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The difference between the two interpretations is in spacetime - namely the energy-momentum relationship - some correspondence with two conflicting attitudes to- and Maxwell equations emerge in an algebraically sim- wards the wave function. Some scientist believe it is a ple and elegant way from a reinterpretation of Eq. 1, i.e. mere mathematical tool while others tend to interpret its from a proper distinction between those quantities that components as the true dynamical variables. We will ar- change and those that stay constant. gue that both attitudes miss the point. It is the pure form of physical theories - i.e. the equation of motion - that re- quires the definition of some fundamental variables. And B. The Form of Physics it is a deformation to believe that we can directly assign physical meaning to these variables. Hence the “classi- Usually textbooks on (classical) physics begin with the cal” form of physical theories is - though mathematically equations of motion of point-masses, the definition of po- sound - conceptually inconsistent or at least incomplete. sitions, velocities and accelerations etc. This means that Let us briefly explain why this is so. the fundamental variables of the theory are usually di- Eq. 1 tells us that the wavefunction of a system with rectly charged with physical meaning, a meaning that energy changes at every time. Instead of postulating the is supposed to have its origin in our “classical” macro- validity of Eq. 1, we again reverse the argument: We pos- scopic experience. However this “method” implies that tulate that change is immanent to physical reality [25]. we have to presuppose a considerable number of mechan- The permanent change is the physical mode of existence. ical concepts. A proper definition of these concepts how- Again metaphorically we say that material objects exist ever relies on and refers to an elaborated theory. Like in time. Any fundamental physical model of reality has Baron Munchhausen, who claimed to have pulled him- to represent this continuous change. This is the essence self by the hair out of the swamp, the theory is based of Eq. 1: Variables ψ that represent existing fundamen- upon notions that become meaningful only in the con- tal quantities have to vary continuously. We show in text of the spelled-out physical theory [27]. It is the fate the following that Eq. 1 can be derived from this pos- of human existence to be thrown into an already exist- tulate. But before doing so, let us briefly describe why ing world and it seems to be the fate of physics students classical physics is conceptionally incomplete. The most to be thrown right into a framework of concepts with- trivial flaw is the inability of classical physics to give an out the chance for a stepwise systematic and logical (re- account of its measurement standards. The most triv- )construction. The classical theory implicitly claims that ial being the length. Classically we take the existence of these concepts are in some way ad-hoc derivatives from a solid rod for granted. Einstein made an enlightening everyday experience - though seen by light Hamiltonian remark: “One should always be aware that the presup- and Lagrangian mechanics can rarely be applied directly position of the existence in principle of rigid rods is a to everyday empirical evidence. presupposition suggested by approximate experience but We invite the reader to follow a different path and to is, in principle, arbitrary”[26]. Why is this so? Because put on hold with the conceptions of mass, point particles, classical physics taken as classical mechanics, electrody- wave functions and also spacetime. A renewed analysis namics and relativity can not explain the existence of a of spacetime is required as soon as we understand that finite and fixed atomic radius and hence can not explain the rod of constant length - and with it spacetime - is a the existence of a measuring rod. Bohr’s orbital theory mere postulate. But if quantum mechanics is required to tried to establish fixed radii by an ad-hoc postulate - but explain the existence of rods of constant length, then the the idea of definite orbitals conflicts with electrodynam- same is true for the concept of spacetime: It requires an ics. We will not derive a finite radius in the following, explanation - in the optimal case in form of a derivation 3. instead we will make the presumption explicit and incor- Physics is an “empirical” science, based on objective porate it as a general principle in what follows. measurements. A measurement is the comparison of vari- On the fundamental level existence implies continuous able properties of objects (measurands) with the corre- change. But though all fundamental quantities continu- sponding constant properties of reference objects. Be- ously change, physics as an experimental science requires fore we are able to measure distances, we need to have the constancy of measurement standards. Without con- stant reference standards we could not test physical mod- els. These two apparently trivial facts are the starting 3 There are more good physical reasons to reconsider the concept point of the ontology of time. Our line of reasoning is in of spacetime. The general theory of relativity (GTR) claims that contrast to concepts that postulate spacetime to be fun- the geometry of spacetime is determined by the distribution of damental. Eq. 1 does not require a concept of spacetime. the masses “in it”. If this is true, then matter and spacetime are If, in the following, we speak of time then it is always not separable entities. Secondly, it is a well-established part of 2 quantum mechanics that the wavefunction can either be defined understood as a rate of change . However we shall ar- as a function of space and time or equivalently as a function of gue in this essay that and how the conjugate concept of energy and momentum. Both representations of the wavefunc- tion are related to each other via a Fourier transformation. If the most fundamental theories suggest that spacetime can not be consistently understood independent from matter and its dy- 2 In accelerator physics this is called phase advance. namics, then we should not ignore this fact. 3 a ruler, for instance a solid rod. Certainly we can think tative description of reality. Whatever a physical theory about (the concept of) length without having a ruler, considers to be fundamental must essentially be repre- i.e. we can develop geometry. But we can not perform a sentable by quantities. We call these quantities “vari- measurement nor predict a measurement outcome with- ables”. The mathematical model of these fundamental out the ruler. The material object that is used as a ruler entities is hence based on a list of variables ψ of (yet) (the rod) is the fundament of objectivity of measurement. unknown dimension, which all continuously change at all Only with the possibility in principle of the existence times. But if the components of ψ change at all times and of reference objects like the prototype meter it becomes if they are fundamental, then there is no way to define physically meaningful to speak about the length of an a measurement standard for them. Nevertheless there object or the distance between objects, i.e. space. In might be functions of the fundamental variables that are practice we say almost unreflectedly that the length L is constant in time: such constants are called constants of x inch, meter or lightyears. However the production of a motion (COMs). These functions may not include linear ruler is the first thing we have to do before we can mea- terms - otherwise it is easy to show that the functions sure length. It is not sufficient to define it - it has to be can not be constants 6. Hence the dimension (unit) of physically made. It is a necessary though not a sufficient these functions is different from the dimension (unit) of condition for a measurement. The handbook of metrology the fundamental variables. And therefore they can not and testing describes a measurement as follows: “Mea- serve as rulers for the considered fundamental variables. surement begins with the definition of the measurand, The logical consequence is that a direct measurement of the quantity intended to be measured. The specification the fundamental variables of ψ is not possible. The above of a measurand requires knowledge of the kind of quan- considerations explain why the components of the quan- tity and a description of the object carrying the quantity. tum mechanical (QM) wave function have to vary at all When the measurand is defined, it must be related to a times, why the quantum mechanical wave function im- measurement standard, the realization of the definition plies an interpretational problem and why QM has to of the quantity to be measured.” [28]. Theory should fol- postulate that measurement devices must be “classical” low practice: Before we postulate spacetime, we should or “macroscopic”: Due to the absence of constant rulers, explain how a measurement of distances is at all possible. the fundamental variables (the components of the wave- This article is not about metrology. But what is im- function) can not be directly measured 7. portant about the measurement standard is that it must The definition of general and abstract quantities like be realized and may not change with time. The mea- the fundamental variables and their evolutions in time surement of a property of a physical entity requires the will not suffice to make up a physical theory. At some existence of material entities where the respective prop- stage we need an interpretation that maps quantities of erty stays constant (the rulers) and other entities - with the model to measurable quantities in the world in order the same type of properties - (the measurands) where this to obtain physical meaning. The physical meaning can property varies or at least can vary. If the measurand then only be induced from the mathematical structure of changes with time, it is the purpose of a physical model the physical quantity. The interpretation can only refer to predict the time-dependence of the measurand. When to already known physical laws in order to identify pos- we say that the ruler must not change with time, then sible interpretations of the quantities and the relations we include other invariances as well. Wigner pointed this between them. We can only recognize fundamental phys- out by referring to invariance as a fundamental princi- ical meaning of certain algebraic relationships because an ple [29]. Hence we can say that physics is irresolvably elaborated theory of elementary particles that has been committed to time, i.e. to continuous change and to con- shown to be experimentally successful in the description stancy. This is not postulate, it is a conditio sine qua of fundamental physical phenomena is already available. non4. The unity of nature is next major premise: To be part of the physical world implies the possibility in principle C. Outline of (direct or indirect) interaction with all other things that are part of this world and therefore we (have to) The paper is organized as follows: In Sec. II we de- presume the existence of a common denominator, a fun- scribe our understanding of fundamental physical enti- damental level5. Physical models are based on a quanti- ties. In Sec. III we derive Hamilton’s equations of motion (EQOM) and the basic properties of the symplectic unit matrix. In Sec. IV constants of motion are introduced 4 Sir Hamilton was not only aware of the intimate connection between physics and time - he even had the vision to develop algebra as the science of pure time [30] and it was his deep belief that “that the intuition of time is more deep-seated in the human 6 Dragt et al have shown in Ref. ([32]) that there are no non-zero mind than the intuition of space” [31]. first order moments in linear Hamiltonian systems. 5 The basis of this believe is much the same as most theorists 7 Or with the words of S. Fortin and O. Lombardi: “The difficul- believe that it should finally be possible to describe all forces of ties can be overcome once it is recognized that classicality is a nature within a grand unified theory. property of the observables” [33]. 4 with the help of Lax Pairs. We review important alge- ables. Since quantum theory postulates that the wave- braic relations that indicate the construction principles function is fundamental, it is nearby to assume that we fi- of spacetime: The required congruence of the algebraic nally have to identify the components of the wavefunction structure of (skew-) Hamiltonian matrices with the basic with (parts of) our list of “fundamental variable(s)”. The elements (i.e. generators) of Clifford algebras. We de- Copenhagen interpretation avoids the reference problem scribe the basic measurable entities in a world based on by assigning a mere probabilistic meaning to the wave- time: (Second) moments of fundamental variables. Since function, i.e. by postulating that a) the wave function second moments can be represented by expectation val- has only a probabilistic meaning and b) the wavefunction ues of matrix operators, the relations between these ma- gives a complete description of reality, i.e. is fundamen- trices are the relations between the observables (i.e. the tal. Most discussions of the Copenhagen interpretation laws of physics). focus on the question, if the probability interpretation In Sec. V we derive the conditions for the emergence is correct, or whether the probability is “classical” or of a geometric space from symplectic dynamics. Ob- “quantum” in nature and what the ontological status of servables are expectation values of matrix operators and the wavefunction is. We do not address these questions. hence spacetime geometry should be representable by a Instead our claim is, that if the wavefunction is truly system of basic matrices with certain properties. Such fundamental, then a meaningful physical interpretation matrix systems indeed exist and are known as (represen- of the wavefunction is impossible. This does not imply tations of) Clifford algebras. that it is less real than the macroscopic quantities derived In Sec. VI we describe symplectic transformations as from it. However a meaningful answer to the question structure preserving transformations. These transforma- what varies requires the existence of some constant en- tions are isomorphic to Lorentz transformations and are tity for comparison, i.e. a reference for a measurement. the basis for the apparent geometry of spacetime. We ex- If we say that an entity represented by a variable in our plore the meaning of structure defining transformations. equation is a length then we need to have, in principle, a In Sec. VII we give a short overview over the basic unit (for instance the meter) to refer to and we need to properties of Clifford algebras in general and especially have an idea of spacetime. Not only that this reference of ClN−1,1(R). We derive conditions that limit the pos- has to be of the same type (i.e. dimension), but it also sible dimensionality for an emergent spacetime. Further has to be macroscopic as “meaning” can only be derived analysis of the properties of these Clifford algebras al- from everyday (macroscopic) experience. Physical mean- lows to restrict spacetime to 3+1 dimensions. Hence the ing and objectivity depend on a constant quantity of the appropriate algebra is the algebra of the real Dirac ma- same dimension that can be used as a reference. We have trices. an idea and a concept of length as we are surrounded In Sec. VIII we describe Lorentz transformations - by objects with this property. The lack of reference is boosts and rotations - as structure preserving (symplec- according to our premises a logical consequence of fun- tic) transformations. We present an interpretation of the damentality. With respect to the fundamental variables Dirac matrix system called the “electromechanical equiv- there is no meaningful answer to the question what varies, alence” (EMEQ). Guided by the EMEQ we derive the but only how they vary9. Lorentz force and in Sec. IX we derive (Quantum-) elec- This is a logical limit in physics: A fundamental physi- trodynamics. We give arguments why momentum and cal theory can describe how entities change but it can not energy should be related to spatial and temporal deriva- tell what the entities are that are changing. Hence the tives and sketch the path towards the Dirac equation, only possible physically real rulers, the only possible mea- a representation of massive spinors and the describe the surement standards are constants of motion: the proto- significance of CPT-transformations in our approach. type meter is an object composed of atoms and molecules Sec. X finalizes the discussion and a summary is given which consist of “elementary particles”. The motion of in Sec. XI. these components is stable and the length of the reference object “meter” is a constant of motion of these particles. But at the level of these particles - or at the level of their II. OBLIGATIONS OF A FUNDAMENTAL constituents - at some level we face the pure variation of THEORY abstract fundamental variables. We conclude that if it is possible to directly measure (the value of) a variable, it The basic variables of a fundamental theory are con- can not be fundamental. tinuously varying quantities 8 - they are dynamical vari- There is no doubt that we can measure distances and angles since we have rods and goniometers. Hence space (or spacetime, respectively) is not fundamental. Space- 8 The physical constants (ε0, µ0, c, ~ ...) as listed in handbooks time is a construction for our theoretical needs - it is are not material but rather theorectical objects. Some of these are merely conversion factors for units. If we refer to “constants”, then we address properties of material objects that are poten- tially useful as measurement standards. For instance the orbital 9 In other words: “The wave function does not describe matter, it radii of hydrogen atoms. describes how matter behaves” [34]. 5

