon Snyder space, perhaps for lack of a guiding quantization principle. A theory like Snyder’s had been discussed by Pascual bJordanc and bvon Neumannc in 1938 without publication [79]. Later von Neumann expressed skepticism about such approaches and instead proposed to eliminate infinities with a bcontinuous geometryc for quantum physics. Continuous geometries are more infinite than classical space-times, not less, in that even the dimensions of their flats have a continuous range. Von Neumann supposed that continuous geometries might permit a finite physics because they lack “points”, projectors of a minimum non-zero dimension; but physics blows up at points of space-time, not at points of the vector space. The existence of sharp vectors is part of the solution, not the problem.

5.2.5 Segal space bSegalc (1951) suggested a useful heurism for the evolution of physics [66]. He pointed out that quantum mechanics and special relativity both result from previous theories by homotopies that carry singular compound Lie algebras toward simple ones. He showed that simple – and even semisimple — Lie algebras are structurally stable against infinitesimal homotopies of their Lie product. He implied that canonical quantization is only the beginning of a process of stabilization by simplification, and that the process should be completed with simple groups. He gave two examples of simplification, h(1) ⇐ su(2) and iso(3, 1) ⇐ so(6; σ); generalizations are clear. This implies that the quantum spaces of physics based on singular Lie algebras are singular limits of more physical quantum spaces based on regular Lie algebras [66]. This includes all the quantum phase spaces resulting 196 CHAPTER 5. QUANTUM SPACE-TIMES from canonical quantization or Bose statistics, for example. In one illustration, here called the three-dimensional Segal space, Segal regularized the canonical commutation relations for one coordinate variable q and momentum p [66]. In the canonical theory i is a central element of the real three-dimensional bcanonical Lie algebrac h(1) : qp − pq = i, iq − qi = 0, pi − ip = 0. (5.12) An irreducible representation of this Lie algebra in a complex Hilbert space defines what may be called a bcanonical quantum spacec. Segal simplified the canonical Lie algebra h(1) to a special orthogonal Lie algebra

so(3) : qp − pq = r, rq − qr = αp, pr − rp = βq (5.13) with structure coefficients α, β. Any faithful irreducible representation of this Lie algebra defines a Segal space. The extra variable in (5.13) is r, which must freeze to a large value in the canonical limit. Similarly Segal suggested simplifying the Poincaré-invariant bcanonicalc algebra h(4) = µ a(x , pµ, i) to an so(6; σ) algebra fixing a quadratic form of unspecified signature σ. Let the six real basis vectors supporting this defining representation of so(6; σ) be designated by hα with α = 1, 2, 3, 4, 5, 6. When r freezes and breaks so(6; σ) symmetry, the first four variables will undergo the Lorentz group, the last two the SO(2) group of the complex plane. µ The usual canonically dual variables x , pµ relate to the dimensionless generators of so(6; σ) through a quantum of time X and a quantum of energy E:

µ E µ5 xe = X ω , µ E µ6 pe = E ω ; (5.14) and obey commutation relations

ν µ νµ E 65 [xe , pe ] = 2XE δ ω . (5.15) This suggests that bi, the fully quantized variant of i, is a component of angular momentum, suitably normalized: 1 bi = ωE 65 = ΣE−CωC 65 (5.16) l where −l2 is the maximum eigenvalue of [ωE65]2 . Presumably the real quantum time bt is found by factoring the imaginary operator bi into the skew-symmetric operator et, the skew-time. This results in the anti-commutator X t = [ω65, ω45] . (5.17) b 2l −

Segal’s proposal influenced the classic retrospective studies of Inönüand bWignerc on contraction and the Galilean limit [47], and Gerstenhaber’s cohomological theory of Lie 5.2. EARLIER QUANTUM SPACE-TIMES 197 algebra stability [44]. These in turn had numerous consequences [36]. It seems that Segal published no further thoughts along this line. Others subsequently found the bSegalc simplification h(1) ← so(3) independently of Segal [5], [50, 52, 51],[59], [74]. It should still be verified that the chosen representation of so(6; ?) is a physically useful approximation to the canonical representation of the canonical Lie algebra h(4; 2). This is taken up in §5.3.2. This left open such questions as how to choose a representation of the simple Lie algebras that arise in this way, how to cope with instabilities on other strata, how to express quantum fields over such a quantum space-time, and how to express the dynamical relations in the resulting kinematics. These questions are answered here by full quantization.

5.2.6 Penrose space

Following a separate path, Roger bPenrosec pioneered the quantum combinatorial approach 2 to space-time by analyzing the sphere S , with its infinitude of infinite tangent planes, into a singular limit of an assembly of spins 1/2 with Bose statistics, representing the rotation group SO(3) in a double-valued way [60]. bPenrose spacec can be regarded as a non- relativistic variant of Feynman space, in that its point is a Bose catenation of Pauli spins 1/2 instead of Dirac spins 1/2, and its coordinates are direct sums of Pauli spin matrices instead of Dirac spin matrices. It too is quantum on only one stratum. But there is no experimental distinction at any one time between an infinite plane and a sufficiently large sphere, and of the two it is the sphere that has the simple group. From the viewpoint presented here, therefore, Penrose quantized not only S2 but also its 2 singular limit, the plane R . The conversion from dimensionless Pauli spin operators to lengths introduces a small length X. The expression of the coordinate variables

N X xk = Xσk (5.18) 1 as sums of N spin variables, fixing the representation, introduces the large integer N. Then a Penrose space too has a fundamental quantum length and a large integer. NX sets the scale of the radius of the S2. This work catalyzed many others. One effort reconstructed Minkowski space-time as singular limit of a Fermi assembly. It had the strata needed for field theory but lacked structural stability and a quantization process connecting to the standard model [38]. They are provided here.

5.2.7 Palev statistics bPalevc (1977) took a major step toward a generic post-quantum physics by simplifying the compound algebra of Bose statistics [59]. A bPalev statisticsc is one deﬁned by the 198 CHAPTER 5. QUANTUM SPACE-TIMES commutation relations of a classical Lie algebra in a representation that approximates the Bose statistics Lie algebra; or more formally put, in a sequence of representations that has the Bose commutation relations as a singular limit.

A bPalev statisticsc can be substituted for Bose statistics everywhere with only small experimental consequences in the present experimental regime. The most conspicuous effect of this substitution is to cap the occupation numbers of one-quantum vectors. The size of this upper bound is a parameter that can be adjusted to fit the data. The Lie algebra of bPalev statisticsc defines a quantum space on which it acts, a bPalev spacec. The Palev statistics in the orthogonal or D series looks like the real quantum theory of a high-dimensional angular momentum in an irreducible representation. The imaginary i must arise from one antisymmetric component of angular momentum, say ω56 to conform with (5.16). ω56 cannot be diagonalized in the real theory. Define the non-negative symmetric operator p 2 . |m| := + −(ω56) = 0, 1, . . . , l. (5.19)

2 (No operator m is deﬁned.) (ω56) has a spectrum of the form

2 2 . 2 (ω56) := −m = 0, −1,..., −l (5.20) with multiplicities

d0 = 1, dm = 2 for m = 1, . . . , l (5.21)

The operator i must have two-dimensional invariant subspaces in the real theory. For real Palev statistics to approximate a canonical quantum theory, which is complex, l must be very large, and the canonical quantum theory must work well enough in a subspace with m ≈ l. Call this subspace the bcanonical subspacec Vcan. Its dimension must be large enough to pass for inﬁnite and small enough for m ≈ l to hold throughout it. Let P [...] designate the projector on the eigenspace of the operator relation [...]. Then, for example, the subspace 2 2 2 Vcan = P [l − l < −(ω56) < l ]V ⊂ V, (5.22) meets these criteria. This raises the question of what physical processes make m ≈ l. A reasonable hypothesis, based on experience with organized matter, especially ferromagnets, is that organization of many small elements has occurred. This requires, ﬁrst, that the angular momentum ω56 be composed of a great many much smaller spins, and that these are aligned in the vacuum.

The quantum event of bVilela-Mendes spacec (§5.2.8) has a bPalevc coordinate algebra. The bPalevc Lie algebra is the second-grade part of a Fermi or Cliﬀord algebra, so bPalevc events can be formed out of pairs of Fermi events. It is diﬃdently assumed here that this is the origin of all bosons. (§3.4.3, §6.4.1). 5.2. EARLIER QUANTUM SPACE-TIMES 199

5.2.8 Vilela-Mendes space

Inspired by the Gerstenhaber theory of rigid (structurally stable) Lie algebras, bVilela- Mendesc constructed a simple quantum space based on an so(6; σ) Lie algebra with generating variables ωβα ∈ so(6; σ). He used multiples of the ω’s to approximate Minkowski µ space-time coordinates x , the canonically conjugate momentum vector pµ, the infinitesimal Lorentz transformation Lνµ, and the imaginary i, and discussed dynamics in such a space [75]. This simplifying group was also proposed briefly by Segal [66]. The resulting bSegal-Vilela spacec is thus a regularization of the singular Poincaré-Heisenberg quantum α space. It may have one, two, or three timelike dimensions among its six x (or pα). To construct a quantum space of the Segal-Vilela family, one must fix a high-dimensional representation of so(6; σ) that approximates the infinite-dimensional SPH representation closely enough. To represent a physical space-time a Segal-Vilela space also needs both a probability form on its vectors and a causality form on its differentials. These are provided in §5.3. This then provides a single-stratum quantum space for a single event, not a space-time. A higher-stratum quantum space-time is constructed in Chapter 6. We index the usual four axes of Minkowski space time with µ = 1, 2, 3, 4 and the two complexual axe with indices α = X,Y . The index α ranges over 1, 2, 3, 4,X,Y .A convenient complete set of Segal-Vilela-Mendes admissible coordinates is

µ µ X xe = Xω , peµ = EωµY , Leνµ = xeνpeµ − xeµpeν = XEωνµ, −1 i = N ωXY (5.23) with scale factors X and E having the units of time and energy. The quantum number 2 N is the square root of the maximum eigenvalue of −(ωXY ) in a matrix representation RJ of so(V ), not fully pinned down by either Segal or Vilela-Mendes. The contraction to classical space-time includes the limits

X, E → 0, N → ∞ (5.24) and the freezing 2 2 (ωXY ) ≈ −N , (5.25) which can be treated as a subsidiary condition, restricting the system to a relatively small sector of S. The physical origin of such a condition is taken up in $5.3. The use of so(6; σ) in Segal-Vilela-Mendes space should not be confused with other uses of so(6) that have been suggested, such as the E numbers of Eddington [28]. Speciﬁcally, the so(6) of Segal-Vilela-Mendes space mixes x and p as discussed in §6.8. To attain structural stability, bSegalc and bVilela-Mendesc combine and unify homotopies separately introduced by bEinsteinc, bHeisenbergc, bde Sitterc, and bSnyderc. The 200 CHAPTER 5. QUANTUM SPACE-TIMES

bSegal-Vilela spacec is a Matrix Geometry [27] without its connections and bgravityc. It is more matrix than the Banks Matrix Model [6] in that its time variable too is a matrix. It has a fundamental time, the chron X, that fixes the quantum scale of space-time, and suitably large integers that fix the representation of so(6R; σ). Other classical Lie algebras besides so(6; σ) also have ph(4) as a singular limit. The events of a Segal-Vilela space could be realized by Palev collections of simpler spinlike quantum elements, generalizing the Penrose sphere, whose point is a Bose series of spins 1/2. In general, if a configuration space has n dimensions then the classical phase space and its tangent bundle have the dimensionality 2n, but this phase space is the singular limit of a regular quantum space with Lie algebra so(n + 2). One need not double the dimension in this case, it suffices to add 2. The two added dimensions provide an axis of symplectic rotation that couples coordinates into momenta and conversely.

5.2.9 Baugh, Shiri-Garakani spaces More recently, Baugh (2004) simplified the Poincarégroup to a special unitary group SU(n) instead of an orthogonal group. The quantum event of Baugh space can be represented as a pair of Palev sub-events with vector spaces 6C [8]. Structural stability forbids the Newton commutation relation [d/dt, t] = 1 of the usual dynamics. Shiri-Garakani (2005) simplified the linear dynamics of a harmonic oscillator [68]. At the extremes of system time t, when t ∼ ± max |t|, the multiplicities of the eigenvalues of |t| typically vary rapidly, namely linearly in t ∓ max |t|, and unitarity is a bad approximation, but in the middle times, when |t| max kt|, unitarity can still be a useful approximation. The usual singular limit keeps only the middle times as the extreme times approach infinity, and so can be unitary.

5.3 Fully quantum event spaces

The primary quantum variable considered in this section is the event, a queue of stratum E corresponding to a space-time point of the canonical theory. Under sufficient resolution the “field”, the system of stratum F, is seen to be a queue of stratum E events. Each event of E in turn becomes a queue of elements of stratum D, “differentials”. The differential stratum has dyadics generating the group of stratum C. These include the space-time position operators, energy-momentum operators, and angular momentum operators for the particle associated with the event, as well as charges of the standard model. 16 stratum-D dyadics belong to stratum C as well. These include charge/spin operators generating the group of stratum C. 5.3. FULLY QUANTUM EVENT SPACES 201

The bself-Grassmann algebrac S makes it natural to specify the representation of Segal- Vilela-Mendes quantum event space that Segal and Vilela-Mendes left open. The vector space of the event E is taken to be a Clifford algebra V E = S[E] as high in S as necessary for bulk. Segal-Vilela-Mendes space can be accommodated within a fully quantum Clifford theory by first imbedding the Lie algebra so(6R; σ) as a Lie subalgebra within the real linear Lie algebra so(S[3]) = so(10, 6) of the 16-dimensional stratum 3 subspace of S. so[S[3]] is isomorphically represented as Lie algebra by the second-grade subspace of the Clifford algebra S[4] with the commutator as Lie product. This choice means that we identify stratum C with stratum 3 of S. The reduction 16R = 4R⊕2R⊕10R then reduces so(16) = so(6) ⊕ so(10) into the Segal-Vilela-Mendez combination of external quantum space-time and the complex plane, and an internal quantum space like that of the Grand Unified Theory. Charge stratum C = 3 is the first rank able to support the so(6) of a fully quantum space-time. Event stratum E = 5 is the first able to represent these operators quasi- continuously. If one adopts tentative stratum assignments C = 3 and E = 5 then there are 120 independent Lie algebra elements ωEβα = ΣE−Cω[C]βα (5.26) representing so[S[C]] on S[E]. Four of the 120 must approach continuous variables in the canonical limit, one must approach the imaginary constant i, and the remaining 115 must approach 0: µ Eµ5 µ xe = X ω → ix , 1 i = ωE56 → i, NEX µ Eµ6 pe = E ω → 0, . . Leνµ = EX ωEνµ → 0. (5.27) The normalization factor N is needed to compensate for the many terms in i. The Minkowski causal metric form igh is then isomorphic to a singular limit of the bKilling formc ikh of the Segal-Vilela-Mendes Lie algebra so(6; σ), suitably restricted (As- sertion 6). This agreement in form can be used to represent the physical Lorentz group within stratum C. It is tempting to use this agreement in form to represent the Minkowski space causality form g as well, but first a chasm between the meanings of igh and the Minkowski gνµ would have to be crossed. igh is a quadratic form on quantum 4-dimensional vectors; if it is used as a probability form, it is usually normalized to unity. gνµ is a quadratic form on classical 4-dimensional space-time tangent vectors, defining the square of a proper time, a clock-reading in a frame where the space-components of the differential dxµ are 0. 202 CHAPTER 5. QUANTUM SPACE-TIMES

The canonical Lie algebra h(N ) has essentially one faithful representation R~ by differential operators, and it is infinite-dimensional. The linear Lie algebra so(6; σ) resulting from Segal’s algebraic simplification has an infinite number of different faithful representations RJ so(6) labeled by a triad of quantum numbers J = (J1,J2,J3). In what follows, a circumflexed variable designates the RJ representative of the un-circumflexed variable. Physics requires a representation with dimension much greater than that of the defining representation. Here this results from exponentiation, passing to a higher stratum by bracing and catenation.

5.3.1 Fully quantum spaces Here it is hypothesized that a physical fully quantum space is a queue, whose stratum-L vector algebra is a subalgebra of the Grassmann power space over its stratum-L − 1 vector algebra. It follows that the vector algebra of each stratum L is a subalgebra of S[L]. Classical space-time physics distinguishes between the Minkowski coordinates on any one tangent space to space-time and the more general coordinates on the gravitational manifold. For brevity we call these special and general coordinates. They seem to have natural fully quantum correspondents: D A bspecial coordinatec on the diﬀerential stratum S D is one that is induced by a coordinate of stratum C, and so is of the form

ΣD−Cω for some ω ∈ so(C S). (5.28)

Special coordinates of the event E are deﬁned analogously, in the form

ΣE−Cω for some ω ∈ so(C S). (5.29)

A bgeneral coordinatec of the event E is of the form

ω = 4s, s ∈ so(ES). (5.30)

Canonical commutation relations among these coordinates arise as singular limits of the so(W ) commutation relations. In the terms of bsupermanifoldsc [23] and bsupersymmetryc theory, the first-grade [C] elements Lsα ∈ S define odd coordinates Lsα and the second-grade sβα define even coordinates ωβα = 4sβα. A Clifford space has less symmetry between odd and even coordinates than a supermanifold: The even are polynomials in the odd. There is still a symmetry group that mixes them: the isometry group SO(W ). The usual space-time and energy-momentum coordinates for Minkowski space-time are [C] singular limits of special coordinates ωβα = 4sβα ∈ so , much as in §5.2.8. The fully quantum theory ϑo provides the odd-grade generators sα as well as the even. Such odd variables can be useful for the construction of fermion field variables, so we 5.3. FULLY QUANTUM EVENT SPACES 203

[C] accept them as vectors. The 16 sα in S are coupled into each other by the Lie algebra so(S[C]) = Grade2 S[C]2. The earliest stratum that will accommodate both the Poincaré group and the standard model group is C = 4, with Dim S[C] = 16. For simplicity we arbitrarily suppose that all the dimensions of S[C] have physical meaning. The symmetry Lie algebra so(S[C]) couples the 16 monadic generators sα ∈ S[D] and seems to be broken in higher strata. The 16 sα divide into 10 spacelike and 6 timelike generators. Four second-grade generators sβα give rise to long dimensions, the runners in the space-time truss, and others give rise to the short dimensions, the spreaders in the cosmic truss. One s65 gives rise to the frozen constant coordinate i. The four vectors sα used to construct generators of long variable (Minkowskian) dimensions are designated by sµ. The two generators of the two (Argand-Euler) dimensions of the complex plane of the quantum imaginary i = sYX are designated by sζ , ζ = {5, 6}; these being the first two with negative signatures beyond the sµ. The 10 sα used to construct the short variable (Kaluza-like) dimensions are designated by sκ, with κ = 5+, 8+, 9−, 10+, 11+, 12+, 13−, 14+, 15+, 16−, the subscripts giving the signatures (Table 1.1). In a model of the present vacuum, the dimensions X and Y associated with space-time and momentum-energy are long (external; longitudinal relative to the cosmic btrussc dome). When the corresponding sums ΣF −C ωCµ6 include enough spins, they become effectively central. This may not be true for the dimension XY associated with i, since i is effectively the same for all events. To be sure, to be approximately central, ΣF−CωC56 too must include a macroscopic number of spins, isomorphs of ωC56. But these could arise from a sum over a space-time region of a great many transverse spins in parallel, as well as from a sum of a great many longitudinal spins in sequence.

5.3.2 A fully quantum space-time A fully quantum theory must first account for the great success of the canonical quantum theory and then do better, just as the canonical quantum theory accounted for the great success of the classical theory and did better. In the best case, all the elements of the low strata of S would have physical meaning; let us examine them with this possibility in mind. A natural basis B for S is that fully generated without the use of addition. Then every basis element sα ∈ B ⊂ S is a product of monads sβ,α, whose number is the bgradec of sα, and each monad sβ,α belongs to some stratum Lβ,α, whose supremum over β is the stratum Lα of sα. A typical physical field-stratum vector sf must be a product of a vast number of 64K monads se, one for every quantum event or cell in space-time. Rank 5 contributes 2 polyadic events, which suffice for now. The vast number must belong to rank 6 or higher. Let us stop provisionally at F ≈ 6 for economy. If we open one of the monads of stratum F, we find a polyad of stratum 5 that we 204 CHAPTER 5. QUANTUM SPACE-TIMES can analyze again. Its monads are braces of polyads of stratum 4 proper. They generate a typically high-dimensional representation of SO(216) and provide a candidate for the quantum coordinates of event space. One may now form quantum space-time events. The canonical Lie algebra a(x, p, i) of µ space-time and momentum-energy is generated by nine elements x , ∂µ, 1 (µ = 1, 2, 3, 4) subject to the canonical relations, with 1 central. One full quantization approximates this Lie algebra within a higher-stratum representation of the isometry Lie algebra so(10, 6) = so[C] of stratum C=3, the earliest possibility. This Lie algebra is represented by generators of the form

D ωβα = Σ4sβα (5.31) in the second-grade subspace of the next stratum S[D], with D = C + 1 = 4. The Clifford 4 algebra S[C] of stratum C has 4 first-grade generators hsS[C]α, 2 dimensions, and a mean square form iHh of signature 22 used both as causality form and probability form. The 16 Clifford algebra S[D] has 16 first-grade generators hsDα, 2 dimensions, and a mean- 8 square probability form i‡h of signature 2 . The elements s1, s2, s3, s4, s5, s6 have signatures +, +, +, −, −, − (Table 1.1). We associate the basis element s4 ∈ S[C] ⊂ S[D] with time and energy, and the basis elements s5, s6 with real and imaginary axes in a symplectic- complex plane. The dimensions of

4,5 4,6 XY X s = et, E s ∼ E,e s ∼ i (5.32) are assumed to be longitudinal in the cosmic dome relative to the vector hΩ of the vacuum [C] queue. A symbol like et designates the predecessor in stratum C of a higher-stratum E variable, here the skew-symmetric time variable et of stratum E. The deﬁning representation of so(3, 3) has skew-symmetric generators 1 ωβα = 4[sβ, sα], where α, β = 1, 2, 3, 4, 5, 6. (5.33) 4 Rank C = 4 supports the adjoint (commutator) representation of so(10, 6). We may therefore take the elements of the Lie algebra so[C] as the origins of the space-time and momentum-energy variables of stratum E, setting

µ E−C [C]5µ µ E−C [C]6µ −1 E−C C56 xe = X Σ ω , pe = E Σ ω , i = N Σ ω . (5.34) We now examine the conditions that must be met for these variables to be quasi- continuous. For deﬁniteness, consider the time operator ωE54 ∈ so(10, 6). The time dyadic t[C] := iωC54 = 4(is5s4) has spectra (−1, 0, 1). By the exclusion principle the polyadics of stratum D have at most one monadic factor γ5 and at most one γ6, so the spectrum of the cumulative representation iωD 54 := iΣωC54 (5.35) 5.3. FULLY QUANTUM EVENT SPACES 205 is still {−1, 0, 1}, but the multiplicities of these values are much greater than on stratum C. Of the 24 basic monomials in S[5], those that contain just one of the generators s4 and s5 transform according to a spin-1 representation of ω54; the other monadics transform according to spin 0. There are thus (24)/4 isomorphic replicas of spin 1 in stratum 5 and none of higher spin. TO DO: Check and ﬁnish. 080905 Rank E = 5 has basic monomials with from 0 to (24)/4 spin-1 factors, and therefore has a spectrum of ∼ 24 = 64K eigenvalues, still not adequate for the current resolutions of space-time-energy-momentum measurements. TO DO: Complete this estimate. These variables of stratum C are represented in stratum E by

Eβα E−C Cβα ωe := Σ ω . (5.36)

The generator s4 is the ﬁrst in S of negative square, and so is natural for the time- energy axis. Reserving the generators s1, s2, s3 for the space-momentum axes, and s5, s6 for the complex plane axes, we set

−1 E65 µ E54 µ Eµ,6 i = N ω , xe = X ω , pe = E ω , µ = 1, 2, 3, 4. (5.37) This uses the time quantum X, the energy quantum E, and the maximum N of |ωEYX |. Present experience, where the canonical relations work, is with a part of the spectrum . of |i| near the maximum value |i| = N, and yet this narrow band must have a multiplicity that passes today for inﬁnite. It is possible to meet both requirements. For example only, the band √ N − N < |ωEYX | ≤ N (5.38) √ is both narrow and populous: |i| √departs from 1 by about one part in N ≈ 0, and the multiplicity of this band is about N ≈ ∞. In the singular limit i → i, E → 0, and (supposedly) (ωYX )2 → −N 2, its extreme eigenvalue, so that the classical time axis emerges. This and the approximate validity of the canonical commutation relations imply

XEN = , N 1, NE ≈ 0, ~ ≈ 0. (5.39) ~ X

The conditions NE, ~/X ≈ 0 mean that these energies are below the present stratum of detectability in the particle laboratory. To represent the physical theory of the time axis of which the singular limit is an approximation, one must freeze i, eliminate E as a dynamical variable, and leave a large, ﬁnely spaced, quasi-continuous spectrum for t. Following the precedent set by classical Euclidean geometry according to the hypothesis of §5.2.1, E is eliminated by making it close enough to 0 to be ignorable. 206 CHAPTER 5. QUANTUM SPACE-TIMES Chapter 6

Fully quantum kinematics which describes the queue.

6.1 History space

At this point we are like molecular geneticists of an earlier period in hypothesizing without convincing evidence that beneath the great complexity of macroscopic nature lies a microscopic stratum of great simplicity in which are encoded some of the order observed in higher strata. The codes of Table 1.1 are one proposal for this structure, now of sub-queues rather than genes. We also have a limited correspondence between some hypothetical lower- stratum structure — “genotypes” — and higher-stratum data — “phenotypes”. We have the enormous algebra S of microscopic queue descriptors, but know the meanings of only (say) 120 second-grade elements representing position, momentum, angular momentum, µ α and charge coordinates x , pµ,Lνµ,Q , mostly frozen in the vacuum. The semantics for canonical instant-based quantum kinematics is known. The problem is to extend the semantics to Q history-based descriptions of quantum scattering and production experiments. The predicted amplitudes for such experiments are then traces of the descriptors. We must therefore cross two signiﬁcant conceptual crevasses by reversing two singular limits

CI ←CH ←QH. (6.1)

We pass from the canonical instant-based kinematics (CI) to the canonical quantum history- based kinematics (CH) in the next section §6.1.1; and thence to a fully quantum history- based kinematics (QH), in the following section §6.1.2.

207 208 CHAPTER 6. FULLY QUANTUM KINEMATICS

6.1.1 Canonical quantum histories The Dirac concept of system history is first illustrated for a quantum mechanical system with a one-dimensional space-time with discrete time coordinate t, to which is attached one coordinate q. The history probability-amplitude vectors of the system form a linear associative algebra A, whose sum is quantum superposition, the addition of relative transition-probability amplitudes, and whose product is catenation of experimental processes. H[A] stands for the quantum system with history-vector algebra A. It is assumed that the generators of this algebra carry their own time-stamps, so that when they are catenated they define their own chronological order in the product, which is not necessarily the order of writing. For example, if hI is a fermionic input, hIhI is not an input of two separate consecutive fermions, but 0. The corresponding identity for Clifford statistics is hIhI = khIk. The written order may still be the order of occurrence of control processes relative to the metasystem. A chronological structure of history is defined for each frame by a time-shifting energy † 0 † operator ∆E = ∆p0 = ∆E and an energy-shifting time operator ∆t = ∆x = ∆t , canonically conjugate in the sense that

[∆E, ∆t] = i~. (6.2)

∆E is the departure from energy conservation for the history and ∆t is the duration of the history; hence the ∆’s. The instant construct of binary observable or projector gives rise to three history constructs, representing the input, ﬁltration, and output of a kind of quantum system. [Conjecture:] They are associated with history vectors E, t in algebra A, which may be regarded as brief histories, through

∆E = 4E, ∆t = 4t (6.3)

One can express the dynamics of quantum mechanics equivalently in terms of two instant vectors or in terms of one history vector with two instants. The development of a statistical distribution looks like the motion of a material distribution but the meanings are distinct. In instant terms, one writes a dynamical development as ψ2 = Uψ1, meaning that an input at time t1 with vector ψ1 followed by the dynamical development U is equivalent, for ﬁnal measurements, to an input at t2 with vector ψ2. In history terms, one writes the amplitude for the briefest possible experiment history as

A = Tr DhE = ψ2hUhψ1,D = U, E = ψ2 ⊗ Ψ1. (6.4) This is as if a history is input with dynamics vector D = U and output with experiment vector E = ψ2h ⊗ hψ1. 6.1. HISTORY SPACE 209

For a slightly longer history, with four equally spaced instants t1, t2, t3, t4, a typical dynamical history vector has the form hDh = hUh ⊗ hUh, and an experimental history vector might have the form E = ψ4h ⊗ h3 = 2h ⊗ hψ1. (6.5)

The factor ψ1 represents an initial input at time t1. The factor h3 = 2h is an identity operator representing a direct connection from time t2 to t3 that means “do nothing” to the system in the specified time interval. h3 = 2h is a primitive representation of a vacuum. The factor ψ4 represents a final output at the time t4. The Schrödingerequation assigns a vector ψ(t) to each time t. It should not be concluded that ψ(N) ⊗ ... ⊗ ψ(2) ⊗ ψ(1) is the resulting history. This would represent performing an input operation ψ(t) at each time t, while in the actual experiment there is just one input operation, followed by a sequence of waits, followed by the output operation. It will be assumed here that each monadic factor in a polyadic history carries its own time information within itself. Then the order of factors in a catenation need not be the order of occurrence of the corresponding operations on the system; and the square of a monad in a queue does not indicate consecutive reiteration but is just the statistical norm of the monad. And linear operators, assembled from monads and dual monads, likewise carry time information. 080619 ...... [Define E in general. Prove CI ≡ CH.]

6.1.2 Fully quantum histories The queue vector space used in the following is SF. An experiment vector is a history vector hE ∈ SF describing the experimenter’s actions during the experiment. The dynamics vector hD ∈ SF is another history vector analogous to the operator hUh, representing the dynamical law that operates between experimental actions. The relative transition probability amplitude is then A = Tr DhE. (6.6) The vacuum vector hΩ is a special case of hE representing a grand do-nothing for the entire experiment: nothing in, nothing out. Usually all transition amplitudes are expressed as vacuum expectation values. The construct of the vacuum is dependent on the dynamics, so it will be taken up later.

6.1.3 Physics without functions The classical construct of a relation can be fully quantized; but that of a functional relation, bfunctionc, or mapping cannot without breaking quantum invariance. This is expressed in more detail as follows. A classical relation is a set of pairs. If X and Y are state spaces of classical variables, the state space of the generic relation R between X and Y is R = Rel(X,Y ) := P(X × Y ) (6.7) 210 CHAPTER 6. FULLY QUANTUM KINEMATICS

To form random variables one forms the linear closures of the three state spaces X, Y , R. The quantization is immediate: A quantum relation is a quantum set of quantum pairs, the quantum entity with vector space 2X⊗Y If X and Y are Grassmann algebras it is natural to use a Grassmann product X ∨ Y ⊂ X ⊗ Y instead of the entire tensor product X ⊗ Y . The classical construct of a mapping or function, however, understood to be single- valued, has no such natural quantization. The key obstacle is the concept of the diagonal. Recall that for any state space X, the bdiagonalc of X × X is the graph of the equality relation: 0 0 [=]X := {(x × x ) ∈ X × X : x = x }, (6.8) where × is the Cartesian product. The diagonal in X × X is a subset of X × X that is naturally isomorphic to X. Further, a brelationc between I [X] and I[Y ] is an element r ∈ 2X×Y , and induces a relation r × r between I [X × X] and I [Y × Y ]. Finally, a X×Y bfunctionc f : X → Y is a relation f ∈ 2 with the property that the induced relation f × f of pairs preserves the equality relation [=], mapping diagonal in X to diagonal in Y :

f × f : [=]X → [=]Y . (6.9)

These concepts do not quantize because the quantum correspondent of the classical Carte- sian product X × X of sets is the tensor product Xb ⊗ Xb of vector spaces, which has no diagonal, no subspace that is naturally isomorphic to either factor Xb. There is no meaningful concept of equality between quanta, no useful concept of quantum clone. This means that any construct of function or ﬁeld in quantum theory has to be basis- dependent. In a simple quantum theory with a simple quantum event space, ﬁeld theories may still use the concept of relation, but not the concept of function. The theories using this concept must be singular limits of ones that do not. These facts of quantum life were expressed by Wigner as the impossibility of quantum reproduction, and later, in quantum computation, as the impossibility of quantum cloning. Living systems are not produced by processes represented by vectors but by processes that generate entropy at the same time. The impossibility of quantum reproduction has little relevance to biology.

6.1.4 Field variables

Central space-time coordinates are built into the foundations of canonical quantum bfieldc theory, as part of the strategy of absolute space-time. Consider the theory of bgravityc, for example. In his gravitational action principle bHilbertc varied bgravitational fieldc variables κ gµν(x) without varying space-time coordinates x = (x ). In the resulting bPoisson Bracketc κ µ Lie algebra, gµν commutes with x , and the x are central in the Lie algebra of fields. There are no such central time coordinates in actuality, however. Our actual physical coordinates for a remote event are based on signals, usually electromagnetic, that reach us through the intervening space-time. These signals inform us about the intervening 6.1. HISTORY SPACE 211

bgravitational fieldcs as well as about the remote event. The material lattice of rods and clocks imagined by Einstein to coordinatize events is not different in principle from optical or radar coordinates, since it too is connected by electromagnetic and gravitational interactions. It seems that physical coordinates are more relative than general relativity admits. They are influenced by the ambient fields as well as by the choice of reference frame. Physical fields are non-central and non-local, and so are physical coordinates.

In generalized bKaluzac models like the present one, bfieldc variables are bshortc coordinates of an event, short also in that what are usually described as two different field-values at one event are actually two different events too close together to be resolved by current chronometry. In metrical terms, the ranges of the bfieldc variables we encounter in nature, measured in just noticeable differences, are much smaller than the ranges of external coordinates, in the current vacuum. It is assumed in the rest of this work that this metrical difference is the main distinction between a field variable and a coordinate variable. The usual distinction — that one is the value of the field function, the other the argument — must then arise in a singular limit of the organization of the vacuum. We no longer postulate two different constructs, coordinate variables and field variables, just one construct of event coordinate variables. The field is replaced by the queue. The vacuum is a predicate hΩh of the queue.

The bosonic field variables we must fully quantize are bgravityc and the bgaugec vector potentials of the bstandard modelc. The fermionic fields are the four leptoquarks in their three generations. The bHiggs fieldc we tentatively identify with i, after an earlier model [73]. In a fully quantum theory, events are powers (braced catenations) of differentials. That is, the event vector space is a Clifford algebra over a lower-stratum differential Clifford algebra. Similarly, it is proposed to replace the bfieldc by a queue that is a power of events. The queue vector space is a Clifford algebra, supposedly of stratum F, based on the canonical quantum theory, where a field polyadic of stratum F is a set of pairs of events and field-values of stratum E.