not itself a “thing” but a formalism to express the rela- and legitimate. In the meantime we restrict our con- tions between things. This is not to say that spacetime siderations to the reals. Or more precisely: we demand does not “exist”. If things exist and relations between that all fundamental variables are of the same type, as things exist, then we can speak of the space that exists. we consider it scientifically not legitimate to introduce However it is not fundamental, which basically means an asymmetry like the asymmetry between the reals and that it can be derived. To say this implies the obligation the imaginary numbers without reason. Hence all funda- to derive the dimensionality of spacetime, the Lorentz mental variables are either all real or all imaginary. The transformations and the role played by electrodynamics, latter might be preferable to indicate their unmeasurabil- i.e. the speed of light. We shall argue that spacetime ity, however as long as there is no need to introduce the is based on the isomorphism of a Lie algebra with (reps distinction between real and imaginary variables, Ock- of) real Clifford algebras Cl(N 1, 1). The “perception” ham’s razor demands to stay with reals 11. of spacetime is a result of pattern− recognition in loose It follows from this conception that fundamental “ma- analogy to the perception of acoustic signals in terms of terial objects” like for instance electrons have to be de- sound and music 10. Our ansatz is based on a list of fun- scribed by their structure. Since the location in space- damental and abstract variables, the values of which can time and hence a continuous trajectory is according to not be directly measured. Such a measurement would our principles not fundamental and may hence not serve require constants of motion from “more” fundamental to define or verify sameness, elementary particles have entities. But then the more fundamental variables could no material identity, but only a structural identity. The not be measured, etc. Hence it is impossible to directly only structure at hand is the structure of the variations of assign physical meaning to the fundamental dynamical the variables - the structure of their dynamics. The fun- variables. In this respect fundamental variables are not damental objects (elementary particles) are not directly identical to degrees of freedom (DOFs) in classical me- represented by the fundamental variables, but rather by chanics as DOFs are usually assumed to be measurable at the dynamical structure of the variations, by the patterns least in principle. But in most other aspects fundamen- of motion. Physics can be described as a method to ana- tal variables are similar to (DOFs) in classical mechanics lyze patterns of motion, it is to some degree identical to and at first sight there is no reason to refuse the possi- pattern recognition 12. If “objects” do not change their bility of a classical description of the dynamics of these structure (i.e. their “identity”) in some interaction, then variables. the involved dynamical processes must be structure pre- E.T. Jaynes wrote that “Because of their empirical ori- serving: If a material entity like an elementary particle is gins, QM and QED are not physical theories at all. In defined and understood by its structure, then its contin- contrast, Newtonian celestial mechanics, Relativity, and uous existence requires - besides a continuous variation Mendelian genetics are physical theories, because their of its constituents (the variables) - that the dynamical mathematics was developed by reasoning out the conse- structure must be preserved. However if objects can only quences of clearly stated physical principles from which be identified by a certain dynamical structure then two constraint the possibilities”. And he continues “To this objects with the same structure can not be distinguished. day we have no constraining principle from which one can If we could distinguish particles experimentally that are deduce the mathematics of QM and QED; [...] In other indistinguishable in our physical model, then the model words, the mathematical system of the present quan- would be incomplete. Thus, if quantum mechanics is tum theory is [...] unconstrained by any physical prin- indeed fundamental, then it must logically include a con- ciple” [35]. We agree with this statement insofar as we cept of identical particles - as particles in a fundamental think that it is important to constraint possibilities and physical theory have to be represented by structures. to develop a theory according to well-defined principles. The introduction of the unit imaginary into quantum me- chanics is not of this kind. No physical principle has ever III. VARIABLES, CONSTANTS AND THE been formulated that explains the necessity for the use HAMILTONIAN of the unit imaginary in QM. There are just ad-hoc pos- tulates. We believe that the use of the unit imaginary as A. Variables it is usually done, is a mistake. It introduces a structure without clear and explicite motivation. Hence we will We suggest the following principles for an ontologically avoid or at least delay the introduction of the unit imag- proper basis of physics: inary up to the point, when its use becomes reasonable