The dimensionality of the bstandard modelc gauge group is small enough to be the kinematic group of a stratum C with C < D < E. C = 3 permits the vector space V [C] of stratum C to have 16 dimensions. Since so(5, 1) suffices for the position-momentum variables and i, as in the theories of bSegalc and bVilela-Mendesc [66, 75], it seems possible that so(10, 6), the kinematical Lie algebra of stratum 3, is large enough to support the bgaugec vector potentials of the bstandard modelc as well. Events of stratum E = 5 then have an event vector space S[5] of dimension 2216 , sufficient events for a queue of them to approximate a continuous field. When the simple field is replaced by a queue, namely of quantum events, iterated Clifford statistics produces as many bstratumcs as needed. This results in a generic fully quantum theory with vector space S produced from the empty set 1 by iterating bbracingc I and Clifford bcatenationc operations. The entire simplified physical system under study is 212 CHAPTER 6. FULLY QUANTUM KINEMATICS

the bqueuec, and the vacuum is represented by a queue eigenprojector, of minimum energy in some sense, presumably of several dimensions. The Clifford algebra on SE is generated by first-grade Clifford elements gamma that flip particle-events into and out of existence in accordance with the Clifford law γ2 = kγk. Call the sign of kγk the bsignaturec of the element γ. Fermionic creators and annihilators, obeying γ2 = 0, are linear combinations of anti-commuting Clifford elements of opposite signature. Each generic event carries space-time-energy-momentum variables as well as extra variables for charges. These are all spins of an appropriate orthogonal group. Taken all together, according to the regularity hypothesis, they generate a semisimple Lie algebra, like the so(6) of Segal-Vilela-Mendes space. The enormous difference between long and short dimensions we must encode in the projector hΩh of the vacuum. S[E] is the vector space for the event, stratum E of S. The vector space for the queue stratum F is another subspace of the Clifford algebra,

SF = PF−E SE. (6.10)

This allows for F − E iterations of association, the number F − E being adjustable to provide the necessary large numbers. In the following F − E = 1 suffices. Both bfieldc variables and space-time coordinates of the canonical quantum singular limit derive from queue operators on SE. The classical event of E in turn is an integral combination of classical differentials (or differential events) of stratum D:

SE = PE−DSD. (6.11)

The state space of a typical classical ﬁeld is a set of pairs of ﬁeld values and events,

{F (x)} ⊂ {F × x}. (6.12)

6.1.5 The cosmic crystal To illustrate the concept of the quantum crystal ﬁlm, here is a simple cellular lattice that has both internal (short) and bexternal (long) dimensionsc:

(6.13)

It is sometimes convenient to call monomials in the Cliﬀord algebra S “cells”, their subsets “faces”, their grade-2 subsets “chords”, and their grade-1 subsets “nodes”. Like the cells of a classical mechanical phase space, these cells combine by superposition to form the entire one-monad vector space and by catenation to form many-quantum vectors for the quantum ﬁeld. Cells may also have common elements and be entangled 6.1. HISTORY SPACE 213

by superposition. This one-dimensional btrussc composed of 2-cells stands in for a four-dimensional truss composed of 16-cells

:= hint] ⊗ hext]. (6.14)

Each cell here is the bbracingc of a tensor product of an internal 1-cell, the vertical edge of each square, symbolizing an internal 10-cell, and an external 1-cell, the horizontal edge of each square, symbolizing an external 6-cell. Thus the model event cell is the braced product of 16 monads: 10 internal ones forming an internal 10-cell hint] and 6 external ones forming an external 6-cell hext]. The lower edge of each cell in (6.13) symbolizes all its external or longitudinal dimensions hext]; the left-hand edge, all its internal or transverse dimensions hint]. The other two edges of each square are drawn only to make the cells look cellular. The four vertices of each square stand for the 216 vertices of the 16-cell. Turn now from this toy model to one that is slightly closer to physics. The vacuum truss is a vector Ω supposedly in S[7]. There is no pressing reason to suspend either Lorentz SO(3, 1) or Standard U(2 × 3) symmetry at the cell stratum. The Segal-Vilela-Mendes SO(6) variant of Poincaré symmetry will also be assumed for each cell of a first model, although this symmetry is already broken by the vacuum environment of the cell, and therefore is quite an approximate symmetry of the cell at best. One crude model of a quantum space-time is a bhypercubicc model, a four-dimensional checkerboard, with time axis along a principle diagonal. Each cell represents a Cliff(4R), and adjacent cells share some of their four generators. A more physical model, allowing for internal dimensions and subsequent organization, is composed of 16-dimensional hypercubical cells, still connected into a 4-dimensional array, now forming a thin truss or dome. This 16-hypercube four-dimensional model is an analogue in higher dimensions of the primitive truss of (6.13). Its cell has the group SO(16R), omitting signatures for the moment. This is broken to SO(4, R) × SO(2R) × SO(10, R) by the dome structure, representing the De Sitter variant of the Poincarégroup, the complex phase group, and internal groups. The SO(4, R) × SO(2R) factor is represented by the truss with a spectrum of macroscopic extent, to describe the long dimensions and i. The internal SO(10, R) must be broken to the standard model S(U(2) × U(3)) as in GUT. In a 16-hypercubic model, the prototype cell may result from a vector sC>, the top Clifford element of stratum S[C], representing the input of 16 independent generating units as chords of the truss cell. All the coordinates ωEβα vanish for this cell:

ωEβα sC> = 0. (6.15)

Therefore it can be regarded as an borigin cellc of the vacuum truss. The group of its long dimensions might be SO(5, 1), making the imaginary axes X and Y spacelike, and leaving SO(5, 5) for the short dimensions transverse to the dome. The bKaluzac-Klein-Yang-Mills- bDeWittc middle bgauge groupc is the orthogonal group SO(5, 5) of a transverse face. 214 CHAPTER 6. FULLY QUANTUM KINEMATICS

6.1.6 The organization of the imaginary unit The organization i → i must occur by the event stratum E, whose singular limit has only four dimensions. −1 E−C C65 E In the model ϑo, E = 6, and i is the inﬁnitesimal transformation l iΣ ω : S → SE induced by : S[C] → S[C], and normalized to unit maximum magnitude by a scale factor of its maximal eigenvalue l:

i := l−1ΣE−CγC65, γC65 = γC6γC5. (6.16)

MORE TO DO: Estimate i. The maximum eigenvalue of i2, occurs midway between the top and bottom of at the E 216 grade M = 26 = 2 . The quantized imaginary has the form

i = l−1ΣF−CωCXY , (6.17) normalized to have the extremum eigenvalue −1. The vacuum hΩ is to be an eigenvector of . (i)2 = −1. (6.18) The hypercharge-isospin-color invariance Lie algebra u(2 × 3) is a direct sum; relative quantum phases between isospin vectors of su(2) and color vectors of su(3) have physical meaning. Therefore u(2 × 3) is an invariance Lie algebra of a composite system composed of one system with su(2) invariance and one with su(3) invariance. If we were working within complex quantum theories we could regard the two parts as a 2-cell and a 3-cell. Since a real quantum theory has been posited, u(2 × 3) is represented in the commutant in o(3 × 6) of a generator i that provides a complex imaginary i in the singular limit, and we must assume an organization that permits us to identify the is of color and isospin. The composite system would then be a 9-cell composed of a 3-cell and a 6-cell with no mutual coupling. 080510 ...... [ Define the vacuum fine structure further.] Since the momentum coordinates are dual to the space-time coordinates relative to i, one expects that large masses are associated with the small internal differentials and much smaller masses with the external differentials. The extended btrussc of large masses connected by small ones begins to feel like an extended molecule of nuclei bound to each other by electrons.

6.1.7 Statistical form The probability form H : S → SD of the self-Grassmann vector algebra S must be ﬁxed, especially its signature. Then the available signatures must be allocated to the various Lie algebras of the fully quantum theory. 6.1. HISTORY SPACE 215

The construction of the Grassmann algebra 2V makes no mention of a probability form on V . It is an affine construction. It builds on the defining contractions of vectors in V with dual vectors in V D. However if there is a probability form H on V , it can be used V to define one on 2 , still designated by H. And there is a natural one on C, the absolute norm kzk = |z|2, to begin the recursive definition. One can proceed as follows: D One posits the canonical commutation relation (4.16) between I and I ; and the product relation H(u ∨ v) = (Hv) ∨ (Hu). (6.19) Note that this is not about the hermitian adjoints of operators. It relates the hermitian adjoints of input vectors, which are output vectors. If u, v ∈ V are anticommuting vectors of norm +1, then their Clifford product has negative norm:

H(u t v)(u t v) = (Hv t Hu)(u t v) = −Hv t Hu u t v = −1, (6.20) then kI(u t v)k = (H(Iu t Iv)(Iu t Iv) = ((HIv) t (Hu)) (Iu t Iv) (6.21)

√This leads to the signatures indicated by tildes in Table 1.1. The signature of stratum 3 is 16 = 4. If 6 dimensions of S[4] are used for the long dimensions of the cosmic dome, of neutral signature, then the internal groups must be accommodated within the remaining 10 dimensions of signature 4, defining the algebra su(7, 3). The reduction su(7, 3) → su(7) ⊗ su(3) easily leaves room for color su(3) in one factor and the remaining internal groups within the other. The dramatic change is in the enlargement of the Lorentz group to quantize the canonical i1. One may look to the transition from the Galilean group to the Lorentz group for some guidance. There were in principle two possible signatures, the definite and the Minkowskian. Lightspeed settled the question in favor of Minkowskian. This answer was confirmed by the fact that Newtonian kinetic energy p2/2m is positive rather than negative; an orthogonal rotation would reduce the energy — the timelike component of momentum- energy — below its rest value, not increase it. The law of transformation of the electromagnetic field provides further confirmation: the invariant is the indefinite E2 − B2, not the definite E2 + B2. The variation from the canonical to the fully quantum is more involved than the variation from the Galilean to the Lorentz, in that it not only adds dimensions to the µ E space but also adds an index to the canonical coordinates: x ∼ ωC ∈ so[E], with C = µ5 in an appropriately adapted frame. The canonical theory has a Minkowskian metric on 4R; the fully quantum theory has correspondingly an invariant metric on so[C]. In the singular limit the square of one skew- [C] symmetric element ω56 has a frozen value of cosmological magnitude and all other basic 216 CHAPTER 6. FULLY QUANTUM KINEMATICS

[C] elements ωβα are small by comparison. Such vectors represent unitary rotations in a plane close to the 56 plane. What is regarded in the canonical theory as an inﬁnitesimal unit translation peµ becomes a small unitary rotation in a distant µ6 plane:

[C] peµ ⇐ (ξ/X)ωµ6 (6.22)

µ Translations xe of momentum, similarly, are singular limits of rotations in a remote µ5 plane: µ [C] xe ⇐ (ξ/X)ωµ6 (6.23)

In order that i be skew-symmetric, γ5 and γ6 must have the same signature. The unitary rotation in (say) the µ6 plane will respect the value of

−1 −1 −1 Hω65H ω65 + Hωµ6H ωµ6 + Hωµ5H ωµ5 = const. (6.24)

It can significantly increase the last two terms relative to their initial values while decreasing kω56k imperceptibly. The usual four differentials dxµ5 ∼ ωµ5 are the ones ordinarily perceived as event coordinate differentials, inherit the Minkowski signature, and give rise to long dimensions of the cosmic dome. The remaining transformations ωρσ (ρ > σ ) whose signatures are in question ordinarily not seen as event differentials. The probability form on the Fermi algebra S[L] of stratum L is designated by H[L]. [L] [L] [L] The Killing form on su is designated by k . Relative to any basis sα for S , a basis for su[L] can consist of operators

ω[β,α], ω{β][αk, (6.25) in which [...] indicates skewsymmetry and real matrix elements, and {...} indicates symmetry and imaginary matrix elements. The hypothesis that the canonical unit 1 ⇐ r is the singular limit of ω65 implies that the 5 and 6 axes of the space S[C] have the same signature. Suppose C = 4, so that S[C] has 16 dimensinsions. The deﬁnite nature of the metrics underlying the standard model groups implies that the remaining axes 7, 8,..., 16 of S[C] have the same signature. There are thus three signatures to be combined: ±2 for the axes 1, 2, 3, 4, ±2 for the axes 5,6; and ±10 for the remaining 10 axes. The intrinsic signature of S[C] is 4 if C = 4. It does not accommodate the signatures inferred from the canonical theory.

6.2 Fully quantum scattering

In a scattering experiment, quanta of known properties come together and interact, and particles of changed properties are produced and are measured, for comparison with theoretical prediction. 6.2. FULLY QUANTUM SCATTERING 217

In canonical quantum theory it is assumed that the transition amplitude for a scattering experiment can be expressed as A = [DhE] (6.26) in terms of a dynamics vector hDh representing effectively rigid external influences on the system, and an experiment vector hE] representing irreversible influences on the system, such as creation or annihilation with registration. Both [Dh and hE] are poorly defined singular expressions because the underlying events have continuous spectra. hE] is often assumed to have the form

hE] = ss0 . . . s00hΩ] (6.27) where hΩ] represents a vacuum and s, s0, . . . , s00 represent input and output quanta of determinate properties. The vacuum hΩ] is supposed to be determined by the dynamics hD]. When hD] has the Hamiltonian form

hD] = . . . e−iHδt . . . e−iHδt ..., (6.28) where each exponential connects two adjacent times, and H has a unique eigenvector hω] of minimum eigenvalue, with projector hωh, then

hΩ] = ... hωh ... hωh ... (6.29)

Due to spontaneous symmetry breaking by organization, the vacuum may be described by a a projector hΩh of dimension greater than 1 instead of a vector hΩ]. Then none of the vectors in the subspace ΩS need have the full symmetry of [Dh. This description assumes an experimenter who can resolve the degrees of freedom that distinguish the various eigenvectors of the vacuum projector hE]. MORE TO DO The Q amplitude has the same form A = [DhE], but its quanta have discrete spectra and the vectors [Dh and hE] are regular. In the example of Segal-Vilela space, the vacuum hΩh is assumed invariant under the skew-momenta QµY but not under the skew-coordinates QµX or the symplectic spin QYX . The dynamics [Dh is supposed invariant under all the QC . A two-event propagator (or correlator) is the amplitude for experiment projectors of the form 0 0 hEh = hsEhsEhΩhsEhsEh, (6.30) 0 in which each of the operators hsEh, hsEh represents a terminal operation (a superposition of input and output) and hΩ] represents a vacuum. In the c and q theories, the one-quantum terminal processes are conveniently chosen to be eigenvectors of the momentum-energy pµ. Since pµ is conserved for each quantum in the absence of the other, and so can be readily prepared by the experimenter at a great distance from the scattering interaction. 218 CHAPTER 6. FULLY QUANTUM KINEMATICS

It is natural to do the same in the Q theory, but the components of the Q momentum energy do not commute. The be joint eigenvectors of the momentum-energy, the scattering vectors MORE TO COME Canonical quantum theories compute transition amplitudes between momentum-energy eigenvectors to predict experimental scattering cross-sections. Here a fully quantum correspondent is constructed for a momentum-energy eigenvector. [Do 080808 To be continued.]

6.2.1 Experiment time The experiment vector must specify the duration ∆T of the experiment, from input to output. In the canonical quantum theory ∆T is allowed to approach infinity, on the grounds that the input and output take place so far from the target, relative to the scale of the system, that their exact times are unimportant. The assumption ∆T → ∞ incorporates the assumption that the spectrum of time is homogeneous and unbounded, and neither assumption holds in a fully quantum theory, nor presumably in nature. If this limit ∆T → ∞ were taken too literally, in nature the experiment would run afoul of the Big Bang. The vector eigenspaces of the end times t = ±T have few dimensions, even if those for the middle times |t| T have enough dimensions for a scattering experiment. What passes as an infinite time lapse ∆T in the canonical quantum theory is an infinitesimal one ∆T T from the point of view of the fully quantum theory. To deal with this inhomogeneity of time in the fully quantum theory ϑo, a fixed ∆T is assumed to exist that is large compared to relevant scattering times and small compared to relevant metasystem times, and experiments are supposed to have proper duration ∆T as a working√ approximation. In the canonical limit ∆T,T → ∞. A plausible first guess is ∆T = XT , the mean proportional between the quantum of time X and the maximum time T .

6.2.2 Momentum vectors A canonical quantum momentum(-energy) vector hk has the factored form

µ ik xµ hk = he h0p. (6.31)

µ ik xµ The operator he h translates momentum by ~kµ. Its operand h0p is a ﬁduciary eigenvector with the properties of 1. zero momentum,

2. zero angular momentum: Lorentz invariance,

3. totally uncertain position, and 6.2. FULLY QUANTUM SCATTERING 219

4. projective invariance under i (implicit).

Such a vector is called a bmomentum originc here. One such vector is the bprinciple vectorc µ h1x of a space-time x frame [24], whose components in that frame are all 1: h1x = h0p. In the Dirac theory it is automatic that the operator i commutes with the translation operator and ﬁxes the ray of the momentum origin vector. In a real version of a fully quantum theory, i invariance is contingent. It will be arranged ad hoc until a theory of the organization of i can be worked out. Both the momentum-translation operator and the momentum-origin vector are singular in the canonical quantum theory. Regular variants are required in the fully quantum theory. The theory ϑo assigns the role of the fully quantum variant of i to the spin component E E ei := ω65/l acting on S . For such spin components to act as inﬁnitesimal isometries of E Grade1 S they must act as the bcommutator representationc

E E ωβα = 4sβα (6.32)

E This ei fails to commute with either the momentum components ∼ ωµ6 or the position E components ∼ ωµ5. E E The momentum components ωµ6 span a linear subspace of 4 Grade2 S that is not E ∼ a Lie subalgebra but generates a Lie subalgebra sop = so(3, 2) called the bmomentum E subalgebrac of S . There is a canonically conjugate position subspace that generates E ∼ another Lie subalgebra sox = so(3, 2), the position subalgebra. The two together generate an so(3, 3) ⊂ soE. The integer l > 0 is determined so that the maximum eigenvalue of −ei2 is 1; and in the canonical limit only vectors with −ei2 ≈ 1 are used, and only operators that respect this condition. A probability operator rho of the fully quantum theory that commutes with ei has the form † ρ = ρ− + ρ−ei, [ρ±,ei]± = 0. ρ± = ±ρ±. (6.33) (6.31) is an instant vector, not a history vector, and so need have no natural full quantization. But a history vector of the form

E(k, k0) = k0h ⊗ h1h ⊗ ... ⊗ h1h ⊗ hk (6.34) describes an experiment with an input of momentum k and an output of momentum k0 and so should have a fully quantum correspondent Eb(bk, bk0). E A fully quantum variant h0p ∈ S would have at least the properties

2 hpeµh0p = 0, i h0p = −h0p, (6.35) useful for a fully quantum scattering theory, with large uncertainties ∆xµ and zero uncertainties ∆pµ. 220 CHAPTER 6. FULLY QUANTUM KINEMATICS

E E (6.35) implies that all four ωµ6 ∈ so , for µ = 1,..., 4, annul hb0p. This implies that E their six commutators, the ωνµ, also annul hb0p. The Schrödinger generalization of the uncertainty inequality is 1 1 (∆A)2(∆B)2 ≥ |h[A, B]i|2 + |h{A − hAi,B − hB i}i|2, (6.36) 4 4 in which A, B can be any two operators, and their expectation values hAi, hB i and disper- sions ∆A, ∆B refer to a given vector s. This assumes a positive-definite probability form. It is therefore still valid if the probability form is indefinite and the orbit of the vector s under the algebra generated by A and B lies in a positive-signature subspace of the vector space. The uncertainty inequality (6.36) clearly excludes simultaneous eigenstate for the canonical quantum position and momentum, which support a definite probability form, but not for the fully quantum ones, since their commutator is another spin component and can have eigenvalue 0. The top element sE> ∈ SE is invariant under all of soE and so has exactly zero momentum. It also has exactly zero position, and it does not maximize but annulls −i2, so it is no candidate for a momentum origin but a kind of vacuum, which can be useful for making one. The eigenvalue of −i2 can be raised from 0 with a ladder operator. For example, the component Lz of angular momentum is raised by the ladder operator Lx + iLy. To make a ladder operator, however, an imaginary operator ei that can play the role of i is necessary. It should have the properties

1. ei = −ei†,

2. ei2 ≈ −1 for enough vectors,

3. ei commutes with all the other operators among the ωEβα that enter into the ladder construction, and

4. ei → i in the canonical limit. By “enough” vectors is meant, enough to simulate the low-energy sector of the Hilbert space of the canonical limit. A product of regular monadics will be imaginary if and only if the number of monadic factors with positive square is even and the number with negative square is odd. This is readily proven case by case. To raise ωE65, a ladder built of ωE45 and ωE64 is useful. An imaginary that commutes with these three is required for such a ladder. Such an imaginary is readily available on stratum C, where the spins in question originate. The polyadic [C] [C] [C] [C] ei := s2 s1 = s21 (6.37) 6.2. FULLY QUANTUM SCATTERING 221 commutes with ωC65, ωC54, and ωC46. Such an imaginary ei can be used to construct the four ladder operators and their adjoints E E † E E Λµ := eiωµ5 + ωµ6, Λµ := −eiωµ5 + ωµ6 (6.38)

E that increment ω65 by ei or annihilates its operand:

E h E i ω65 [ΛµΨ] = Λµ ω65 +ei Ψ. (6.39)

080621 ...... [Construct a momentum origin!] E E−C [C] The cumulative spins sβα = Σ sβα generate a representation on stratum E of the E E relativity group of stratum C Among the left multiplications ωβα = Lsβα are both the momentum and coordinate operators of the fully quantum theory. The catenation [C] [C] [C] [C] s4 s3 s2 s1 (6.40) E is annulled by each momentum component peµ = 4sµ6, µ = 1,..., 4. E The four momentum operators γµ6 relative to one frame generate a Lie subalgebra ∼ E so[E|pb] = so(3, 2) ⊂ so(3, 3) that contains approximate correspondents peµ ∼ γµ6 ∈ so[E|pb] E of the canonical momentum variables, and angular momentum operators Leµν coupling them to one another. Call this the (fully quantum) momentum subalgebra for stratum E, relative to its frame. Its infinitesimals include cumulative spin angular momenta on [C] [C] stratum E of individual spin angular momenta γµ6 = Lsµ6 on stratum C, four of the 15 spin components that generate so[C]. [C] There is no monadic vector s of stratum C which the spin angular momenta γαβ all annul and which could therefore provide a bmomentum originc. The top polyadic vector C> [C] [C] . s ∈ S is simultaneously annulled by all 15 γαβ = 0. In particular the eight mean coordinates and momenta all vanish exactly on this vector, in consequence of the Clifford- algebraic commutation relations. This is not the metaphoric North Pole of Sphereland but the center, so central that it cannot be shifted by any relativity transformation in SO[C] and so s useless as fiduciary vector. The more restricted catenation

[C] [C] [C] [C] [C] γ6 γ4 γ3 γ2 γ1 (6.41)

[C] is invariant under all the fully quantum momentum operators γµ6 and not under the [C] fully quantum space-time coordinate operators γµ5 . Therefore its brace is adopted as momentum origin for stratum D:

[C] [C] [C] [C] [C] D hb0p := I(γ6 γ4 γ3 γ2 γ1 ) ∈ Grade1 S . (6.42) 222 CHAPTER 6. FULLY QUANTUM KINEMATICS

This monadic of SD also belongs afortiori to SE, and the E stratum coordinate and momentum operators act upon it. 080616 ...... [What is the preferred E-stratum isomorph of the C-stratum first-grade [C] generators γα ? They themselves? Or their double-bars? Or their Σ’s? And what is the preferred E-stratum isomorph of the grade-16 top C Clifford polyadic? There are even more possibilities for the top polyadic than for the generating monadics.] The experiment description is ignorant about the remote past and the distant future of the system, the times outside the interval ∆T . In a canonical quantum theory this ignorance is consigned to infinity and oblivion. In the fully quantum theory, however, our ignorance concerns almost the full range of time, and cannot be left out of the picture. One way to express this ignorance is to assume a sharp description confined to the interval ∆T . A full quantization of (6.34) that is restricted to ∆T will suffice. This is formulated factor by factor. One fully quantum correspondent of the translation operator factor replaces the translation of canonical quantum momentum-energy space through k by a rotation of its corresponding angular momentum quantum space through an angle θ:

Eµ5 µ θµ5ω ik xµ hθ = he ⇒ hk = he h0k. (6.43)

As the four angles θµ5 vary, hθ ranges within the subgroup SO(3, 2) ⊂ SO(3, 3) that ﬁxes the 6 axis. This group has 10 parameters instead of the 4 of the translation group or of µ the four θµ5. However the four parameters θµ5 are related to the momentum energy k in the singular limit by 1 θµ5 → lXkµ, ωEµ5 → xµ5 (6.44) lX where l is the maximum eigenvalue of : iωE65 :. The other 6 parameters of SO(3, 2) enter Eµ5 as commutators of the operators θµ5ω with one another, in the higher-order terms of the exponential in (6.43). These higher-order terms are small for ordinary values of kµ due to the large denominator lX. It is necessary that the large number l overwhelm the small Heisenbergchron X, making the product lX large on the laboratory scale. 080612 ......

6.2.3 Quantum topology One does not use the heuristic cells of phase space to infer the topology of the canonical quantum phase space with coordinates (q,b pb); nor can one use the heuristic cells of a quantum event space to infer its topology. A topology can be defined by the algebra of its admissible coordinates. The algebras of continuous, differentiable, or analytic functions on a manifold define corresponding topologies. There are then at least two quite different senses in which a topology can be quantum or non-commutative. A topology whose coordinate algebra is the operator algebra of a vector space, here an event Clifford algebra, is 6.2. FULLY QUANTUM SCATTERING 223 quantum in a weak sense. A topology is quantum in the strong sense if it itself is a quantum variable, necessarily of a higher stratum than the event, defined, say, by a Clifford algebra over an event Clifford algebra. Evidently a fully quantum theory can describe quantum topologies in the strong sense.

6.2.4 Origins of the coordinates The coordinates on canonical space-time are supposed here to be classical limits of quantum operators of a fully quantum stratum E. One must distinguish between space-time coordinate operators of one elementary particle and those of astronomical bodies. Parti- cle coordinates belong to flat Minkowski space-time, supporting the Poincarégroup, and they and their canonically conjugate momentum-energy variables might have origins in the so(16) of stratum C, with representations on all higher strata. Astronomical voordinates on curved Einstein space-time support the diffeomorphism group, which is first approximated on stratum E. One maximal commuting set of event variables in so(16; σ) is

(ω12, ω34; ω56; ω78, . . . , ω15 16). (6.45)

In the 120 spins ωβα one ﬁnds the

ωµ5, µ = 1,..., 4 (6.46) which provide four Heisenbergexternal coordinate generators, a quantum imaginary generator ω65 = li whose representation in stratum E is a large but frozen pure imaginary li, and all the remaining spins, whose representations in stratum E become 0 in the classical limit. 080612 ...... [Check whether the signatures matter.] Coherent, perhaps superconducting, propagation occurs along the Heisenbergexternal dimensions. The great differences between internal and external masses and coupling constants are differences between propagations in the long and short dimensions. The quantum event cell I[X] has a vector space X. A natural candidate for a quantum field is a quantum variable I[Y ] with vector space

Y = 2X , (6.47) the real Cliﬀord algebra over X. This formula can be read classically too: if X is the sample space for an event, 2X is the sample space for space-time as a queue of events. If X is simple, the random object 2I[X] is a random set.

6.2.5 Fully quantum fermions Here the fully quantum correspondent of a quantum field of leptons or quarks is formulated. In the quantum kinematics neither field values nor coordinates commute, and the concept 224 CHAPTER 6. FULLY QUANTUM KINEMATICS of functional relation among non-commuting variables is not well-defined in any invariant sense. A spinor field of the standard model can have the general structure ψτ (x), accepting external event coordinates x and an internal basic vector sτ specifying Lorentz spin, hy- perspin, isospin, and chromospin, to deliver a fermionic bgeneratorc ψ for a fermion with the specified properties. The corresponding fully quantum operator ψ should be a fermionic annihilator ψ for an event with specified coordinates and spins. It is assumed that sτ belongs to stratum C, sξ belongs to stratum E, and the operator ψ belongs to stratum F. It is convenient for this unification of internal spin and external coordinates that S[C] ⊂ SE; the internal spin operators of stratum C are among the spin operators of stratum E, and so are the event coordinate operators. 080522 ...... If bfield variablesc and coordinate operators merge into one operator algebra, one must explain the canonical quantum limit, where some coordinates xµ commute and others yα are functions of them determined by the dynamics. Organization can do this. For example, consider a gas of particles with coordinates xn, yn, zn at one time t. The class of all these triples defines a ternary relation among the three variables x, y, z, but not a functional dependence of two coordinates on the third. To be sure, all three are functions of n, but the label n is arbitrary, without physical meaning, so this function has no physical meaning either. Suppose however the gas condenses into a droplet, thread, or bubble, organizations with 0, 1, or 2 long dimensions and 3, 2, or 1 short dimensions, respectively. If enough particles are so organized, 0, 1, or 2 manifold coordinates emerge that are approximately classical variables, even though the particles are quantum, and the atom variables may be smooth functions of these manifold coordinates. The condensation then creates new collective physical variables, makes them effectively classical, and establishes a functional relation of the old quantum coordinates on these new collective variables. Space-time and fields seems to originate when cells organize into a quantum crystalline structure with some long dimensions for space-time and some short ones for the field. One can formulate how this process might take place in a quantum cell system. Clifford n bfull quantizationc converts the bgaugec potential into a cumulative coordinate Σ ωαµ, a E µ n µ5 representation in G of the Lie algebra ωαµ ∈ so[D], converts the coordinate x → Σ ω , and converts the momentum pµ into ωµ6, up to a sign. The stratum difference F − D is at least 2. The action contains the commutator

n µ6 n n [Σ ω , Σ ων6] ∼ Σ ωµ,ν], (6.48) which becomes a derivative with respect to xµ in the classical limit. One can expect that owing to such derivatives in the action, discontinuities in the classical limit result in unbounded momentum and kinetic energy, and so are energetically suppressed. 6.3. FULLY QUANTUM GRAVITY 225

Therefore in the canonical quantum limit a bfieldc F arises that is effectively differentiable with respect to the classical space-time point of X. In general the effective field will be multivalued. If the bfieldc is double-valued at a point, it will therefore be double-valued in a neighborhood for the same energetic reason. Hopefully, then, an n-valued field is effectively the same as n single-valued fields on the same region of space-time. If single-valued fields have a significant zero-point energy in the canonical quantum limit, there will be energy-gaps between the single-valued, double- valued, ..., fields. The multivalued field will require more energy than the single-valued. Energetic considerations then favor single-valuedness much as they favor differentiability in the classical limit.

6.3 Fully quantum gravity

To fully quantize gravity one may analyze the global field into events and local fields at events, fully quantize both the event and the local field, and then assemble them into the global gravitational field. These steps are taken in §6.3.1– 6.3.3.

6.3.1 Fully quantum events It has been posited that space-time coordinates, momenta, and i emerge in a stratum E as cumulative representations of swaps ωβα ∈ so(10, 6) of stratum C. This assumption is tested here. For it to hold, a vector subspace ΨE ⊂ SE must exist with a nearly constant value i2 ≈ −1 where i = N −1ωE56, macroscopically large and variable values for the space- µ Eµ5 time coordinates xe = Xω , negligible values for the momentum-energy coordinates pµ = EωEµ6 ≈ 0 and the other generators ωEβα and for the uncertainties in all these operators. D D The ωβα act on the 16-dimensional vector subspace S1 ⊂ S as commutators with D elements sβα ∈ S2 : 1 ωβα := 4s . (6.49) 2 βα Here

s1 := I1 = 1, s2 := Is1 = 1, s3 = Is2 = 1, s4 = I(s1s2) = 1 1,..., (6.50) with s5, . . . , s16 given in Table 1.1. This is the deﬁning representation of so(10, 6). These operators induce operators on the next stratum by the Lie homomorphism (3.22). βα E Each ω of stratum C acts on Iψ ∈ S2 as Eβα E−C Cβα D ω := Σ ω I (6.51) Since the vector space SD is real, antisymmetric operators like ω21 on this space cannot be diagonalized. But looking back to the condensation already required by a real 226 CHAPTER 6. FULLY QUANTUM KINEMATICS

Segal-Vilela-Mendes space, we can set aside s56, which commutes with s12, s23, s31, as an “incipient i” and use it to construct “incipient eigenvectors” of (say) ω12, anticipating that these will give rise to the physical i and proper eigenvectors after organization on a higher −1 F −D stratum F centralizes i = N Σ s56/l → i. Then the Cliﬀord element s31 + s56s23 is an 12 incipient eigenvector of ω with incipient eigenvalue s56:

1 ω12 (s + s s ) := [s , s + s s ] = s (s + s s ) . (6.52) 31 56 23 2 12 31 56 23 56 31 56 23

These are the seeds of the event coordinates of the next stratum D:

βα Eβα E−D Dβα E−D x ∼ ω = Σ ω = Σ 4sDβα (6.53) up to dimensional constant coeﬃcients. This is a concrete algebraic realization of the Segal-Vilela-Mendez commutation relations for so(6; sigma). In fully quantum physics the system interface is shifted to move some of the basic space- time variables x from the metasystem to the system, where high-resolution measurement exposes their non-commutativity and discrete spectra. In the fully quantum theory, the components of x do not commute, and there is no basis of simultaneous eigenvectors hx. It is supposed here that the fully quantum correspondent of x is, up to dimensional constant factors like X and ~, the many-quantum operator

F F−E E F−C [C] ωβα = Σ ωβα = Σ ωβα (6.54)

F [C] on S induced by the operator ωβα ∈ so(10, 6).

6.3.2 Time form

The spin form γµ of the Dirac equation defines the causality form by its self-anticommutators and defines the Lorentz group by its self-commutators. To quantize gravity it seems enough to quantize the spin form field γµ(x). The classical spin form maps each vector in a certain vector space, originally a 4- 4 dimensional Minkowski tangent space R , into a first-grade Clifford element. The brace I too maps vectors into first-grade Clifford elements. The restriction of I to any stratum L is [L] [3] designated by I . Then I is isomorphic to the spin form of a 4-dimensional Minkowski 4 [4] [3] space-time R and I , which includes I , is the spin form on a 16-dimensional space of signature 4. These spin forms are native to a fully quantum theory and do not have to be imported. In the following constructions, the Clifford algebra S4 of stratum C is used as [3] vector space for a lower-stratum seed of a space-time cell, I is the seed of the gravitational [4] field, and I is the seed of all the standard model fields as well. 6.3. FULLY QUANTUM GRAVITY 227

6.3.3 Fully quantum gravitational potentials It is reasonable within the conceptual framework of full quantum theory to seek the roots of both the causality form and the probability form in lower strata of the system queue. It seems doubtful, however, that a linear space of low dimensions can support two different metrical forms without leading to a clash of symmetry groups. The physical interpretations of the forms, moreover, make it implausible that they are independent. Their functions overlap when the probability form is indefinite like the time one. The causality form distinguishes timelike translation of negative norm from spacelike translations of positive norm. Operators can be used reflexively, to transform from the initial experimenter to another, or transitively, to transform the system; sometimes these are called passive and active interpretations. Timelike translations are feasible both transitively and reflexively; spacelike ones are feasible only reflexively. An indefinite probability form also distinguishes by its sign between transitively feasible or unfeasible actions, namely the input-output actions represented by vectors. This means that these two forms must be related, although one concerns classical elements — space- time events — and the other quantum events — quantum input-output generation. It is natural to explore the assumption that Assumption 8 (Chronometrical hypothesis) On some stratum C < E, the causality form is the probability form: ig[C]h = iH[C]h (6.55) 080211 ...... [Segue] Classical gravitation theory has a sequence of increasing groups with correspondents in fully quantum theory proposed in §4.4.1. To represent the classical gravitational field, schematically speaking, one combines Dirac spin variables γµ and classical coordinate variables x into a variable field function γµ(x) that defines the chronometrical tensor field 1 g (x) = [γ , γ ] . (6.56) νµ 2 ν ν − µ It is supposed that in the fully quantum theory corresponding operators xe and γ are to be combined in a corresponding way. The fully quantum field is a descendant of stratum C:

SF = ΠF−CS[C]. (6.57)

It therefore has the natural spin structure

F−C [C] γalpha := L Σ sα, sα ∈ S. (6.58) By the chronometrical hypothesis (Assumption 8) the probability form of stratum C deﬁnes the causality form of that stratum and all higher ones, and none needs to be attached. Like 228 CHAPTER 6. FULLY QUANTUM KINEMATICS the Cartan repere mobile, the quantum cell provides a local metrical structure; and then the connection between cells deﬁnes the global one. 080319 ......