11 For discussions about the use of real and/or complex numbers 10 We should be aware that “physically” there are just wavelike generally in physics and specifically in quantum mechanics, see density fluctuations in air. It is our mind that processes the also Refs. [14, 36–38]. perceived data and interprets them as sound, language or music. 12 Recall that the birth of modern physics is connected with names This interpretation is indeed adequate. Nevertheless it remains like Kepler, who recognized the “true” pattern of planetary mo- an interpretation. tion. 6

1. Existence happens in time. The “time” that is is the basic form of the symplectic unit matrix. Accord- meant here does not have to be identical to the ing to the axioms the matrix must have full rank 2 n k, time that an observer would measure using a clock, i.e. all variables vary and hence there are no constants≤ but we insist that there is a bijective functional re- in the state vector such, i.e. lationship 13. T Q SQ = diag(η0, η0, η0,... ,η0) γ0 . (9) 2. Existence in time is manifest by variation. All ≡ quantifiable properties of all fundamental physical The transformation Eq. 9 is a linear change of variables entities continuously change and are representable which is used to find (or recognise or define) the “natu- by real quantities that continuously vary, i.e. by ral” or normal variables. We call such transformations “variables”. structure defining (see below and App. (E)) and rewrite Eq. 5 accordingly 3. Measurements require constant references (rulers), i.e. a physical model requires constants. dH = (Q−1 )T QT SQ(Q−1 )=0 dt (10) = (Q−1 ∇ H)T γ (Q−1 )=0∇ H . From these axioms it follows that ∇ H 0 ∇ H 1. There are no other (physical) constants than con- In the following we assume that the transformation into stants of motion (COMs). the normal variables has been done such that the normal variables are given by ψ and that S has between trans- 2. The fundamental variables have no measurement formed into γ0 so that standard, i.e. can not be directly measured. ψ˙ = γ0 ψ . (11) The value of a single variable varies. Hence a single ∇ H variable can not generate constants of motion. Therefore The matrix γ0 has the even dimension 2n 2n: in any we start with an arbitrary number k > 1 of dynamical time-like physical world the dynamical variables× come in variables 14. According to the axioms there exist one (or pairs 16. It is therefore sensible to refer to a pair of a set of) constant function(s) (ψ , ψ ,...ψk) = const of variables when we speak of a degree of freedom. We call H 1 2 the dynamical variable list ψ. Hence we may write (with ψ2j = qj the j-th canonical coordinate and ψ2j+1 = pj the ∂H T the presumption = const we imply here ∂t = 0): j-th canonical momentum, i.e. ψ = (q1,p1,...,qn,pn) , H but this nomenclature is purely formal as long as the d ∂ ∂ ∂ variables ψ are fundamental. It is just the structure of H = H ψ˙1 + H ψ˙2 + + H ψ˙k =0 , (4) dt ∂ψ1 ∂ψ2 ··· ∂ψk the matrix γ0 that leads to this distinction. The matrix γ is called the symplectic unit matrix and the equations or in vector notation: 0 of motion (EQOM) have Hamilton’s form: d = ( )T ψ˙ =0 . (5) ∂H H ψ q˙k = dt ∇ H · ∂pk ∂H (12) p˙k = . The (simplest) general solution is given by − ∂qk Whenever we have constants (of motion) of the dynam- ψ˙ = S ( ψ ) , (6) ∇ H ical variables, we can derive Hamilton’s EQOM in some where S is a k k non-singular skew-symmetric real ma- way. We can interpret the abstract and intrinsically un- trix, i.e. ST =× S. The skew-symmetry of S is sufficent measurable basic variable list ψ as “spinors”, i.e. as to solve Eq. 6 so− that is a constant of motion. Accord- “objects” in a phase space. Just formally we call the ing to linear algebraH there exists a non-singular matrix components “coordinates” and “momenta”. But in fact Q such that 15: every ψ is a point in an abstract 2n-dimensional phase space 17. However the Hamiltonian formalism itself does T Q SQ = diag(η0, η0, η0,..., 0, 0, 0) (7) not require that the variables are fundamental and there- fore this formalism can be applied to any system with where dynamical constants. The difference to the “classical” 0 1 η = , (8) 0 1 0  −  16 The unit imaginary has indeed significance in quantum mechan- ics as it ensures an even number of fundamental variables and a continuous variation of these variables due to Eq. 1. However 13 We explicitly include the possibility of regularizing transforma- our approach contradicts the frequently expressed opinion that tions [39, 40]. QM inherently requires the use of complex numbers in the sense 14 Ockham’s razor commits us to determine the minimum number that it could not be formulated without the unit imaginary. of variables required to generate objects with spacetime proper- 17 In classical statistical mechanics this kind of space is called Γ- ties. space [43] but the intimate relation to quantum mechanics is 15 See for instance Ref. ([41]) and App. (E 3,E 4) known as for instance described by Kim and Noz [44]. 7

Hamiltonian formalism is solely that the classical state The Hamiltonian EQOM (11) can then be written as a vector ψ describes the (average) properties of systems product of a matrix F and the vector ψ: as the position of the center of mass or the average mo- ψ˙ = γ ψ = F ψ , (16) mentum and does not consist of fundamental variables. 0 ∇ H Thus the components of a classical state vector ψ can where we defined the matrix - at least in principle - be measured. But if the state vector represents fundamental quantities then the com- F γ0 A . (17) ponents can not be directly and individually measured. ≡ Before we are able to present an interpretation of ψ we first have to construct constants of motion and the corre- IV. CONSTANTS OF MOTION sponding observables. One constant of motion has been introduced already: The Hamiltonian function . In H We define the dyad Σij ψi ψj which can be under- classical physics this function most often represents the stood as a matrix of all possible≡ quadratic forms. If we energy of a physical system. We will suggest a similar optionally consider some type of averaging, then this ma- interpretation in what follows. trix becomes a matrix of second moments Σ:

Σ= ψψT , (18) h i B. The Hamiltonian where the angles indicate some sort of averaging. We hi define the matrix S by S Σ γ0, so that: We introduced the “Hamiltonian” as (an arbitrary) ≡ H T T constant of motion. Typically there are several constants S˙ = Σ˙ γ0 = ψ˙ ψ γ0 + ψ ψ˙ γ0 T T T of motion and hence the function is not yet well- = F ψ ψ γ0 + ψ ψ F γ0 (19) H T defined. In the following we assume that is positive = FS S γ0 F γ0 , (semi-) definite with respect to the variablesHψ. This re- − 2 T striction is neither arbitrary nor weak: If the constructed where we use the fact that γ = 1 and γ0 = γ . From 0 − − 0 constant of motion is a reference of existing measur- Eq. 17 it follows that H able things, then there must be a Hamiltonian function T T T 2 that reflects the amount of something, for instance F = A γ0 = (γ0) A γ0 = γ0 F γ0 , (20) theH amount of substance. In a physical theory that is where we used the fact that A is symmetric. We then free of ad-hoc postulates the constant existence of a sub- obtain stance must be represented by a positive (semi-) definite Hamiltonian as positivity is immanent to the notion of S˙ = FS S F . (21) substance. − We write the “Hamiltonian” as a Taylor series of That is, if F and S commute, then the matrix S is con- H the 2 n variables ψk: stant, while ψ varies. Operators S and F that fulfill Eq. (21) form a so-called “Lax Pair”. For such pairs it T 1 T 1 can be shown that (ψ)= 0 + ε ψ + ψ A ψ + Bijk ψi ψj ψk + ..., H H 2! 3! k (13) T r(S ) = const , (22) where ε is a 2 n-dimensional vector and A is a symmetric 2 n 2 n matrix and B is a tensor that is symmetric holds for any natural number k [45–47]. Hence the bilin- in all× indices. We assume that has a local minimum ear form (18) is the basis for a set of constants as given H by Eq. (22). Note that the equation of motion (Eq. 21) somewhere. Further we may set 0 = 0 as it has no influence on the equations of motion.H A typical method of the second moments fullfills the requirement to repre- S in physics is to stepwise study the solutions of Eq. 13, sent observable properties of a physical system. If and F starting with low amplitudes first. Hence the first step are similar, i.e. if they share a system of eigenvectors, S is to neglect higher order terms and to focus on small then is constant. If this is not the case, then we still amplitude solutions, i.e. to write have the constants of motion given by Eq. 22 as reference quantity. 1 The average of the quadratic form equals a matrix (ψ) εT ψ + ψT A ψ . (14) of second moments, if summed over an ensemble of N H ≈ 2! spinors:

After the truncation to second order, an offset ψ˜ = ψ ψ0 N −1 ˜ − 1 T of size ψ0 = A ε (ψ = ψ ψ0) enables to get rid of Σ= ψ ψ , (23) − − N k k the linear term, so that: k X=1 1 or - which is a complementary description - if we use a (ψ) ψT A ψ (15) H ≈ 2 “density” ρ(ψ) to describe the distribution of states in 8

Γ-space. In this case the matrix of second moments is these names are misleading, the former mainly because given by the matrix F does not appear in the Hamiltonian and the latter since F is neither symplectic nor infinitesimal. ρ(ψ) ψ ψT d2nψ Therefore we use the term symplex (plural symplices) [48, Σ= . (24) ρ(ψ) d2nψ 49]. R A cosymplex (or “antisymplex” or “skew-Hamiltonian” Eq. (21) holds also for aR single Γ-space trajectory ψ(t), matrix) is a matrix C that holds 20: but for a single trajectory one finds for all k that k T T r(S ) = 0, so that the constancy is trivial: The eigen- C = γ0 C γ0 . (27) − values of a matrix of second moments Σ depend on whether they are computed for a single vector ψ (which A cosymplex C can always be written as a product of γ0 results in a matrix Σ with vanishing determinant and and a skew-symmetric matrix. The sums of (co-) sym- vanishing eigenvalues), for two linear independent vectors plices are (co-) symplices, i.e. the superposition principle (ψ1, ψ2) (Σ has vanishing determinant, but two non-zero holds. Hence (co-) symplices form a linear vector space eigenvalues) or 2 n linear independent vectors ψk as in and any (co-) symplex can be written as a linear combi- Eq. (23). Only≥ in the latter case, Σ is non-singular 18. nation of “basic” (co-) symplices. The algebra of (co-) This implies that our ansatz unfolds to full generality symplices is the Lie algebra sp(2 n). only if we consider ensembles. Or more precise, the ob- Denoting symplices by S and cosymplices by C (op- servables are averaged over ensembles. tionally with subscript) it is easily shown that the anti- Since S (and F) are by definition the product of a commutator of two symplices is a cosymplex: symmetric matrix Σ and the skew-symmetric matrix γ , 0 S S S S T ST ST ST ST it follows that ( 1 2 + 2 1) = 2 1 + 1 2 = γ0 S2 γ0 γ0 S1 γ0 + γ0 S1 γ0 γ0 S2 γ0 T r(S)= T r(F)=0 . (25) = γ0 (S2 S1 + S1 S2) γ0 . − (28) so that the simplest meaningful constants of motion are The following general rules can be derived: given as S1 S2 S2 S1 2 − T r(S ) = const . (26) C1 C2 C2 C1 − symplex CS + SC  ⇒ Eq. 15 describes a n-dimensional harmonic oscillator. S2 n+1  The truncation of the Hamiltonian to second order is not  arbitrary - it guarantees stability. It is well-known, that S1 S2 + S2 S1 (29) non-linearities yield (in many or even most cases) unsta- C1 C2 + C2 C1 ble or chaotic behavior. Therefore in order to establish a CS SC  cosymplex −2 n  ⇒ system of stable references linear systems are preferable. S  This restriction is per se not problematic, since the equa- Cn tions that we will derive are also linear 19. We might also   interpret this restriction as a low-energy approximation  as in Ref. ([2]). The relations Eq. (29) define the structure of a specific The constants of motion that can be used as references algebra, called the Lie algebra sp(2 n) of the symplectic are quadratic forms (22). If a reference is based on a group Sp(2 n). As the matrix of generators F, also the quadratic form, the variable compared with the reference matrix of observables S of Eq. (22) is a symplex. Ac- (the measured variable) also must be a quadratic form. cording to Eq. (29) odd powers of S are symplices with Therefore the “dynamical variables” qi and pi cannot zero trace. Hence the non-zero (i.e. useful) constants of be measured directly, but only (functions of) quadratic motion given by Eq. (22) are of even order. forms based on these variables. We call these (functions of) quadratic forms observables. Matrices that fulfill Eq. 20 are called “Hamiltonian” or 20 sometimes “infinitesimally symplectic”. In our opinion The nomenclature of (co-) symplices combines the terms “sym- plectic”, “(co-) sine” and “matrix”. The connection to (co-) sine will become obvious in Sec. VI A. Furthermore, in geometry a “1- simplex” is a line, a “2-simplex” is a triangle and a “3-simplex” is a volume (tetrahedron). Equivalently the elements of a Clif- 18 In accelerator physics the eigenvalues of S =Σ γ0 are the “emit- ford algebra (for instance the Dirac algebra) include vectors (ver- tances” (times the unit imaginary). It is evident, that the posi- tices), bi-vectors (edges) and tensors (faces). Pascal’s triangle tion of a single particle at time t defines a point but not an area can be used for both - the number of k-simplices of an n-simplex in phase space [48, 49]. (k <= n) and to the number of traceless k-(co-) symplices of 19 We refer to the well-known regularization methods that allows an N-dimensional Clifford algebra. Hence the Clifford algebra to map the Kepler problem and a large variety of other central Cl3,1 represented by the real Dirac-matrices can be compared to potential problems to the harmonic oscillator [50]. a regular 3-simplex (tetrahedron). See Fig. 3 in the appendix. 9