Canonical quantum bfield theoriesc use a complex vector space, sometimes with an indefinite Hermitian form, as in the theories of Gupta and Bleuler. To generate the complex Lie algebra of such a vector space from an underlying real Lie algebra, however, we must include an bimaginaryc i among the generators. This creates a radical and makes the canonical Lie algebra structurally unstable in principle, though it seems to result in no infinities. For the sake of structural stability a real finite-dimensional vector space is therefore assumed. To create an approximately central i near a singular limit, a global organization that freezes a generator i suffices. to convert a real quantum theory into a complex one [72]: If A is (at least) an algebra and e ∈ A then

A\∆e := {a ∈ A : ∆e · a = 0} (6.59) is called the bcentralizerc (or bcommutantc) of e (in A). If A is a real algebra and i ∈ A has square −1 then A\∆i (often designated by i0) is a complex algebra with imaginary unit ∼ i. A can be factored as A = A0 ⊗ Alg 2, where Alg 2 consists of 2 × 2 real matrices, and e = 10 ⊗ , where 10 is the unit operator in A0 and

= 0, −11, 0 . (6.60)

Any two-dimensional projector ρ ∈ A\∆e can be written in the form

† ρ = ψψ , ψ = ψ0 ⊗ 12 + ψ1 ⊗ e, ψk ∈ A0. (6.61)

Organization is the most plausible physical origin of such centralization in the present context.

We have provisionally assumed that every bstratumc has the same real Cliﬀord statistics. Therefore every bstratumc has an invariance Lie algebra so(N+.N−) belonging to the D sequence of classical groups, until broken by organization or a singular limit.

Conveniently, the signature of the bstratumc of S with dimension n+ + n− is √ n+ − n− = n+ + n− for n+ + n− > 2. (6.62)

Therefore the so(4; σ) of stratum 3 is the Lorentz Lie algebra; while so(n, σ ) for n, σ → ∞ n n√ is asymptotically neutral as required for bFermi statisticsc, in that signature σn = n = o(n); and a Pavel variant of Bose statistics is deﬁned by the Lie algebra of the second-grade elements of each stratum. 6.4. CONSTRUCTION OF THE VACUUM 229

6.4 Construction of the vacuum

In canonical quantum theories like the standard model, the space-time continuum is built into the algebra of the theory and the vacuum is the instant vector with minimum energy. In the canonical theory one can specify a time to fix an instant and then specify three space coordinates to determine an event. A history is a succession of instants; an instant vector therefore says something about the history of which that instant is a part, and can therefore be expressed in terms of history vectors. Formally speaking, from the historical perspective an instant vector seems local in time and grossly submaximal in its description, and so might be expected to correspond to a projector of high dimension in the space of history vectors. In fact the canonical theory usually supplements the explicit local information in an instant vector with enough tacit global information to define a unique history vector. It is always understood that the specified instant occurs between the input and output phases of a experiment, and that the system is adiabatically shielded during the experiment. Then a fixed dynamical history vector hD = hUT h ⊗ ... ⊗ hUth ⊗ ... ⊗ hU0h (6.63) governs the development of the system through a series of unitary operators here labeled by time. The background experiment vector hE has the form

hE = φh ⊗ hlT −1h ⊗ ... ⊗ h1th ⊗ ... ⊗ h11h ⊗ hψ (6.64) where each 1t is an identity operator associated with a specific intermediate instant t. When the canonical theory tells us that the vector at the instant t is ψt, the meaning is that for outputs after time t, the transition amplitude from hψ0 at time 0 is the same as the transition amplitude from ψt at time t. In the history theory, the transition amplitude for DhE is also given by D0hE0 with truncated history vectors with input delayed to time t: 0 0 hD = hUT h ⊗ . . . ox < Ut <, hE = φh ⊗ hlh ⊗ ... ⊗ hψt. (6.65) Little of this works in a fully quantum theory. In each frame it is still possible to define a skewsymmetric time coordinate operator bt in each frame, but different values of bt are not connected by unitary or orthogonal transformations. There is a plausible skewsymmetric energy operator Eb in each frame but it does not increment time by a fixed amount but by a third operator, essentially the commutator of time and energy. In a Poincaré-invariant theory the vacuum energy is adjusted to 0 and the vacuum vector is independent of time. In a fully quantum theory, Poincaréinvariance is replaced, for example by an so(6; σ) invariance in the model ϑo, and time is replaced by a skew- symmetric operator like F−C et = TΣ γ45 (6.66) a suitable normalization factor T providing the units of time. 230 CHAPTER 6. FULLY QUANTUM KINEMATICS

Here we consider how to relate the fully quantum vacuum to the dynamics. The bvacuumc is what is left in a target chamber when the air is pumped out and no quanta are injected. Its symmetries are determined by experiment, not definition. It is the ambience of the experiment, including in principle long-range influences of the the experimenter, apparatus, and cosmic environment, which break all symmetries. Clearly the physical vacuum is only approximately Poincaréinvariant. so(6; σ) invariance or so(10, 6) invariance may be a better approximation on a cellular scale. The vacuum is represented here by a projector hΩh. In the main application the vacuum has a non-trivial multiplicity or “degeneracy” Tr hΩh 1. It is convenient to reach the vacuum from the dynamics vector in several stages in a canonical quantum theory. First, to pass from the history mode to the instant mode of representation, one factors the dynamics vector into a time-ordered product of infinitesimal unitary transformations e−iHdt/~ and defines a category of unitary operators U(t0, t). This recasts the canonical quantum action principle in Hamiltonian form, describing the temporal development of any instantaneous variable Q = Q(t) with no explicit time dependence: [E − H,Q] = 0 (6.67) with Hamiltonian and energy operators d ∂L H = −L + ,E := i d/dt. (6.68) dt ∂t ~ Then to define the canonical quantum vacuum one seeks instantaneous vectors of minimal energy eigenvalue :

hHhΨ = hΨ for minimum . (6.69)

If Ψ(t) is a time-dependent Schrödinger vector, it would be a mistake to think of the succession of its values as a history vector. It is a history of a vector but not a vector of a history. To be sure, a classical time-dependent state q(t) defines a history. The function Ψ(t), however, describes a single input process, carried out at any one of the times t, not a sequence of input processes at all the times. After the input process Ψ(t), all that happens at later times t0 > t, before the final output process, is a passive transport, represented by a sequence of unitary transformations. Ψ(t0) is not another input process besides Ψ(t) but an alternative input process that could be performed instead of Ψ(t) and have statistically indistinguishable consequences for future times t” > t0. When the canonical quantum theory is based on a dynamics vector of the singular form (1.39), the classical action density L acquires a quantum interpretation. In a finite- element approximation, the space-time can be divided into finite hypercubes or 4-intervals ∼ I4 of 4-volume ∆x. Then each factor ei∆x L/~ in the dynamics vector hD represents a quantum transition probability amplitude for a process represented by a vertex with 6.4. CONSTRUCTION OF THE VACUUM 231

8 lines or generation processes, one through each 3-face of the I4. The Galilean space- time dissection parallel to the coordinate axes results in 6 spacelike lines representing infinite velocity and two timelike lines representing zero velocity, a relic of pre-relativistic thought. In a less unphysical dissection appropriate to special relativity, the hypercubes, diagrammatically speaking, stand on one corner instead of one 3-face. The normal to each 3-face is a null vector, representing generation at light speed rather than infinite speed or zero speed. Now the eight lines are clearly divided into four inputs and four outputs for each vertex. The four input vectors form a basis of null vectors for the entire space. In this null basis the Minkowski metric form is a matrix with 0 on the diagonal and (say) 1 everywhere else. A man in the game of checkers or draughts moves along such null diagonals in the plane. The proposed dissection of space-time defines a 4-dimensional checker-board, with checkers in the center of hypercubes and moves across 3-faces. Projected on a 3-space of constant time, the four null directions appear as four unit vectors from the center of a regular tetrahedron to its vertices. This classical cellularization is highly artificial, a computational expedient. A fully quantum dynamics vector hΨ does not have to be artificially cellularized, however, since it is composed of finite quantum elements in stratum E from the start: hΨ ∈ SF = 2SE . Each monadic factor ψ in Ψ counts as a cell or vertex. Each cell ψ of the vacuum in turn has several monadic factors of stratum D within it. These represent generation processes connecting the cell to other cells. The canonical dynamics is a development in time, one variable, generated by a canonically conjugate variable, the energy E = −i~∂t. Both variables have fully quantum counterparts. One must not proceed too formalistically, however. The time at which the energy of the system is determined is not determined by the system but by a clock in the metasystem. One determines system energy at a metasystem time. First one forms the fully quantum correspondents of the system energy and time. In a fully quantum theory, canonical conjugacy is not a binary relation but a ternary one, involving a quantum imaginary. A canonical conjugate is not unique and or absolute but is relative to a choice of i. For the choices of (5.23), the canonical conjugate of imaginary time XωE45 is EωE46, which is therefore the imaginary energy. The imaginary unit is i := N −1ωE65. It must be used to convert imaginary time to real time, a symmetric operator, that can be used to define instants by its eigenvectors. The imaginary energy suffices to generate the time development. The fully quantum imaginary unit i and imaginary time et do not commute and their product is not a symmetric operator; the easiest symmetric operator that approaches the product in the canonical limit is the anticommutator. Therefore the fully quantum real time on stratum E is defined by the anti-commutator

X Eb = {ωE45, ωE65}. (6.70) 2 232 CHAPTER 6. FULLY QUANTUM KINEMATICS

Now the general history must be partitioned into instants, slices of constant time, and a unitary development U(tn+1, tn) from one instant to the next must be formed from the dynamics vector. The fully quantum development is closest to unitary for the two central time intervals. Then the vacuum energy condition can be expressed as

E Eb hΩ = {ωF , ωF }hΩ = hΩ, for minimum . (6.71) 2 56 46 080603 ...... [Segue.] The quantum imaginary is the i that appears in both the Heisenberg equation of motion and the quantum action principle. It is used to convert elements a of the Lie algebra to observable variables ia that are conserved when the dynamics is invariant under i, one of the main principles of quantum kinematics. It has three defining characteristics: Its square is −1; it is central; and it is a bmeta-operationc, in the folllowing sense. When we compose two systems into a product system, important quantities compose in one of three ways. We add corresponding infinitesimal operators (like translation generators), multiply corresponding finite operators (such as parity), and equate corresponding meta-operators (those which act on the metasystem) . The main meta-operator is time t in elementary quantum mechanics. When we combine two particle systems, each with a time variable t, the composite system still has only one time variable, inherited from both subsystems. This is because the composite system has but the one metasystem for both of its parts, and time is read from a clock in the metasystem, not from the system. The imaginaries of the two subsystems are also equated. Moreover under time reversal, i is replaced by −i. It seems clear that i too is a meta-operator. In stratified quantum theory we quantize part of what was previously metasystem. This part must have a non-central operator i, the quantized i, that becomes the central i in the singular limit of canonical quantum theory. It is necessary to give a concrete expression for i to make the theory definite, and for i to approach a unique central i in the canonical quantum limit. It is common in canonical quantum theories for observables to be bilinear in variables of the system and the metasystem. For example, when we speak of the energy of the system in special relativity we mean the bilinear expression

µ E = V pµ (6.72)

µ where V is the experimenter’s time axis and pµ is the momentum-energy of the system. We cannot make maximal quantum determinations on the experimenter without disrupting the experiment, but we can measure quantities like the world velocity V µ and position x of one experimenter with respect to another without harming either, if we do not exceed macroscopic precision. This respect for the experimenter introduces errors that are usually negligible compared to those we make anyway. They permit us to regard V µ as central, 6.4. CONSTRUCTION OF THE VACUUM 233 even though the laws of quantum theory tell us that V µ = P µ/M is a multiple of the momentum energy of the experimenter and fails to commute with the centroid coordinates Xµ of the experimenter. Evidently the central i is an operator like V µ, determined by the metasystem. In the classical limit all variables commute and the energy of a mass m is positive and bounded away from 0: q 2 2 E := + −∂0 > mc > 0 (6.73) √ where + is the positive square root. Then we may take i to be the phase of the antisymmetric time-translation generator ∂o: ∂ i ∼ o , (6.74) E which is trivially central. Simple quantization replaces this i by a generator of a simple Lie algebra a on a deeper stratum, for example the generator ω56 of Segal and Vilela-Mendes. A candidate is needed within fully quantum theory for i, the quantum i that is used to form observables from generators on stratum F. i is required to transform like ω56 under the simple Lie algebra F−C a. ω56 acts ﬁrst on stratum C, and induces a transformation Σ ω56 on stratum F of the metasystem, that, suitably scaled by physical constants, is the proposed i. It seems permissible that i belong to the metasystem, since it is used to form observables, and observations are interactions between system and metasystem. The i of the experimenter also has the important property of near centrality relative to the algebra of the system, and so seems to be the natural choice for forming observables from system generators.

6.4.1 Bosonization

It is especially easy to construct excitation quanta with bPalevc statistics and even spin from a Fermi aggregate of quantum entities with spin 1/2: Assertion 4 The algebra of a Fermi aggregate of spinorial quanta includes the algebra of a bPalevc aggregate as subalgebra.

Argument The basis vectors ψα ∈ V deﬁne creators ψbα = Lψα ∈ Fermi V in the Fermi algebra over V annihilators β † ψb := ψbα . (6.75) β If Λ = (Λ α) ∈ Alg V is any linear operator on V then its many-quantum representative on PV is β α ψbβΛ αψb (6.76)

By (3.22) these obey bPalevc commutation relations for sl(V ) generators and have bosonic generators as singular limits. 234 CHAPTER 6. FULLY QUANTUM KINEMATICS

To actually combine bPalevc quanta represented by such second-grade operators, one cannot merely brace the pairs and then multiply; bracing produces grade-1 monads, not Palev quanta, which have grade 2. One cannot simply multiply the second-grade operators; that would scramble the pairs. However if the two ﬁrst-grade elements in each Palev quantum have a coordinate in common that distinguishes them from the other pairs, the product can always be unscrambled. In that case one may combine Palev quanta by Cliﬀord multiplication. One may go from one such quantum to a catenation of many by applying the Lie algebra homomorphism Σ, which preserves the commutation relations and bPalevc statistics. Σ may be iterated to produce larger assemblies.

6.4.2 Antiparticles 6.4.3 Flavor [Do: Since the one-quantum Dirac equation includes transitions from positive to negative energy, can it be a good approximation for an electron moving over the vacuum Dirac sea, which ﬁlls the negative energy strata? Does this question persist in the normally ordered theory? Or must something like the Foldy-Wouthuysen equation, without such transitions, replace the Dirac equation at a deeper stratum than usual?]

6.5 Fully quantum gauge theories

A canonical quantum bgaugec theory has a variable covariant differentiator DµM (x) on a given bundle with a given lower-stratum gauge group in the bundle group and a given c space-time manifold for base. Its classical space-time destabilizes it. Here it is fully quantized stratum by stratum. The gauging of a canonical quantum field theory is a heuristic process that converts a quantum bfieldc theory with a lower-stratum gauge group G(C) represented on stratum F to a richer quantum field theory with a much larger stratum-F gauge group GF consisting of group-valued fields: sections of the principle bundle with fiber G over the base E. Gauging has developed along three quite different lines. All set out from Einstein’s theory of gravity, both have a boson vector field Bµ, the bgauge bosonc vector field, as a basic variable, and both have several descendants along the line. In the Weyl line, which includes theories of Yang-Mills and the standard model, the gauge boson vector Bµ is a connection analogous to the Christoffel vector-connection Γµ of gravity. One introduces a covariant differentiator Dµ = ∂µ − Bµ as basic field variable, making the replacement ∂µ ⇐ Dµ throughout the ungauged field theory. This is a bgauge connection theoryc. In the Kaluza line, which includes theories of Klein, DeWitt, and others, the base manifold is a Cartesian product EM ⊗ G, where EM is a pre-gauge event manifold and G 6.5. FULLY QUANTUM GAUGE THEORIES 235 is the lower gauge group. Vectors in E ⊗ G have a composite index (µ, α) of a Minkowski index µ and an index for a basis in the gauge Lie algebra dG. Therefore a causality form igh on E ⊗ G has a Minkowskian external term gνµ, an internal gauge term gβα, and a chronometrical term gµα coupling internal and external differentials. On this line

Bµα = gµα. (6.77)

This is a bgauge metric theoryc. In the BRST line, the usual differential operators representing translation are supplemented by nilpotent operators that can be interpreted as monadics of never-observed quanta, the ghosts. The higher gauge transformations become translations of the polyadics of a new field, the BRST field. Under full quantization, neither the lower nor the higher gauge transformations remain local. In general relativity the stratum F gauge group can be taken to be the diffeomorphism group G[E] of the space-time event manifold E, with a Lie algebra a(E) defined by singular canonical relations like

ν ν ν λ ν [∂µ, x ] = δµ, [δµ, x ] = 0, [δµ, ∂µ] = 0. (6.78)

There are several choices for the lower bgaugec group whose gauging gives the diffeomor- 4 phism group. For now we adopt Feynman’s choice, the translation group R . Every bgaugec theory has an equally singular relation of much the form (6.78), and a replacement ∂µ ⇐ Dµ. (6.79) µ The coordinate x can be treated as a trivial scalar field. Therefore the simple bquantizationc of bgravityc can be a guide for that of the other bgaugec fields, especially if the other bgaugec fields are simply aspects of bgravityc in higher dimensions, as bKaluzac suggested.

6.5.1 Fully quantum gauging

Einstein created general relativity to provide a bfieldc theory of gravitational interactions that could replace the theory of action at a distance of Newton, who had explicitly declared his theory of gravity to be unphysical on grounds of its non-locality. To quantize his theory is to synthesize Einstein’s bfieldc theory with Newton’s particle theory. By gauging we mean a heuristic process for converting a special bgaugec theory to a general one. The gauging process of general relativity is a prototype for gauging in general. It represents the Minkowski space-time differentiator as singular limit of a covariant differentiator: ∂µ ⇐ Dµ = ∂µ + Γµ (6.80) We construct a simple quantum counterpart, having the usual gauging as a singular limit; call it fully quantum gauging. 236 CHAPTER 6. FULLY QUANTUM KINEMATICS

Gauging begins by enlarging the group of allowed coordinate transformations from a Lie group to a function group. The allowed coordinates of special relativity form a vector space and a commutative Lie algebra a(xµ). The allowed coordinates of general relativity form instead a commutative algebra Alg(xµ), with the same elements as a(xµ) in a certain representation. To fit this into a simple framework one drops commutativity and provides a fully quantum interpretation for the gauging process, as follows. First the commutative Lie algebra of the four basic coordinates, a(xµ), is simplified to so(4, 2); σ) ⊂ so(10, 6), whose defining representation consists of mappings 6R → 6R. This 6R with its metric of signature σ is interpreted as a subspace of the vector space S[C] of the hypothetical quantum element of stratum C. The Lie algebra so(6; σ) is interpreted as defining the statistics of this quantum. The defining 6 × 6 representation consists of certain spin variables of the q differential. Aggregating these variables by means of the E−C iterated bcumulatorc Σ results in an isomorphic Lie subalgebra of a much larger Lie algebra, that of the event stratum E: ΣE−D’ so(6; σ) ⊂ so(V E). (6.81) The associative algebra Alg V E algebra is interpreted as the kinematical algebra of the event. The commutative algebra Alg(xµ) of general relativity is a singular limit of Alg V E. According to simple quantum relativity, then, space-time is a catenation, probably multiple, of q differentials, self-organized to freeze out one variable ωb56 that serves as i, and approximated in a singular limit where Eωbµ6 is ignorable [75]. The transition from special to general relativity is a singular limit of a transition from the kinematics of a q individual to that of a q catenation. Gauging is catenation seen in a singular limit. In what follows hνµ is again the Minkowski metric, gνmu(x) is the Einstein metric, Dµ is the covariant differentiator, G is the dimensionless form of the Newton gravitational constant, and Lg is the Hilbert gravitational action. Gauging replaces Poincaré-invariant structures that break general covariance by local generally covariant structures, replacing hνµ by gνµ(x), and replaces the differentiator with respect to special (inertial) coordinates by the covariant differentiator with respect to general coordinates. If one simply views special relativity in general coordinates, the Minkowski constant metric becomes a variable ν metric field, hνµ ⇒ gνµ(x), and one that is flat, obeying R µλκ = 0. Einstein replaced the fixed metric by a dynamically variable metric field, and replaced flatness by the Einstein ν ν equation R µ = κT µ, based largely on analogy with electromagnetism. At the same time, the constant differentiator ∂µ keeps its form while that for vector fields is replaced by the covariant derivative, ∂µ ⇐ Dµ = ∂µ − Γµ(x). Γµ(x) is a Lie- algebra-valued vector field Γµ(x) = (Γµλκ(x)), and transforms so that Dµ transforms as a vector. The condition of metricity, Dµgλκ = 0, was assumed mainly for convenience in the absence of experimental evidence to the contrary. In the present context, the absence of experimental evidence is reason to assume non-metricity, since non-metricity is generic and metricity is not. 6.6. FULLY QUANTUM METRICS 237

The quantum vectors of stratum D are quantum correspondents of classical space- time tangent vectors, with extra components for binternal dimensionsc. The invariance Lie algebra of SD as bilinear space is, however,

so[D] := so(216; 64)n, Dim so[D] = 215(216 − 1) (6.82)

The simple theory unifies the space-time and a differentiator (or momentum) in the Lie [C] algebra so(6) of of dimension 6, imbedded here in S = 16R. The covariant differentiator is an extension of the differentiator from scalars to vectors. These are singular limits of operators already included in the representation algebra of so(6). In the canonical quantum theory the variable differentiator Dµ(x) is a bfieldc of local differentiators. In the fully quantum theory the field is replaced by a queue. Association takes us from the representation of so(6) on stratum C to that on stratum F. Concretely, let the c space-time coordinates of the event be xµ and the differentiator be ∂µ. For any operator X : S[T ] → S[T ], ΣX : S[T +1] → S[T +1] designates the induced representation of X one stratum higher. Then the full quantization proposed is

so[C] ⇐ so[F], D F σβα ⇐ σba, (6.83)

The indices a, b label basic monadic generators of SF, and have enormously greater range than the indices α, β, which label basic monadics of SD. Indeed, the matrix of variables D F σβα is a small corner block in the matrix of variables σ aba. This implies the correspondences

F −C [C] xµ ⇐ Σ ωµ5 , F −C [C] Dµ ⇐ Σ ωµ6 . (6.84)

6.6 Fully quantum metrics

This chapter fully quantizes the Einstein kinematics of bgravityc; the the Einstein-Hilbert dynamics is fully quantized in Chapter 7. In his famous inaugural dissertation, bRiemannc noted that discrete multiplicities have natural metrical structures determined by counting, while continuous multiplicities have no natural metrics but must import them from outside themselves[61]. As though obedient to the continuum clause of Riemann’s Principle, bEinsteinc and bCartanc imported a metric µ0 ν “from outside”, a real quadratic form g(v) := v (x)gµ0µ(x)v (x) on space-time vectors v, interpreted as the square of the proper time differential. The language Cartan chose for differential geometry was the exterior calculus, an instance of Grassmann algebra. The exterior algebra, however, does not describe a metric or linear operators on itself. Cartan resorted to a foreign language, the tensor calculus, to represent the metrical form. This 238 CHAPTER 6. FULLY QUANTUM KINEMATICS could also be used to express linear operators on the exterior algebra. A more economical extension that provides a metric is a Clifford algebra. If one asks Riemann’s Principle whether metrical structures of quantum multiplicities are to be imported or domestic, doubt arises because quantum multiplicities are neither purely discrete nor purely continuous, but quantum, having aspects of both: Their Hasse (lattice) diagrams are horizontally continuous due to superposition and vertically discrete due to dimensionality. Riemann himself seemed to prefer the discrete path and the domestic metric, due to its conceptual unity, but the conflict between the known continuous symmetries of nature and the discrete symmetries of any hypothetical discrete spaces closed this path to physicists before the quantum theory. The Clifford algebra language brings in a metric form, and so is not as totally inade- quate as the exterior algebra language, but its metric form is constant for a given algebra, appropriate only for a flat space. The self-Grassmann algebra S, however, constant metric forms, enough to approach the variable form of a Riemannian manifold as a singular limit. Riemann’s principle works for topology too. Ordinarily a topology is defined by a set of sets of points; for example, by the set of all closed sets. Thus topology bridges three strata. A continuous manifold must import a topology, for example that of its imported metrical structure, and a discrete set can have a natural discrete topology like that of a checkerboard. In a fully quantum theory, a continuum topology emerges as a singular limit of infinitely many points. Finally, Riemann’s dichotomy applies to dynamics too, which may be imported or domestic. Classical dynamics too has a kind of metric, the action integral of a history. A classical dynamics must be imported for continua but may be domestic for discrete multiplicities. In quantum theory the dynamics of the isolated quantum system can be domestic. The present fully quantum dynamics employs natural topological and metrical constructs for the isolated quantum system. On the other hand the experimenter must be allowed to influence the dynamics from outside the system, or measurement would be impossible. Since the probability form seems conceptually prior to a chronometric, in that it operates on a lower stratum, it is constructed next.

6.6.1 Fully quantum probability forms Higher strata provide reducible representations of the isometry groups SO(V ) of lower strata. There are physical variables besides the symmetry generators, describing internal structure. But such a reducible representation of SO(V ) can have several quadratic forms invariant under SO(V ), so more physics must be injected to ﬁx a probability form. For example, a Cliﬀord algebra C has a commutator Lie algebra, and therefore a 2 2 bKilling normc v ikhv := Tr(∆v) , as well as a bmean-square normc v iHhv = Tr v / Tr 1. 6.6. FULLY QUANTUM METRICS 239

The physics of Fermi statistics is useful at this point. There a many-quantum vector hΨ ∈ W0 is a Grassmann polynomial in one-quantum vectors hψ ∈ V . Its Fermi dual D D HΨh ∈ W0 is the transposed polynomial in the dual vectors Hψh ∈ V . The probability form deﬁnes and is deﬁned by the quadratic form

kΨk := ΨiHhΨ = ΨiHhΨh. (6.85)

A corresponding procedure works word-for-word for Clifford algebras as well and produces a probability form that reduces to the Fermi form in the singular limit. In the Clifford case a probability form is built into the algebra 2V by Clifford’s Clause. Taking λ = 1, we have ∀ψ, φ ∈ V : ψHψ = ψ2. (6.86)

It follows that Hψ = Grade0 ◦Lψ: to act with the breversec of ψ, multiply by ψ from the left and then take the scalar part. The probability form H has a natural extension from V to 2V : V ∀Ψ ∈ 2 : HΨ = Grade0 ◦LΨ (6.87)

This defines H in terms of left multiplication L and taking the scalar part Grade0. Moreover, if H is positive definite on V then H is positive definite on W = 2V . However an algebra A, Lie or associative, generally has many bnatural quadratic formsc. The H just constructed is one of many that have the proper specialization to Fermi statistics, but is the most plausible and is tentatively adopted. Briefly,

H = Grade0 ◦L. (6.88) In other words, the probability form of a queue is assumed to be the mean-square form (3.10).

6.6.2 Fully quantum causality form Now we turn to a construction of the fully quantum variable corresponding to Einstein’s gνµ(x). In physical and mathematical practice, the system always has an underlying probability form, which enters into the construction of the chronometric. One such two-metric space is M 4, the 4-dimensional Minkowski tangent space, consisting of tangent vectors vµ at some arbitrary space-time origin O. The more familiar metric of the two is Minkowski’s bcausality formc ν µ v ighv = v gνµv (6.89) giving the squared proper time of the displacement vµ. Implicit, however, is a classical probability form deﬁning the Boolean logic of predicates about diﬀerentials. This is the 4 singular bprobability formc on M given by

v0 iHhv = δ4(v0 − v); (6.90) 240 CHAPTER 6. FULLY QUANTUM KINEMATICS the transition probability amplitude from one classical tangent vector hv to another hv0 is 0 unless they are identical. This is singular because these tangent vectors are classical objects with commuting coordinates and can be observed without changing them. The probability norm deﬁned by iHh is then the ill-formed expression

v iHhv = δ4(v − v) = δ4(0) (6.91) telling us only that all the unnormalizable orthogonal vectors hv have the same infinite norm. The charge stratum C is surely at or below the differential stratum D. These forms differ so much on higher strata that they must migrate upward quite differently. In classical general relativity the causality form gν,µ(x) is the main dynamical variable. In the classical theory each value of gν,µ(x) defines a Clifford algebra C(x) over the tangent space at each event x. The C(x) are all isomorphic to each other. For each value of µ = 1, 2, 3, 4 the tangent vector ∂µ at x is written as γµ(x) when it is used as a Clifford algebra element. The remaining classical variable is a connection (form) Cµ(x) used to construct an isomorphism from the Clifford algebra at one event to that at any nearby event,

µ Cµdx : C(x) → C(x + dx). (6.92)

It is important to fix a physical interpretation for these quantities. The relation of the causality form to chronometry is not a useful clue for the microscopic interpretation, since we have no microscopic chronometers, but must emerge in the macroscopic limit. This is an important condition on the interpretation. The components gνµ(x) are variables whose values are numerical functions of x in the classical theory but operators in the canonical quantum theory, creating and annihilating gravitons in the linearized approximation. This means that in classical gravity the γµ are not regarded as quantum objects but as classical tangent vectors, provided with a product that encodes both the Grassmann product and chronometrical information; or as Clifford put it, both topological and metrical structure. But it is also true that if a quantum electron is place in the classical gravitational field the γµ serve as operators in the kinematical algebra of the quantum spin. It is inferred that the γµ(x) Clifford algebra of the classical space-time manifold is like the Poisson bracket of classical mechanics: a tip of the buried quantum structure that protrudes into the classical surface structure. It is not merely a variable to be quantized, like the space- time coordinates, but a piece of the quantum theory to be respected in the quantization process. One did not quantize the Lie product of classical mechanics (the Poisson Bracket) but kept it intact as the quantum theory beneath it was excavated; one can do the same with the Clifford product of special and general relativity. The bchronometricc form igh on the field stratum F, the gravitational potential, is taken to be the form in stratum F induced by the probability form iHh of stratum C. 6.6. FULLY QUANTUM METRICS 241

080601 ...... [Express the classical gravitational potential field gµν(x) is a singular limit of igh. Show that the classical limit of the fully quantum graviton annihilator defines a causality form like Einstein’s. ] This relation between the time and probability forms is another balgebra unificationc, echoing the one in §3.3.3 that occurs in canonical quantum theories. This one provides a deep bunificationc of general relativity and quantum theory. It uses the uncertainty on a deeper quantum stratum C to define a clock and a meter stick for classical dynamics on the field stratum F. For a more symmetric choice of constants, one may replace the pair (~, X) by a pair (E, X) of natural energy and time quanta related by

EX = ~. (6.93)

ϑo has two adjustable quantum parameters: the quantum of action ~ and the quantum of space-time X. In addition it has a large pure number N that characterizes the speciﬁc representation of its Lie algebra. In the singular limit, X → 0 and NX → ∞. Existing theory also contributes relevant parameters c and G of special and general relativity. Since a non-relativistic limit is not taken here, one sets c = 1. G enters into the Hamiltonian and thus into the dynamical Lie algebra of the canonically quantized bgravitycof bArnowittc,bDeser, and cbMisnerc [2], but not into the kinematic Lie algebra of the functional quantum theory, which is based on history vectors rather than instantaneous ones. Since the functional precedent is followed here, the constant G ﬁrst appears here in the dynamical history tensor. Therefore a quantum event space SE need not be metrized from outside like Einstein’s singular space-time. Its special coordinates form a simple Lie algebra, for example so(DS) in one model. In any case the Lie algebra of special coordinates has a natural causality form derived from the probability form of stratum C. In general relativity the proper time between two points 1 and 2 is a “continuous sum”

Z 2 µ ν 1/2 τ = |dx|, |dx| = [gνµ(x)dx dx ] (6.94) 1 of contributions from each Minkowski form gνµ(x) along the path from 1 to 2. All these forms are isomorphic but there is no natural isomorphism among them, before the path defines one. This is expressed by general bgaugec invariance, invariance under a translation x 7→ x + a(x) that varies from event to event. Similarly the Klein-Gordon action term R 4 νµ (dx) g pν(x)pµ(x) is a continuous “sum” of bilinear catenations of many momentum vectors at different events x. A catenation of quantum differentials with Clifford statistics, and therefore with event vector space ES = PDS, was already assumed in ( 4.33 ) as quantum correspondent to the catenation of Minkowskian tangent spaces in a manifold. Now its metric tensor is constructed. 242 CHAPTER 6. FULLY QUANTUM KINEMATICS

Einstein replaced the Minkowski hνµ ⇐ gµµ(x) to make a theory with general covariance. Now we have one with quantum covariance constructed in the following strata:

1. The causality form igh and the probability form iHh on stratum C are both the bmean square formc iHh.

2. The bKilling metric formc ikh deﬁnes an operator in any faithful irreducible representation of so(n), namely the quadratic Casimir operator bk, which now represents a one-quantum observable.

3. Since so(n) acts on SD, so does its Casimir operator bk.

4. The classical metric ﬁeld arises from a many-quantum catenation whose vector space is SF.

5. Then the bquantum chronometricc operator is the many-quantum operator

F−C D(F−C) F F gb = I bk I : S → S (6.95)

induced in stratum F by bk in stratum C.

In words, to apply the quantum chronometric to a vector in stratum F, one unbraces and factors as many times as necessary to reach stratum C, applies the Killing metric, and remultiplies and rebraces to return to stratum F. Currently the exponent F − C is tentatively assumed to be 3. Thus if special relativity is regarded as having one quadratic form, that of Minkowski, then general relativity has an infinity of quadratic forms, one on each tangent space, all un- naturally isomorphic; and simple quantum relativity has a large finite number of quadratic forms, one for each quantum event. Schematically speaking, we can pass from the quantum back to the classical as follows. The quantum metric form Σbk(L) is again a polynomial in a great many momenta pµ ∼ ωµ6. Each momentum is accompanied by coordinates xµ ∼ ωmu5. In a singular limit these p, x variables can be treated as commutative and assigned simultaneous eigenvalues. If the density of values in the spectrum is low, most values of xµ do not occur, and most of those ν that occur do not occur twice. Then the collection of pµ, x pairs nearly defines a partial function pµ(x), a co-vector field. If the dynamics prevents large changes in pµ for small changes in xµ, a smooth function can be formed from this partial function by interpolating between the values already defined. Then the quantum metrical form approaches the classical general relativistic one. When two events in a Clifford vector are exchanged, the vector merely changes sign. The same values of the coordinates are paired with the same values of the momentum vector, so that the field pµ(x) is unaffected. The surface appearance of Bose statistics in 6.7. COVARIANT DIFFERENTIATOR 243

bgravitational ﬁeldc theory is an artifact of the singular limit. It has nothing to do with the deep statistics, which is Cliﬀord. Let 1µ(x) be a tangent vector to the µ coordinate axis at the event x in classical relativity. The c metricity and atorsional conditions are

Dµgλκ = 0,Dµ1ν − Dν1µ = 0 (6.96) respectively. They reduce all the degrees of freedom of the c differentiator Dµ to those of the c metric gλκ. They are, however, unstable and so are not expressed in the generic Lie algebra FG of stratum F. We must see what variants of metricity and atorsionality the action principle implies. Moreover in the canonical quantum theory Dµ and gνµ are Bose fields. Therefore their fully quantum correspondents are assumed to be bPalevc catenations of Clifford pairs.