Since the matrix A is symmetric (Eq. 15), the maximal with some arbitrary (non-zero) factor λ. Vice versa - number ν of free parameters of a symplex for n degrees if two (normalized) states φ and ψ may not “generate” of freedom (DOF) is non-vanishing expectation values of the cosymplex type, then is follows that: 2 n (2 n + 1) νs = . (30) φ = ψ . (38) 2 ± If the matrix F can be expressed as a sum of basic ma- Since we assume that physical reality can be described by the totality of all fundamental variables, the coefficients trices, the basis νs elements. A basis for all real 2n 2n × of the matrix F are functions of (internal or external) matrices is completed by νc cosymplices: measurable quantities, i.e. of some symplex S generated 2 n (2 n 1) from fundamental variables. And since cosymplices have ν = − . (31) c 2 a vanishing expectation value, they can not contribute. We will use this important result in Sec. IX in the deriva- tion of MWEQ. Furthermore, if a structure that repre- A. Measurable Quantities: Observables sents a “particle” is preserved in a physical process, then the involved driving terms must be symplices. Otherwise Usually the second moments are assumed to be average the evolution in time would not be symplectic, i.e. struc- values of N identical systems as in Eq. (23), but the equa- ture preserving. And vice versa: If the particle structure tions hold equivalently for a single system. However it (particle type) is transformed in a process, then there should be noted that for a vector of 2 n fundamental vari- must be a contribution from cosymplices as for instance ables, 2 n linear independent “samples” are required, if axial vector currents as in the V-A theory of weak inter- action. the matrix S =Σ γ0 is supposed to have a non-vanishing determinant. Later we will come back to this point. This means that if we focus on a closed system, then We can define the Σ-matrix as well using a (normal- (the expectation values of) all cosymplices have to vanish. ized) probability density ρ: The properties of “open” and “closed” have a relative meaning which depends on the context. If we consider two separate “particles” described by state vectors ψ and Σ= ρ(ψ) ψ ψT d2nψ ψ ψT , (32) ≡ h i φ and let them get in contact, then we would for in- Z stance combine the state vectors into a larger state vector with the normalization according to: Ψ = (ψ, φ)T . In App. (C 2) we give the explicit form of general 2 n 2 n-symplices. It is shown that non-diagonal sub-blocks× may per se have arbitrary terms. Hence the 1= ρ(ψ) d2nψ , (33) question, if a (sub-) matrix is a symplex or a cosymplex Z depends on the context. However all quantities - includ- i.e. the density ρ(ψ) is defined as a function of the fun- ing the matrix F - must by some means result from “fun- damental variables or phase space variables. damental variables”, i.e. from state vectors. Also a static We introduced the (“S-matrix”) S = Σγ0 and derived electromagnetic field has its origin in charges and cur- Eq. (21). Let the adjunct vector ψ¯ be defined by rents of “particles” that are described by state vectors. If reality is described by the totality of the fundamental ¯ T ψ ψ γ0 , (34) F ≡ variables, then also the symplex must somehow be gen- erated from ψ. Since the matrix of observables S =Σ γ0 so that S = ψ ψ¯ and the expectation value of an oper- | ih | is a symplex as well, the functional relationship might be ator O: simple: consider (for instance) a Hamiltonian of the form O ψ¯ O ψ . (35) ψT Σ−1 ψ , (39) h i ≡ h i H∝ −1 Now consider that O is a cosymplex, i.e. can be written which implies F γ0 Σ . In this special case a Boltz- mann distribution∝ ρ(ψ) exp( H ) is identical to a as γ0 B with some skew-symmetric matrix B, then the k T multivariate Gaussian density∝ distribution− [51]. In gen- expectation value of O = γ0 B vanishes: eral the functional relationship between F and S might O = ψT γ2B ψ be more involved, but obviously there is a kind of dual- h i h 0 i (36) = ψT B ψ =0 . ity between the matrix of observables S = Σ γ0 and the −h i matrix of generators F = γ0 A that can be illustrated by Hence only symplices represent measurable properties Eq. (39). This duality is well-known in classical mechan- of (closed) systems. Cosymplices C only yield a non- ics. The Hamiltonian is an observable and the generator vanishing result when they appear between different (i.e. of translations in time (evolution in time), the angular linearily independent) states: momentum is an observable and the generator of rota- tions, the momentum is the generator of spatial transla- φ¯C ψ =0 φ = λ ψ , (37) tions [52]. 6 ⇒ 6 10

The time derivative of an expectation value is (assum- where the division by 2 n is introduced to let the unit ing the observable O does not explicitly depend on time): matrix have unit norm. The matrix γ0 hence has a norm of 1, i.e. the norm (42) is not positive definite. − O˙ = ψ¯˙ O ψ + ψ¯ O ψ˙ Two elements of an algebra are said to be orthogo- h i h T i h i = ψ˙ γ0 O ψ + ψ¯ O F ψ nal, if their inner product vanishes. Since we have no h T T i h ¯ i inner product defined (yet), we use an alternative geo- = ψ F γ0 O ψ + ψ O F ψ (40) h T 2 i h ¯ i metric definition which we consider to be equivalent in = ψ γ0 F γ0 O ψ + ψ O F ψ = hψ¯ (O F FO)iψ h i flat spacetime - the Pythagorean theorem. Two vectors ~v h − i and ~w are orthogonal, iff [53]

~v + ~w 2 = ~v 2 + ~w 2 , (43) This general law connects the commutator of an operator k k k k k k O and F with its time derivative. This is the linear form which implies (in conventional vector notation) that the of the Poisson bracket and it is purely classical. One inner product vanishes: conclusion from Eq. 40 in combination with Eq. 29 is that the time evolution is such that (co-) symplices stay (~v + ~w)2 = ~v2 + ~w2 +2 ~v ~w. (44) (co-) symplices. They belong to different domains. · For two elements A and B of the Lie algebra sp(2 n) the use of Eq. 42 then yields