6.7 Covariant diﬀerentiator

In the simpliﬁed quantum theory, as in bSegal-Vilela-Mendes spacec, the events have momentum coordinates which are uniﬁed with their space-time coordinates as generators of the same Lie algebra

so(6) ⊂ so(S[C]) ⇒P so(SD) ⇒P so(SE) ⊂ Alg SE. (6.97)

The distinction between space-time coordinates and momenta returns in the singular limit of physics. The extension of the classical differentiator from scalar to multivector fields is the exterior differentiator. This extension is purely algebraic, introducing no new element of geometric structure. The same is assumed here for the quantum theory. A covariant differentiatior Dµ(x) is an extension of the ordinary differentiator ∂µ from scalar to the most general tensor fields, preserving the Leibniz product law. It is not Λ unique but a dynamical variable. It defines a tensor connection ΓK M, where K, Λ, M,... are collective tensor indices, and

Dµ(x) = ∂µ1 + Γµ; (6.98) here ∂µ is the Lie diﬀerentiator with respect to the vector ﬁeld 1µ tangent to the µ coordi- N nate axis, 1 is the unit operator on the local tensor space, with elements δ M, and Γµ(x) is generally a non-trivial Lie algebra element acting on the local tensor space, with elements Λ Γµ K(x). The gauge variable Dµ(x) is usually subject to the conditions

1. Dµ(x) is a tensor diﬀerentiator: that is, it is a linear operator on the space of tensor ﬁelds, transforms as a co-vector, and obeys the Leibniz Relation. 244 CHAPTER 6. FULLY QUANTUM KINEMATICS

2. Equivalence principle: At any event x0 there is a suitable general coordinate transformation that transforms Dµ(x0) ⇒ ∂µ at that point.

Here a factor of a unit operator is implicit. 1. implies that Dµ(x) restricted to scalar fields is indeed ∂µ. 2. implies that Dµ(x) restricted to vector fields differs from ∂µ by a metrical and atorsional Levi-Civita connection. As Einstein set out for general relativity from the Minkowski space of special relativity, we start out for simple quantum relativity from Segal-Vilela-Mendes space.

6.8 Reciprocity

Let us call the x–p symmetry reciprocity, though bBornc’s breciprocityc acted only on a higher stratum, that of dynamical law [15]. Since xµ is local and pµ, being off-diagonal in µ x , is slightly non-local, breciprocityc breaks blocalityc in the quantum theory. To describe the field organization that breaks reciprocity and restores blocalityc we D may define a blocality algebrac homomorphic to 2R ⊗ 2R , a real version of the Pauli spin algebra. It contains the seeds of both locality and reciprocity as anticommuting operators.

The blocality algebrac is a 2 × 2 real matrix subalgebra of the algebra with basis matrices 1 0 0 1 0 −1 1 0 L = ,L = ,L = ,L = (6.99) 0 0 1 1 1 0 2 1 0 3 0 −1 regarded as deﬁning a quantum space. In a basis of eigenvectors of blocalityc like t and E, the reciprocity operator L2 generates the transformation

µ µ µ µ L2 : δx = p , δp = −x . (6.100) and the locality operator L3 generates the transformation

µ µ µ µ L3 : δx = x , δp = −p . (6.101)

The symmetric operator

L1 = [L2,L3]/2 (6.102) generates blocalityc and breaks breciprocityc. We can call (6.99) the locality basis of the locality algebra, since the locality operator L3 is diagonal in this basis. In the Segal-Vilela-Mendes so(3, 3; R) Lie algebra, we may reserve the ﬁrst four basis vectors 1A (A = 1, 2, 3, 4) for the usual Lorentz so(3; 1; R) Lie algebra and the last two (A = X,Y ) for the blocality algebrac. Since the anti-symmetric operator LXY interchanges µ µ x and p , it represents breciprocityc L2; the Segal-Vilela-Mendes basis is a blocalityc basis. 6.9. FULLY QUANTUM COVARIANCE 245

6.9 Fully quantum covariance

A fully quantum general relativization rule will now be developed that has the familiar classical general relativization rule as a classical limit. The generic event vector space SE has high dimensionality

Dim SE = PE−DDim SD Dim SD. (6.103)

Like general relativity, simple quantum relativity has special coordinates at the differential stratum and general coordinates at the event stratum. The general coordinates form an algebra. The special coordinates do not form an algebra but merely the Lie algebra so(SD). Since the Clifford event has vector space SE, the quantum field history F has vector space SF = PF −E SE = PF −D SD and coordinate algebra Alg SF. A significant lapse of correspondence now appears. The vector space SE is a direct sum of Grassmann products of SD’s. There is no natural isomorphism among tangent spaces; but the factor spaces in a Grassmann algebra are identical to each other. General covariance blocks such an identity between tangent spaces at different events. To be sure, Maxwell(-Boltzmann) statistics has no such antisymmetrization and requires no such identity between factor spaces. Distinguishable particles can form Maxwell catenations, which are merely sequences. Maxwell statistics alone is invariant under independent automorphisms of all its factor spaces. In the classical limit, all forms of statistics approach the Maxwell statistics. Therefore either the proper statistics for quantum differential events is Maxwellian, or general covariance is an artifact of a singular limit. Since no quantum object is known with Maxwell statistics, I conservatively suppose that it again arises in a singular limit. If the proper statistics is Clifford, then to approach Maxwell statistics the differential event should have a vector space of high dimension, approaching infinity in the classical limit, and the number of occupied vectors in that space should be low, so that identity of two occupied vectors becomes an event of low probability that can be ignored. We already have been forced to the high state-vector-space dimension by other requirements. As for the low occupation, the classical limit of a bosonic field like gνµ(x) requires organizing a large number of effective bosons with high occupation numbers of bosonic vectors, although our bosons are pairs of fermions, which exclude each other. These requirements are compatible provided the bosonic pairs of fermions have internal degrees of freedom that are sufficiently excited, like Cooper pairs in superconductivity, so that the exclusion principle rarely applies. Therefore if Clifford statistics holds, there are violations of general covariance far from the classical limit, at short distances and high energies. We already expect such violations due to non-locality, so this is not a serious problem. 246 CHAPTER 6. FULLY QUANTUM KINEMATICS

To put this assumption on a more secure foundation we recapitulate the formulation of simple quantum general covariance for Clifford events. Let a(v; 0, 0) be the Lie algebra of the diffeomorphism group of the c/c event manifold 4 µ E = R . An element of a(v; 0, 0) is a vector field v = v (x)∂µ on E. The product in a(v; 0, 0) is the Lie Bracket [u, v]L. Let a(vb; ~, X) be the simple variant of this Lie algebra that we seek. It is not the diffeomorphism Lie algebra of any space; call it the generic covariance Lie algebra, and rename the usual diffeomorphism algebra a(v; 0, 0) the “general covariance” Lie algebra to indicate correspondence. Let ωβα be the 16 × 16 matrices generating so(C). Let

E−C ωbβα = Σ ωβα (6.104) be the operators on stratum E induced by the ωβα, generating a Lie algebra soc(E) := ΣE−C so(C). Assertion 5 D a(v; ~, X) = ∆ Alg (S ), (6.105) the commutator (adjoint) Lie algebra of stratum D.

µ Argument The generic form of the vector ﬁeld v (x)pµ is Poly(ωbµ5)ωbν6. These polynomi- E als generate the matrix Lie algebra Poly(ωbβα) ⊃ so(S ) with the commutator Lie product. [Do:Check.] Step 1 in bsimple quantizationc is to convert classical event coordinates and diﬀeren- tiatiors into generalized Segal-Vilela-Mendes coordinates in the Lie algebra Σ so(SD) ⊂ so(SE):

µ µ5 x ⇐ X ωb , µ6 ∂µ ⇐ E ωb , µ µ µ5 v (x) ⇐ V (X ωb ), µ µ µ5 µ6 v = v (x)∂µ ⇐ vb = E vb (X ωb )ωb , (6.106) µ where the vb (...) are arbitrary ordered polynomials in the indicated operator arguments. µ Step 2 is to close the vb into the Lie algebra that they generate, the quantum general D covariance Lie algebra. This is the Lie algebra of all ordered polynomials in the ωb(βα)(S ), and must therefore be E ba[E] = so(S ) ⊂ Alg[E]. (6.107) Step 3 is to convert the variable hνµpνpµ of special relativity into the variable gb(L) of simple quantum relativity. [Do: Check.] The singular limit that converts simple quantum relativity to general relativity must undo these steps in reverse order. 6.9. FULLY QUANTUM COVARIANCE 247

To invert Step 3 [Do: Invert 3.] To invert Step 2 and recover the vector ﬁelds

µ µ5 µ6 vb = E vb (X ωb )ωb (6.108) evidently one must let EX → 0 with E ~/X, and keep only the leading term in µ µ5 µ6 E vb (X ωb )ωb . This can be a Lie-algebra homotopy. To invert Step 1 and recover the Lie Bracket, one must freeze the variable ωb65 that µ contributes the right-hand side i of the commutation relation between x and pν, as in (5.23): 2−1/2 i := ωb65/N, N = max(ωb65) . (6.109) In nature, only four dimensions of the hypothetical 15-dimensional so(6) of the differential stratum D organize into macroscopic space-time dimensions of the event stratum E, and the structure of so(6) does not determine which four, so a spontaneous breakdown of so(6) invariance seems to make this determination, leaving an unbroken so(3, 1) Lorentz subgroup in the tangent space at each event. The kinematics permits a different four dimensional subspace of so(6) to be so favored at different events. There is no natural isomorphism between these different subspaces, even of the same dimension, and an approximate general covariance can emerge. 248 CHAPTER 6. FULLY QUANTUM KINEMATICS Chapter 7

Fully quantum dynamics for gravity and the standard model.

7.1 The history vector

In classical mechanics the dynamical development is described by an equation of motion of the Lagrangian form Z δS = δ Ldt = 0. (7.1)

In the canonical quantum theory we learn that this is not the exact dynamical law, but results from a deeper quantum dynamical law of a more statistical nature as an approximate condition for constructive interference. The deeper dynamical law merely assigns a quantum probability amplitude to each history. In the usual formulation, pioneered by

Dirac, a certain history vector hD~ (of the form (1.39) deﬁnes the quantum dynamics. This is less singular than the dynamics of the form (7.1), since it softens the commutativity of the coordinate with the momenta with the parameter ~. But the form hD~ is still singular. It preserves the commutativity of the coordinates and of the momenta separately and implies a measure on the space of histories that does not exist. It is a research program, not a mathematical statement. This leads to the expectation that the history vector hD~ is a singular limit of a deeper regular dynamical law hD~X. An exponential form −Se hD~X = he (7.2) will be assumed for the dynamics vector in a fully quantum theory as in the canonical theory. A factor i has been absorbed into the skew-action Se, which is an antisymmetric generator, not a symmetric observable. Usually a time-ordering operation T is applied to the factors in hD; this can be omitted here.

249 250 CHAPTER 7. FULLY QUANTUM DYNAMICS

For well-known reasons Einstein chose a classical action density R bilinear in the covariant diﬀerentiator D. What actually appears in the physics is the skew-action density iR. The corresponding queue expression is designated by Rb. In Rb the canonical i must be replaced by a suitable dyadic which in one frame is bi = NωE56. Now the skew-action of lowest non-trivial order in the dyadics is trilinear in ωb instead of bilinear. The action trilinear in dyadics hopefully reduces near the vacuum to one that is bilinear in monadics and linear in dyadics, like the usual action for the fermions of the standard model. Recall the Dirac relation between Hamiltonian and Lagrangian,

0 the−iH∆tht0 = Neth iL∆th t (7.3)

0 with a normalization factor N that diverges as ∆t := t − t → 0. In (??)Ψ~X is written as an exponential of an operator, not of matrix elements, to be basis independent, so the exponent corresponds to a Hamiltonian rather than a Lagrangian. In a fully quantum theory, the Lagrangian form of the history amplitude emerges only in the limit of classical time, which singles out a basis in which space-time variables are all diagonal. In the canonical theory the Lagrangian form is invariant under a larger group of space- time coordinate transformations than the Hamiltonian, because L is a scalar and H is one one component of a vector. Here the Hamiltonian form is more invariant than the Lagrangian because it does not break the canonical group, as preferring a time variable over its canonical conjugate, the energy, does. This invariance is possible because the sum in (7.2) is over dynamical elements, which are invariant entities, not over instants of time, which are not. Correspondence requires an approximation, admittedly singular, relating the fully quantum dynamics vector hDX to the canonical one hD0:

ΣHe n n−1 1 0 hDX = e → u(t , t ⊗ ... ⊗ u(t , t ) = hD0; (7.4) u(t0, t) = e−iH(t)(t0−t) is a unitary operator representing a small dynamical development from time t to t0.

One still imposes a condition for constructive quantum interference on Ψ~X to recover the usual dynamics. Then Ψ~X can be regular because its space and time coordinates belong to a simple Lie algebra. This does not break Lorentz invariance, though it imbeds the Lorentz group as a non-normal subgroup in a larger simple group. If one tolerates singular formulas without clear meaning, then a classical dynamics too could be associated with a vector Ψ0, a functional of a classical history, giving inﬁnite probability amplitude to histories obeying (7.1) and 0 amplitude to others. There is then a sequence of singular limits

Ψ~X → Ψ~ → Ψ0 (7.5) that must be reversed to reconstruct the fully quantum theory. 7.2. HIGHER-ORDER TIME DERIVATIVES 251

7.2 Higher-order time derivatives

In the canonical theory it is assumed that the dynamical equations are of first order in derivation with respect to time, ∂t. To be sure, the classical canonical equation of motion, dQ = [H,Q] , (7.6) dt P equating the time derivative to a Poisson bracket, is of first order in ∂t. Since the classical theory is merely an approximation to the quantum theory, all one can deduce from the classical theory is that higher order time derivatives must be a properly quantum effect, vanishing in the limit of classical time. Linearity in ∂t is equivalent to linearity in the energy operator E. Linearity in energy, however, would almost always be a good approximation for sufficiently small energies. In the fully quantum theory it is natural to expect that higher powers of the energy appear in the dynamical equations and can be neglected in the limit of classical time because that is a low-energy limit.

7.3 Spinorial dynamics

A bFermi quantum theoryc is a fully quantum theory whose vector space is a subspace of S of bounded stratum number. The operator algebra is then a Clifford algebra. The vector space is a spinor space for that Clifford algebra. [L] [L] Let sα ∈ S be homogeneous polyadics forming a basis for S . It is convenient to index them by their bBaugh numberc, since they can be naturally generated in that order. Then the first of them is s0 = 1 and the last is the top element sN = s>. [L+1] Their braces Isα form a basis for the first-grade elements of the next stratum S . [L+1] [L+1] Let γα := L{s}α : S → S be left-multiplication by the {sα}. [L+1] Let H be a symmetric form on the Grassmann algebra S in the Isα basis:

[{s}β, {s}α]− = 2hβα = Tr sβsα. (7.7)

α αβ s := H sβ. [L] [L] β o : S → S is an inﬁnitesimal linear transformation with matrix elements o α in the basis sα. [L+1] Isα = {sα} ∈ S are the monadics produced by bracing the sα. 1 ω = σ(o) = γ oβ γα (7.8) 2 β α is a spinorial representation of o, with

∀Ψ ∈ S[L] : oΨ = [ω, Ψ], (7.9) in virtue of the Fermi commutation relations. 252 CHAPTER 7. FULLY QUANTUM DYNAMICS

7.4 Gravity action

[C] In the following, we consider the Lie algebra so[S ], which includes the so[6R] underlying bSegal-Vilela-Mendes spacec, and its representation R by induced transformations of some higher stratum L, constructed by iterating the cumulator Σ the necessary L − C times. [C] [D] Let γβα = −γαβ be the usual basis for the Lie algebra so[S ] represented on S . F−C Let kβ0α0βα be the invariant Killing form of so[C]. Let Σ γβα be the representation on stratum L of γβα, induced by F−C applications of the cumulator Σ. Then the usual Casimir operator of the representation R is, up to a conventional constant, F−C F−C βα K = Σ γβαΣ γ . (7.10) The Casimir operator for an irreducible representation is a scalar. The representation ΣF−C is highly reducible, and K is highly non-scalar and non-central. This permits the following tentative Assertion 6 The operator for fully quantum gravity corresponding to the Hilbert action for classical gravity is the Casimir operator K on S[F] up to a numerical factor. It is unacceptable as a queue action.

Argument In the limit where ωµ6 → ipµ and the other ωβα → 0, νµ K → K1g66g pνpµ. (7.11) To relate this mathematics to the physics of moving bodies and clocks one first observes that the bKilling formc has defining features of the Minkowski quadratic form: It is invariant under the special covariance group, now simple, and it reduces to proper time dτ 2 = dt2 in the rest-frame of the singular limit. Before the singular limit, to be sure, there is no rest frame: the spacelike momentum variables do not commute and cannot all be 0 except when all the ωβα are 0, and in the vacuum the variable ωYX is not 0 but as large as it can be. The primary interpretation of this form is as the bKilling formc, and its further interpretation is fixed by how it appears in the action principle.

7.4.1 Cosmological constant

This raises the question of the contribution of event energy to bdark energy densityc, the bcosmological constantc Λ, which appears in the Einstein equations as a part c4Λ T νµ(Λ) = gνµ (7.12) 8πG νµ of the source stress tensor T . The observed baccelerating expansionc of the universe indicates a dark mass spatial density c4Λ ρ(Λ) = ≈ 6 × 10−27kgm−3, (7.13) 8πG 7.4. GRAVITY ACTION 253 about 75% of the critical value, according to

http://www.astro.ucla.edu/~wright/cosmo_constant.html

[?]. In Planck units this is about 10−123. The edge of a cube containing one Planck mass of dark energy is about 2137 Planck lengths, or 106 m. The size of a light diamond (double 31 −4 cone) containing one quantum ~ of dark action is then about 10 Planck units, 10 m, or 10−12 s. This is to be compared with the contribution of quantum event energy to dark energy density, which depends on the vacuum values of the spatial density of events and the mean energy of each event: e(Λ) = ρ(E) Av E (7.14) A stress tensor calls for a conjunctive evaluation of the energy, momentum, position, and time of a volume element. These are, however, already complementary in canonical quantum theory, and more so in a fully quantum theory. The Einstein Equation must be regarded as a singular limit of a fully quantum equation with a diﬀerent meaning. It is natural to wonder whether a dark energy density might emerge naturally from event energy in the singular limit. Vacuum events cannot all have energy eigenvalue 0. The fact that i ≈ i in the vacuum, and the commutation relation E [Ee[C], i] = t[C], (7.15) NX . imply a non-zero uncertainty product for energy and i unless t = 0. This does not prevent E from having expectation value 0 in the vacuum, however. Experimentally dark energy density is locally small and globally large, dominating other sources on the cosmological scale. The question will be reopened when a vacuum vector is available. 080609 ...... [Find the contribution to the cosmological constant from the vacuum expectation of event energy-momentum ﬂux.] 254 CHAPTER 7. FULLY QUANTUM DYNAMICS Chapter 8

Output which summarizes our results.