V. FROM LIE ALGEBRA TO SPACETIME 2 1 2 GEOMETRY A + B = 2 n Tr (A + B) k k 1 2 2 = 2 n Tr(A + B + AB + BA) = A 2 + B 2 + 1 Tr(AB + BA) . From the (anti-) commutation properties described by k k k k 2 n Eq. 29 it follows that any fundamental system of (co-) (45) symplices requires that the basic (co-) symplices either As we argued above, all elements of the desired algebra commute or anticommute: Only in this case the product either pairwise commute or anticommute. Two anticom- A B is a pure symplex or cosymplex, i.e. either observable or muting elements and with not. If we consider pure (co-) symplices A and B, then Tr (AB + B A)=0 . (46) (AB)T = BT AT = γ B A γ . (41) ± 0 0 are then orthogonal in the sense that they obey the Pythagorean theorem: It follows that the product of pure (co-) symplices is again a pure (co-) symplex, iff AB = B A. This implies A + B 2 = A 2 + B 2 . (47) that all (anti-)commutators of pure± (co-)symplices are k k k k k k again pure (co-)symplices. Hence - if there are systems Hence the (trace of the) anticommutator is structurally of (anti-)commuting matrices, they are expected to play equal to the inner product 21: a fundamental role. AB + B A In the following we will show how spacetime struc- A B . (49) tures emerge from the algebra of (co-)symplices (Eq. 29), · ≡ 2 i.e. we will explain why and how the spinors ψ - which The above arguments and the fact that γ0 seems to have are composed of fundamental variables - are able to pro- the form of a unit vector suggests to search for other duce the relations between observables that represent the unit vectors γk for k [1 ...N 1], especially other unit ∈ − spacetime structures. In order to do this we have to iden- vectors that anticommute with γ0. For such unit vectors 2 tify algebraic objects that represent (unit) vectors for the we find by the use of γ0 = 1: space and time directions (or - equivalently - the energy − and momentum directions). For this purpose we need a γk γ0 + γ0 γk = 0 2 definition of “length” (a norm) and of orthogonality. So γk γ0 + γ0 γk γ0 = 0 (50) far we only introduced time and a single symplex: γ0. γk = γ0 γk γ0 , Apparently γ0 is the natural choice of the “unit vector” so that in accordance with Eqs. 20 and 27 γk (k = 0) in the “time direction”. 6 Usually vector norms in Euclidean spaces represent the is a symplex, iff it is symmetric. It is a cosymplex, iff length of a vector, i.e. a property that is invariant under rotations and translations. Hence we prefer an invariant quantity for the desired definition of a norm. Besides the 21 Since the anticommutator does not necessarily result a scalar, Hamiltonian the only invariants we introduced so far are one may distinguish between inner product and scalar product. given by Eq. (22). The simplest non-trivial k-norm is The scalar product can then be defined by the trace of the inner given for k = 2, i.e.: product: 1 AB + B A (A · B)S ≡ Tr( ) , (48) 1 2 n 2 A 2 Tr(A2) . (42) where the index “S” indicates the scalar part. k k ≡ 2 n 11 it is skew-symmetric. This fact is remarkable. Since up N There are k possible combinations (without repe- to now we just defined time, it is nearby to interpret γ 0 tition) of k elements  from a set of N generators. Then as the unit vector “in the direction of time”. Then from N Eq. 50 we can take, that all other unit vectors which are there are 2 products of 2 basic matrices (named bivec- perpendicular to γ0 and that correspond to driving terms tors), N products of 3 basic matrices (named trivec- of the Hamiltonian, are symmetric and hence formally 3 different from “time”. In other words - there is only one tors) and  so on. The product of all N basic matrices is observable direction of time. Using the algebraic identity called pseudoscalar. The N anti-commuting generators are the (unit) vectors. The total number of all k-vectors T 2 22 Tr(A A )= aij then is : i,j N X N =2N . (54) - where aij are the elements of the matrix A - it follows, k k that (with an appropriate normalization) symmetric unit X=0   “vectors” A square to +1 and skew-symmetric B square As we have no reason to allow more elements in the alge- to 1: bra than 2N , the vector space sp(2 n) and Cl have − N−1,1 2 T 2 the same dimension: Tr(A ) = Tr(A A )= aij 0 i,j ≥ (51) 2N = (2n)2 , (55) Tr(B2) = Tr(BBT )=P b2 0 . − − ij ≤ i,j which - if fulfilled - is nothing but the condition for com- P Therefore there are two types of measurable unit vectors: pleteness. γ that represents time and squares to 1 and a number If such a basis of anticommuting unit matrices exists, 0 − of symmetric symplices γk (anticommuting with γ0 and we know that with each other) that square to +1. We interpret the 2 2 γk = 1 for all k [0,..., 4n 1] latter (if existent) as the unit vectors in space-like di- ± T ∈ 2 − γk = γk for all k [0,..., 4n 1] rections. Skew-symmetric elements that are orthogonal γ γ = ± γ γ for all j =∈ k, [0,..., −4n2 2] . to γ are cosymplices and have a vanishing expectation k j j k 0 ± 6 ∈ − (56) value. From Eqs. 56 above we readily conclude: In summary: if a (flat) spacetime emerges from dy- T namics, all symplices that represent orthogonal directions γ0 γk γ0 = γk have to anticommute. Therefore the obvious choice of T ± (57) γk γ0 γk = γ0 , the inner product or scalar product of two elements of a ± dynamically generated geometric algebra is the anticom- so that all γk are either symplices or cosymplices and they T mutator. The unit vector in the time-direction squares to are either symplectic (γk γ0 γk = γ0) or “cosymplectic” T 1 and all vector-type symplices γx which are orthogonal (γk γ0 γk = γ0). Since Det(AB) = Det(A)Det(B) and − 2 − to γ are symmetric and square to +1. since γk = 1 it follows that all basic matrices have a 0 ± Given that we can find a (sub-) set of N pairwise determinant of 1. ± orthogonal symplices γk with k [0, 1,...,N 1] in Given such a matrix system exists for some n and N, sp(2 n), then the product of two of∈ them holds: − then it has remarkable symmetry properties. Note that we derived these properties without fixing the size of the γi γj +γj γi γi γj −γj γi γi γj = 2 + 2 matrices. If (as we demand) Eq. (55) holds, then the γi γj −γj γi (52) = 2 , system is a complete basis of the vector space of real 2n 2n-matrices and therefore these matrices M can be for i = j. It is easy to prove that the product γi γj with × 6 written as a linear combination of the γk according to: i = j is orthogonal to both - γi and γj . Hence we identify 2 the6 exterior product with the commutator of A and B: (2n) −1 M = m γ , (58) AB B A k k A B = − . (53) k=0 ∧ 2 X so that with the signature s defined by s Tr(γ2)/(2n) A system of N pairwise anti-commuting real matrices j j j we have ≡ (that square to 1) represents a Clifford algebra Clp,q, N = p + q. p is± the number of unit vectors that square 2n−1 1 M M 1 to +1 and q the number of unit vectors that square to 4n Tr( γj + γj ) = 4n Tr mk (γj γk + γk γj ) k=0 1. As only a single time-direction is possible we can fix   − = sj mj . P q = 1. If (as we presumed) space-time emerges from the (59) dynamics of fundamental variables, then the apparent geometry of spacetime is described by the (representa- tion of) a Clifford algebra Cl . The unit vectors are N−1,1 22 called generators of the Clifford algebra. The case k = 0 can be identified with the unit matrix. 12

Given that γk with k [0,...,N 1] are the gen- are generators of orthogonal transformations, since erating elements of the algebra,∈ then− the vector A = N−1 Rj = 1 cos(ε)+ γj sin (ε) T ak γk squares to a scalar: Rj = 1 cos(ε) γj sin (ε) (64) k=0 T − = Rj R = 1 . P ⇒ j N−1 N−1 2 2 2 2 Hence skew-symmetric symplices are generators of (sym- A = sk a = a a . (60) k k − 0 plectic) rotations. The symplectic transformations gener- k k X=0 X=1 ated by symmetric symplices are essentially boosts. As we Hence the algebra of a dynamical system generates a flat will show in more detail in Sec. VIII A, it is legitimate Minkowski spacetime with proper observables if the cor- to call the structure preserving property of symplectic responding real matrix algebra is a representation of a transformations the principle of (special) relativity since Clifford algebra in which all generating elements are sym- Lorentz transformations of “spinors” ψ in Minkowski plices. The idea to relate Clifford algebras and symplec- spacetime are a subset of the symplectic transformations tic spaces is not new (see for instance Ref. ([54, 55]) and for two DOFs. references therein), however (to the knowledge of the au- The structure of the matrix exponential of a symplex thor) it has not been shown yet that the concept of a geo- F and hence the structure of a symplectic matrix is given metric (Clifford) algebra can be derived from the concept by: of a dynamically emergent spacetime. M(t) = exp(F t)= C + S , (65) where the (co-) symplex S (C) is given by: VI. STRUCTURE PRESERVATION S = sinh (F t) (66) C = cosh(F t) , A. Transfer Matrizes and Symplectic Transformations since (the linear combination of) all odd powers of a sym- plex are symplices and all even powers are cosymplices. The solution of the Hamiltonian EQOM (Eq. 16) for The inverse transfer matrix M−1(t) is then given by: the linearized Hamiltonian (with constant matrix A) are M−1(t)= M( t)= C S . (67) given by the matrix exponential: − − For the matrix of second moments Σ and S = Σ γ0 ψ(t) = exp(F (t t )) ψ(t )= M(t,t ) ψ(t ) . (61) − 0 0 0 0 Eq. (61) and Eq. (62) yield: T T T The matrix M(t,t0) = exp(F (t t0)) is usually called Σ(t) = ψ(t) ψ (t)= M(t,t0) ψ(t0) ψ(t0) M (t,t0) − 2 T transfer matrix and it is well-known to be symplectic if Σ(t) γ0 = M(t,t0) Σ(t0) γ0 M (t,t0) γ0 − −1 F is a symplex (“Hamiltonian”), that is, M fulfills the S(t) = M(t,t0) S(t0) M (t,t0) , following (equivalent) equations [42]: (68) so that the evolution of S in time is a symplectic similarity T M γ0 M = γ0 transformation. Hence all eigenvalues of S are constants. −1 T M = γ0 M γ0 From Eq. 61 it is obvious that the column vectors of −1 T − (62) (M ) = γ0 M γ0 the transfer matrix are the solutions for initial conditions − MT = γ M−1 γ . ψ(0) that are equal to unit vectors. The first column − 0 0 ψ1(t) of M(t,t0) is given by: F Since the matrix exponential of is quite complicated T in general, it is more convenient to consider transforma- ψ1(t)= M(t,t0) (1, 0, 0, 0) . (69) tions generated by basic symplices γj , since all γj square Furthermore we note that if a symplex F is symplectically to 1: ± similar to γ0, then it follows from Eq. (62) Rj = exp(γj ε) F M γ M−1 = MMT γ , (70) 2 0 0 1 cos(ε)+ γj sin (ε) for γj = 1 ∝ = 2 − 1 cosh(ε)+ γj sinh (ε) for γj =1 so that F equals the matrix S = Σ γ0 if the second mo- −1  ments are the average over the orthogonal unit vectors Rj = exp( γj ε) − 2 as exemplified by Eq. (69): 1 cos(ε) γj sin (ε) for γj = 1 = − 2 − 1 cosh(ε) γj sinh (ε) for γj =1 T T  − Σ= MM = ψk(t)ψk (t) . (71) k (63) X The solutions are the sum of the unit matrix and γj with This is another example for the mentioned duality be- (hyperbolic) (co-) sine coefficients. From Eqs. 63 we find tween observables and generators (see Eq. (39) and that skew-symmetric symplices (those that square to 1) Ref. ([52])). − 13

B. Time-Dependent Hamiltonian to Eq. 78 and all symplices change according to Eq. 76. According to classical statistical mechanics, the phase We started out from a Hamiltonian that has no explicit space density ρ(ψ) is a constant, if it is a function of time dependence. Let us see, if we can still use the same the Hamiltonian (or - more generally - of the constants concepts if the driving term in Eq. 15 is time-dependent: of motion) only. In our case we can expect a system in thermodynamical equilibrium, if the density is a function dH ∂H T = + ψ ψ˙ =0 of ψ A ψ, i.e. a function of the bilinear forms that we dt ∂t ∇ H 0 = 1 ψT A˙ ψ + ψT A ψ˙ =0 (72) identified as measurable quantities [43]. In Sec. VIII A we 2 will show that the constants of motion can be interpreted by the mass of the system. We use the Ansatz:

ψ˙ = (γ0 A + G) ψ , (73) C. Cosymplices as Generators of Transformations with a general symplex G and plug it into Eq. 72 The definition of what we consider to be a symplex 0 = 1 ψT A˙ ψ + ψT A (γ A + G) ψ 2 0 or a co-symplex depends on the choice of the symplectic T 1 ˙ (74) 0 = ψ 2 A + A γ0 A + A G ψ unit matrix γ0. Eq. (63) holds also for (skew-) symmetric   cosymplices as it holds for symplices. Consider a skew- T As we know that ψ A γ0 Aψ = 0 due to the skew- symmetric cosymplex γj (which we know to anticommute symmetry of A γ0 A, the remaining matrix has to vanish with γ0), then the matrix exponential is an orthogonal separately, i.e. has to be skew-symmetric. Furthermore similarity transformation, so that the transformed matrix T we have to assume that A˙ = A˙ as the symmetry is γ0 is skew-symmetric and squares to the negative unit logically required in the Hamiltonian for a single system: matrix:

1 ˙ T 1 ˙ T R RT T R T RT 0 = ( 2 A + A G) + 2 A + A G γ˜0 = ( j γ0 j ) = j γ0 j T T 0 = A˙ + G A + A G = Rj γ0 Rj = γ˜0 2 − T 2 − 2 T (79) 0 = A˙ + γ0 G γ0 A + A G γ˜0 = (Rj γ0 Rj ) = Rj γ0 Rj (75) T 0 = γ A˙ G γ A + γ A G = Rj Rj = 1 0 − 0 0 − − F˙ = G F F G − The transformed matrix differs from γ0, but has the same properties. This shows us that - as long as A˙ = A˙ T holds, then If we consider on the other hand a symmetric cosym- remains constant if the time-dependence of F is the plex γ (which we know to commute with γ ), then the H j 0 result of symplectic motion. Since is positive definite, matrix exponential is also symmetric and commutes with H we may write F in the general form γ0 so that

T F BB γ0 T T T T 2 ≡ (Rj γ0 Rj ) = Rj γ0 Rj = Rj γ0 F˙ BB˙ T B B˙ T − (80) = γ0 + γ0 (R γ RT )2 = R4 γ2 = R4 1 ˙ −1 T T T −1 ˙ T j 0 j j 0 j F˙ = BB BB γ0 + BB (B ) B γ0 (76) − F˙ = (BB˙ −1) F F (γ (BT )−1B˙ T γ ) − 0 0 The transformed matrix is still antisymmetric but (in general) it does not square to the (negative) unit matrix. The last lines of Eq. 75 and Eq. 76 are equivalent, if One may conclude that skew-symmetric cosymplices are generators of time-like rotations: They change the T −1 ˙ T G = γ0 (B ) B γ0 “orientation” of γ0, the system is rotated towards a new T −1 ˙ T “time direction”γ ˜ = R γ RT , but remains structurally γ0 G γ0 = (B ) B 0 0 GT = (BT )−1B˙ T (77) similar (though not the same). G = BB˙ −1 The physical meaning of such non-symplectic transfor- mations can be related to a variety of phenomena. In B˙ = GB Ref. ([56]) for instance non-symplectic transformations Hence we again find a spinor-type of linear equation with have been shown to be related to the appearance of vec- a symplex G as driving term: tor potentials and to the transformation into curvilinear coordinate systems. In App. E we give an example which B˙ = GB , (78) is directly related to the structure of a system. Further- more we explicitly give a matrix that transforms between which tells us that symplectic motion can be regarded as the six possible choices for γ0 of the Dirac algebra in a kind of “closed system”, if all spinors change according App. E 4. 14

VII. THE DIMENSIONALITY OF SPACETIME i.e. on the geometric interpretation of the matrix oper- ators that represent the observables. The possible num- A. Clifford Algebras bers of spacetime dimensions N = p + q in this approach is given by the representation theory of real matrices [57]: Sets of anticommuting nonsingular matrices are well- p q =0or2mod8 . (83) known as generators of Clifford algebras. The inner prod- − ucts of the basic elements are the elements of the metric As we have derived above, there is only one single skew- tensor. The vector space constructed in this way is Eu- symmetric (time-like) symplex-generator of the Clifford clidean only for diagonal metric tensors with signatures algebra (i.e. γ0), i.e. we have to set q = 1 and therefore s = s = = sN = 1, i.e. Euclidean unit vectors there are p = N 1 space dimensions so that: 1 2 ··· − square to +1. Therefore generators that square to +1 N 2=0or2mod8 . (84) are spatial or space-like and generators that square to 1 − time-like 23. A Clifford algebra with p space-like and−q Hence the dimensionality of dynamically emergent space- time-like generator matrices is indicated by Clp,q. time geometries are algebraically limited to 1 + 1, 3 + 1, From Eq. 55 it follows that such matrix systems can 9+1, 11+1, 17+1, 19+1, 25+1, 27+1, ... dimensions, only be complete for even N =2 M 24. Then: corresponding to symplices of size 2 2, 4 4, 32 32, 64 64, ... [38, 58]. × × × n =2M−1 (81) Symplices× of size 2 2 have either real or pure imagi- nary eigenvalues and× are therefore not able to represent Any real matrix system that fulfills these requirements the general (i.e. complex) case. But there is another is based on a number of DOFs which is a power of 2. argument, why one degree of freedom is not sufficient. The simplest system hence is the one for which M = 1 A single degree of freedom does not enable to describe (N = 2, n = 1) holds, i.e. it consists of 2 2-matrices. interaction. We can not expect to derive physics based × As mentioned in Sec. IV only those elements γk that on non-interaction. Especially spacetime can only be ex- are symplices have a non-vanishing expectation value plained based on interaction. Furthermore one oscillating ψγ¯ k ψ and are therefore observables. Hence we restrict degree of freedom implies turning points, where the mo- ourselves to Clifford algebras with generators that can be mentum (or kinetic energy) is zero. At these points there represented by symplices. This includes γ0 as the direc- is a moment without change. An hypothetical observer tion of time. Since any generator that anticommutes with could not experimentally decide, whether the direction γ0 is symmetric and therefore squares to +1, we have of time has changed at these points or not. Only in case q = 1, i.e. the Clifford algebra that represents spacetime of two (or more) dimensions there is an additional con- is of the type Clp,1(R). stant of motion - the angular momentum - that (if it is If γ0,γ1,...,γN−1 are the generators of a Clifford al- non-zero) ensures that there is change at all times. In gebra ClN−1,1 with metric case of two dimensions and spin, time reversal is always connected to a non-zero action. Therefore the 2 2 case × γµ γν + γν γµ = 2 Diag( 1, 1,..., 1) , (82) can not be generalized and is rejected as too simple. − Certainly there are several different lines of argumen- then the bivector-elements γ0 γk with k = 1, 2,...N tation, why spacetime has 3 + 1 dimensions. However 1 are symmetric and therefore generators of boosts− the algebraic framework described here has the advan- (“boosters”) while the bivector-elements γj γk with j = k tage that it is under certain constraints maximal abstract and j, k = 0 are the skewsymmetric generators of spatial6 and does not require to postulate and establish an elabo- rotations6 (“rotators” or “gyrators”). We come back to rated physical framework based on which the arguments this in the next subsection. achieve explanatory power. Precisely speaking, our con- siderations are less concerned with “the world” but in- stead with the most simple general interpretation of clus- B. The Number of Space Dimensions ters of variables as objects of “a world”. 1. The eigenvalues of 2n 2n matrices over the reals We have shown that it is possible to construct space- are (for n> 1) pairs of× complex conjugate eigenval- time geometry on the basis of (Hamiltonian) dynamics, ues. Since N is even, all representations for N > 1 have matrix dimensions that are multiples of 4 and can therefore always be block-diagonalized to 4 4- blocks. Therefore higher dimensional cases based× 23 on 4M 4M -matrices can be split into M objects This is only meaningful for real representations, since the case × of complex representations allows to switch the sign of matrices with representation 4 4, i.e. into M objects in × by the multiplication with the unit imaginary. Cl3,1. 24 A Clifford algebra also can be defined for odd N, but these algebras are not isomorphic to a representation in form of 2n×2n 2. The number of variables of the spinor should be real matrices. equal to the dimension of the geometry constructed 15

from it. The Pauli and the Dirac algebra are the positions (or directions), but they have no measurable only cases where the state vector ψ has the same volume. I.e. fundamental (measurable) objects have - dimension as the spacetime constructed from it, i.e. interpreted as objects in spacetime - no volume, they are the only case where 2 n = N so that point-like particles.