Where canonical quantization greatly reduces the radical on one stratum, full quantization eliminates it on every stratum, making a finite quantum theory possible. A bfully quantumc theory is developed and applied to gravity and the bstandard modelc in a loop (2-form) formulation, imbedding both four-dimensional gravity and the standard model bgaugec potentials within a higher-dimensional gravitational potential. It is constructed in nested modular strata, with nested simple classical groups. In any application the stratum construction cuts off at a finite stratum, here the seventh. All fully quantum variables have discrete bounded spectra. Symmetry groups of the canonical theory are preserved, either exactly or approximately.

In the theory ϑo under current study, Fermi quantification relates the strata, leading to a family of nested so(nR; σ) relativity Lie algebras. Six strata A – F suffice. Ranks D, E, and F have familiar classical correspondents, which are the strata of Differentials, Events, and Fields. This raises the question of a physical origin for the causality form g of relativity and the probability form H and imaginary i of quantum theory. The organization of i reduces the algebra so(4R) ⇒ sl(2C). The organization of g reduces the algebra ∼ so(4R) ⇒ so(3, 1). But so(3, 1) = sl(2C). This is taken to indicate that the organization of i is also the organization of g, pointing to a loop (2-form) theory of gravity, where both are expressed as a Hodge star. The space-time coordinates are generators of a representation 216 ∼ of the sl(16R) of stratum C in the sl(2 R) of stratum E. In stratum C, g = H. The g and H of stratum C organize into those of stratum F, of which the standard g and H are singular limits. Events have momentum-like coordinates as well as position-loke coordinates, all generators of an sl(16R). The quantum non-commutativity of the momentum-energy components is the origin of space-time curvature. One may think of the quantum structure of ϑ(20008) as a four-dimensional modular

255 256 CHAPTER 8. OUTPUT

btruss domec composed of a cosmologically large number of simplicial cells each with 16 vertices. Four vertices of each cell fit into long stretchers, two are frozen into a global complex plane, both organizations involving very many cells. The remaining 10 vertices are short internal stiffeners. The cosmic truss dome has four blong dimensionsc with variable coordinates, seen as space-time in a singular limit, two long dimensions with frozen coordinates, seen as the complex plane, and is about one strut thick in each of 10 bshort dimensionsc. A further organization must be invoked for the Higgs. There are no fields in this theory in the usual sense, until a singular limit of classical space-time is taken. The elements of stratum F are not functions on stratum E but quantum aggregates of quantum events of stratum E, with Fermi statistics. Chapter 9

ACKNOWLEDGMENT in which the following are thanked for beneﬁcial and pleasurable communications during this work

Yakir bAharanovc, James bBaughc, Walter bBloomc, Eric bCarlenc, Giuseppe bCastagnolic, Martin bDavisc, David bEdwardsc, Andrei bGaliautdinovc, Tenzin bGyatsoc, Werner bHeisenbergc, William bKallfelzc, Alex bKuzmichc, Garrett bLisic, Danny bLunsfordc, Dennis bMarksc, Tchavdar bPalevc, Aage bPetersenc, HeinrichbSallerc, Joseph bSamuelc Frank bSchroekc, Jack bSchwartzc, Sarang bShahc, Mohsen bShiri-Garakanic, Henry bStappc, Carl-Friedrichs von bWeizs¨ackerc, Eugene bWignerc, and Julius bWessc.

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’ , 31 Bohr atom, 100 H, 21 Bohr, 56, 101, 106, 119, 125, 128, 177 ø , 21 Born, 125, 128, 241 gentle determination, 50 Bose quantification, 154 queue, 169 Brauer, 158 reversal, 73 C gauge group, 94 Grade, 145 C-gauge, 91 SL[V ], 74 Carlen, 253 SO[L], 74 Cartan, 84, 158, 234 Semi, 159 Castagnoli, 253 ϑl, 16 Chevalley, 158 ϑo, 16, 46, 62 Clifford algebra, 38, 146, 156, 160–163, 170 ∨, 46, 175 Clifford form, 145 b-ary expansion, 168 Clifford product, 99 g-adics, 146 Clifford statistics, 149, 162, 185 m-tuplet, 141 Clifford, 77 p-ad, 141 Complete, 62 hHh, 118 Compton, 108 S, 72, 142 Connes, 28, 84 S[L], 142 Copenhagen quantum theory, 132 X, 24, 73 Davis, 253 D I , 35 De Witt, 92 Aharanov limit, 137 DeWitt, 150, 211 Aharanov, 253 Deser, and , 238 Ambarzumian, 28 Dirac algebra, 146 Arnowitt, 238 Dirac, 22, 143, 190 Bacon, 69 Dynkin diagrams, 149 Baugh expansion, 169 Edwards, 253 Baugh number, 247 Einstein Equivalence Principle, 89 Baugh, 168, 253 Einstein, 28, 108, 197, 234 Berkeley, 123 Everett, 136 Bleuler, 22 Fermi algebra, 60, 154, 155 Bloom, 253 Fermi full quantization, 179

265 266 INDEX

Fermi graded Lie algebra, 155 Killing norm, 235 Fermi operator algebra, 41, 76 Killing, 84 Fermi quantification, 154, 158 Kleinian, 76 Fermi quantum theory, 247 Kuzmich, 253 Fermi statistics, 49, 60, 77, 86, 149, 151, 153, Large Number, 52, 53 167, 179, 186, 225 Law of Large Numbers, 135, 183 Fermi(-Dirac) statistics, 146 Lisi, 253 Feynman space, 191 Lunsford, 253 Feynman, 125, 190, 191 MacDowell-Mansouri, 92 Fresnel, 97 Mach, 52 Full Fermi quantization, 85 Malus Relation, 112 GUT, 96 Malus-Born, 23 Galiautdinov, 253 Malus, 108, 125 Galileo, 186 Manin, 120 Gell-Mann, 127 Many World Theory, 136 Giles, 118 Marks, 253 Grade, 31 Misner, 238 Grand Unified Theory, 49 Newton, 97, 124 Grassmann differentiator, 152 Nicolas of Cusa, 124 Grassmann product, 99 Nietzsche, 69 Grgin, 153 Noether, 82 Gupta-Bleuler, 190 Oppenheimer, 193 Gupta, 22 Palev space, 196 Gyatso, 253 Palev statistics, 31, 178, 185, 195, 196 Heisenberg, 56, 106, 120, 125, 127, 128, 178, Palev, 149, 195, 196, 230, 231, 240, 253 186, 193, 197, 253 Pauli, 125 Hertz, 108 Peano, 99 Hessian determinant, 179 Peirce, 52 Hibbs, 191 Penrose space, 78, 195 Higgs field, 83, 209 Penrose, 18, 28, 195 Hilbert, 208 Petersen, 153, 253 Inönü,134 Planck time, 191 Ivanenko, 28 Planck, 108 Jacobi identity, 147 Poincarégroup, 69 Jordan, 193 Poisson Brackets, 58 Kallfelz, 253 Poisson Bracket, 208 Kaluza strategy, 182 Rank, 31 Kaluza-Klein, 92 Regge, 19 Kaluza, 209, 211, 232 Riemann, 234 Killing form, 29, 178, 199, 248 Saller, 253 Killing metric form, 238 Samuel, 253 INDEX 267

Schrödingercat, 136 assured, 114, 144 Schrödinger’scat, 138 atomism, 15 Schrödinger,111, 125–128, 133, 136 bilinear space, 38 Schroek, 253 brace, 20, 33, 35, 46, 170 Schwartz, 253 bracing, 87, 98, 142, 209, 210 Schwinger, 125 c set, 16 Segal-Vilela space, 197 canonical Lie algebra, 188, 194 Segal-Vilela-Mendes space, 150, 182, 240, 248 canonical classical strategy, 57 Segal, 12, 28, 193, 195, 197, 209 canonical codicil, 58 Shah, 253 canonical gauge theory, 79 Shiri-Garakani, 253 canonical group, 69 Snyder, 28, 188, 193, 197 canonical limit, 64 Spinorial Clock, 161 canonical quantization, 14, 40, 45 Stückelberg, 117 canonical quantum space, 194 Stapp, 253 canonical quantum theory, 23 Vilela-Mendes space, 196 canonical subspace, 196 Vilela-Mendes, 188, 197, 209 canonical, 45, 194 Vilela-Mendez, 28 catenation, 209 Volterra-Burgers, 82 causal structure, 25 Weizsäcker, 253 causality form, 21, 25, 73, 236 Wess, 253 cell, 19, 20, 42, 95 Weyl gauge strategy, 91 center, 186 Weyl, 158 centralizer, 225 Wick product, 157 central, 186 Wick, 58 charge level, 77, 99 Wigner’s friend, 138 charge, 17 Wigner, 127, 130, 132, 134, 138, 156, 159, chronometric, 237 194, 253 chron, 33, 95 Yang-Mills, 92 classical logics, 108, 175, 183 accelerating expansion, 248 classical logic, 108, 125, 135 acceleration, 88 classical set, 16 algebra unification, 99, 153, 237 classical system, 22 algebra, 27 class, 115, 141 amonadic, 171 cliffordons, 162 ancestors, 168 coherent superposition, 117 anti-canonical, 45 collapse, 133 anti-system, 110 commutant, 120, 225 apostrophe, 31 commutative reduction, 111 aspect ratio, 61 commutator representation, 147, 216 association, 71 compactification problem, 92, 150 assured transitions, 38 complete, 12, 111, 120, 165 268 INDEX complexual, 19 dyad, 141 compound, 12, 13, 69, 88, 115, 176–178, 185 dynamics vector, 32, 37, 51 constraints, 81 dynamics, 37 contact, 13 eductor, 21 contiguous action, 55 electromagnetic field, 81 continuous geometry, 193 emergent, 25 coordinate state, 120 equivalence principle, 88, 94 correspondence, 119 ether, 97 cosmological constant, 75, 248 event level, 35 cosmological leap, 44, 67 event number, 181 creations, 26 even, 146 creators, 26 exchange parity, 159 crisp, 62, 118 experimental, 26 crystal film, 20 experiment, 118 cumulant, 68 exponential logics, 142 cumulator, 48, 68, 181, 233 exponential, 35 cut, 132 exponentiation, 34 dark energy density, 248 external (long) dimensions, 210 de Sitter, 197 family property, 162 de-brace, 99 field level, 35 defects, 82 field theories, 224 definite, 22 field variables, 221 deformation quantization, 70, 177 field, 39, 48, 49, 60, 85, 170, 181, 187, 188, degravitization, 60 193, 208–210, 221, 231, 232, 234 density matrix, 117, 118 film, 19 derived, 69 filtration, 57, 130 diagonal, 208 forbidden, 144 diffeomorphism group, 69 fragile, 13 diffuse, 62, 118 full quantization, 15, 24, 40, 45, 71, 72, 77, diff, 94 83, 85, 179, 184, 190, 221 doublet, 141 fully quantize, 26 dual space, 144 fully quantum, 25, 26, 251 dual system, 110 function, 14, 207, 208 duality, 118 fundamental length, 193 dual, 75 gauge boson, 231 ductor, 21 gauge connection theory, 231 duplex form, 34, 76 gauge equivalence principle, 89, 92 duplex norm, 76 gauge groups, 43 duplex space, 34, 156 gauge group, 69, 211 duplex vector, 75 gauge metric theory, 231 duplex, 75 gauge potential, 73 INDEX 269 gauge theory, 179 irreducible, 12 gauge, 49, 150, 183, 209, 221, 231, 232, 238, isometry, 23 251 kinematical Lie algebra, 86 gauging, 91 kinematical algebra, 185 general coordinate, 94, 200 kinematical group, 69, 185 general relativization, 85 lattices, 107 generating Lie algebra, 86 lattice, 106 generating vectors, 77 left-multiplication, 147 generations, 99 levels, 28, 99 generation, 46 level, 14, 49, 72, 142, 161, 162, 175, 176, 209, generator, 220 225 generic, 13 linear logics, 142 geometric algebra, 77 locality algebra, 241 ghost particles, 80 locality, 13, 15, 25, 60, 241 grade operator, 145 long dimensions, 252 grade, 35, 141, 146, 201 long, 95 grand, 33 mean square form, 148, 238 gravitational field, 81, 153, 189, 190, 208, mean-square form, 77, 148, 151 239 mean-square norm, 46, 68, 235 gravitization, 64 meta-operation, 229 graviton, 97 metasystem, 50, 101, 112, 115, 118, 119, 160, gravity, 77, 85, 87, 97, 103, 149, 186, 189, 170, 183 190, 197, 208, 209, 232, 234, 238 minimal coupling, 13, 85 growth problem, 93, 150 minimal differential order, 13, 85 harmonic coordinates, 89 miss, 35 high-temperature superconductor, 97 mixing, 118 history, 88 mixture, 117 hypercubic, 211 momentum origin, 215, 218 hyperexponential, 45 momentum subalgebra, 216 iconoelastic, 28 monadics, 36 idol, 69 monadic, 171 imaginary, 224 monadization, 71 incoherent superposition, 117, 118 monad, 15, 141 incoherent, 172 moves, 192 incomplete, 119 multilevel Clifford algebra, 172 induced, 94, 180 multiplicative norm, 147 induce, 73 multiplicity, 110, 141 inductor, 21 natural quadratic forms, 236 input, 95, 109, 110, 112, 143 negative norm, 67 internal dimensions, 233 neutral, 76, 158 io, 144 non-commutativity, 84, 107, 119 270 INDEX normal ordering, 58 quantized imaginary, 83 normally ordered product, 157 quantum chronometric, 239 number operator, 152 quantum generators, 18 observable, 57 quantum logics, 106, 107 occlude, 35 quantum non-commutativity, 84 odd, 146 quantum praxics, 108 ontologism, 100, 127, 135, 136 quantum set theory, 115 ontology, 56, 57, 100 quantum set, 16 orbital part, 91 quantum time, 188 orbital, 91 quantum variables proper, 63 origin cell, 211 quantum, 27 orthonormal, 45 quasicontinuous, 73 output, 112, 143 queue algebra, 72 palevons, 31 queue gauge group, 94 phase space, 100 queue strategy, 77 polyad, 141 queue, 16, 24, 27, 31, 60, 115, 141, 168, 170, polynomial logics, 142 175, 209 polynomial, 94 q, 116 power set, 34, 168 radical, 69 praxics, 106 random set, 40, 169, 172 praxic, 21 rank jump, 171 praxiological, 56, 121 rank, 14, 33, 34 praxiology, 27, 56, 120 reciprocity, 241 predicate, 115, 141 reduced, 181 principle vector, 215 regular, 13, 29, 145 probability form, 21, 23, 25, 38, 48, 73, 112, relation, 208 114, 118, 144, 153, 236 relativity, 26 probability function, 128 reverse, 236 probability operator, 50, 118 rigid, 13 probability tensor, 117, 118 robust, 13 product unification, 153 root vector, 84 projective geometry, 114, 144 scalar, 148 projector, 57 sea, 16 proper state, 120 second quantization, 39, 158 pseudo-bosons, 97 self-Fermi vector algebra, 76 pure sets, 40 self-Grassmann algebra, 40, 49, 62, 198 q bit, 38 self-organization, 75 q set, 16 semi-simple, 84 quantification, 39, 149, 185 semiquanta, 158 quantifier, 152 semiquantum, 154 quantization, 63, 150, 232 semisimple, 12, 69 INDEX 271 semivectors, 76 strongly continuous, 165 semivector, 158 structurally stable, 13 set, 95, 141, 146 structurally unstable, 13, 60, 186 sharp, 37, 62, 118 structure manifold, 13 short dimensions, 252 successor, 99 short, 95, 209 supermanifolds, 200 signature, 210 superposition, 35, 115, 118 simple quantization, 243 superquantification, 158 simple system, 116 supersymmetry, 200 simplex, 19 swap, 17 simple, 12, 84, 85 symmetry, 26 simplicity principle, 16 throughput, 144 simplicity, 15 time level, 224 simplification by quantization, 70, 185 time quantum, 24 simplification strategy, 12 time reversal, 156 singlet, 141 torsion, 89 singular Lie algebra, 29 transformation theory, 143 singularities, 149, 150, 179 transition probability amplitude, 118 singular, 13, 29 transition probability, 118 solvable, 69 truss dome, 95, 252 sortation, 130 truss, 96, 201, 210, 212 special coordinate, 200 unification, 49, 237 special, 13, 89 unimodular diffeomorphism, 89 spin 1/2, 158 unimodular, 89 spin parity, 159 unimodulat covariance, 89 spin part, 91 unit function, 109 spin vector, 47 unitarity, 60 spin-statistics correlation, 159 unitization, 71 spin-statistics, 41, 77, 149, 158–160, 162, 163 vacuum, 53, 226 spinor, 158, 161 variable, 57 spin, 17, 47, 91, 162 vector strategy, 57 standard basis, 149 vectors, 145 standard model, 49, 85, 87, 167, 176, 180, vector, 16, 117, 124, 143, 162, 168 187, 209, 251 vinculum, 171 state space, 100 von Neumann, 106, 107, 130, 193 state strategy, 57 wave function, 125–128 state vector, 113 zero-point action, 44 state, 119, 120 statistical Lie algebra, 86 , 46 statistics, 149 strategy, 100 atomism, 15 272 INDEX curvature, 96 monadic, 46 polyadic, 46 simplicity, 16