2N = N 2 . (85) C. The Real Dirac Matrices This criterium is extremely strong and restricts N to N = 2 or N = 4. We will discuss this point in Sec. IX C. In the previous section we have shown that the case of 4 4-matrices that represent a geometric (Clifford) 3. Since N is even in all possible dynamically gener- algebra× is unique and fundamental. It is the simplest ated spacetime dimensions, the dimension of the and most general algebra that allows for construction of spinor 2 n is always a multiple of 4. Therefore all spacetime. And it is the only algebra that generates a higher-dimensional spinors (and hence spacetime spacetime as we experience it. With respect to the ba- geometries) can be re-interpreted as a number of sic variables ψ it represents systems with 2 DOFs. It interacting 4-spinors. can also be regarded as “fundamental” as it describes the smallest possible oscillatory system with (internal) 4. In 3 space dimensions the number of rotation axes coupling or “interaction” 26. equals the number of spatial directions since one An appropriate choice of the basic matrices of the Clif- can form 3 rotator pairs γj γk (both space-like ford algebra Cl3,1(R) are the well known Dirac matrices. vectors) without repetition from 3 spatial direc- Ettore Majorana was the first who described a system tions. Both - rotators and boosters - mutually anti- of Dirac matrices that contains exclusively imaginary (or commute with each other and hence generate an exclusively real) values [63]. We call these matrices the orthogonal space of the same dimension. In more “real Dirac matrices” (RDMs) and since we are - on the than 3 space dimensions this changes: The number fundamental level - committed to the reals, we will use of boosts stays proportional to the number of spa- the RDMs instead of the conventional system 27. tial dimensions, but the number of (symplectic, i.e. Often the term “Dirac matrices” is used in more re- binary) rotators grows quadratically with the num- strictive way and designates only four matrices, i.e. the ber of spatial directions N 1 since one can select generators of the Clifford algebra 28. The RDMs as we N−1 − = (N 1)(N 2)/2 rotators (see Tab. II). define them here include the complete system of 16 real 2 − − For N 1 = 4 space dimensions this gives 6 rotators matrices and include all elements of the Clifford algebra − and not all of them are orthogonal to each other. In generated by γµ: more than 3 space dimensions some rotators com- mute, for instance in a 4-dimensional space the ro- γ14 = γ0 γ1 γ2 γ3; γ15 = 1 tators γ1γ2 and γ3γ4 commute [59]. This means γ4 = γ0 γ1; γ7 = γ14 γ0 γ1 = γ2 γ3 that (dynamically generated) spacetimes with more γ5 = γ0 γ2; γ8 = γ14 γ0 γ2 = γ3 γ1 than 3 spatial dimensions are topologically not ho- γ6 = γ0 γ3; γ9 = γ14 γ0 γ3 = γ1 γ2 mogeneous. More precisely: D = 3 is the only γ10 = γ14 γ0 = γ1 γ2 γ3 space dimension in which all rotational axis have γ11 = γ14 γ1 = γ0 γ2 γ3 the same relation to all others. γ12 = γ14 γ2 = γ0 γ3 γ1 γ13 = γ14 γ3 = γ0 γ1 γ2 Hence the simplest possible generally useful dimension- 25 ality of “objects” is 3 + 1 . (86) In Sec. IV we raised the question of measurability. We There are 2 n! = 24 possible permutations of the variables emphasized that all unit vectors have to be symplices in in the state vector ψ, but since a swap of the two DOFs order to be measurable. Since all basic elements (unit vectors) anticommute, also the bi-vectors - i.e. the prod- ucts of two unit vectors - are symplices. Geometrically bi-vectors are “oriented faces”. However products of 3 26 The role of the Dirac matrices as generators of the symplec- orthogonal symplices - which would correspond to a vol- tic group Sp(4, R) has been described before, see for instance ume - are cosymplices and therefore have vanishing ex- Ref. ([61]). It has been noted already in Ref. [62]. 27 pectation values (see also App. C 1)). It follows that According to the fundamental theorem of the Dirac matrices by in a dynamically emerging spacetime “objects” can have Pauli, we are free to choose any system of Dirac matrices, since all systems are algebraically similar, i.e. can be mapped onto each other via similarity transformations [64]. The transformation to the conventional system is given in App. E 5. The isomorphism of this matrix group to the 3+2-dimensional de Sitter group has 25 Other authors also considered it worthwhile to analyze aspects been pointed out by Dirac in Ref. [62]. 28 of the dimensionality of space-time from a Lie-algebraic point of Note that we define γ14 = γ0 γ1 γ2 γ3, which is labeled γ5 in view [60]. QED. 16

does not change the form of γ0, there are 12 possible basic reason for the congruence of GA with 3 + 1-dimensional systems of RDMs [48]. Since γ0 is antisymmetric, there Minkowski spacetime. Then the relativistic covariance are only six possible choices for γ0. Each of them al- of the Dirac algebra is the only logical explanation for lows to choose between two different sets of the “spatial” the form of the Lorentz transformations. This implies a matrices 29. The explicit form of the RDMs is given in reversal of common understanding. Neither the experi- Ref. ([48, 49]). The four basic RDMs are anti-commuting mentally found or postulated properties of spacetime nor the postulated constancy of the speed of - yet undefined γµ γν + γν γµ =2 gµν 1 , (87) - light waves are the proper reason for the form of the 30 Lorentz transformations, but the algebraic structure of and the “metric tensor” gµν has the form Hamiltonian dynamics of the fundamental variables. It is a consequence of the physical restriction that constants gµν = Diag( 1, 1, 1, 1) . (88) − of motion are required as references for measurements. Any real 4 4 matrix A can be written as a linear com- Measurable physical properties of physical objects are bination of× RDMs according to those properties that appear or can be related to (func- tions of) constants of motion - in some (other) physical 15 system. Non-measurable (boolean) properties are typi- A = ak γk , (89) cally due to symmetry properties - again determined by k X=0 the properties of the Dirac algebra. We will show this in detail in the next section. where a quarter of the trace Tr(A) equals the scalar com- ponent a15 - since all other RDMs have zero trace. The RDM-coefficients ak are given by the scalar product

2 Tr(γk) A γk + γk A ak = A γk = Tr( ) . (90) · 4 8 VIII. LORENTZ TRANSFORMATIONS ARE SYMPLECTIC TRANSFORMATIONS Any real 4 4 Hamiltonian matrix (symplex) F can be written as a× linear combination in this basis Structure preservation refers to Eq. (7). That is: the 9 transformation of the skew-symmetric matrix S to γ is F = f γ . (91) 0 k k the exact contrary of structure preserving - it is structure k X=0 defining. Only transformations that preserve the form of If Minkowski spacetime is interpreted as a physical real- γ0 are called structure preserving. And since γ0 repre- ity without intrinsic connection to quantum mechanics, sents the form of Hamilton’s equations, structure pre- then it appears to be a jolly contingency that the (real) serving transformations are canonical. Linear structure Dirac matrices have the geometric properties that ex- preserving transformations are called symplectic. actly fit to the needs of spacetime geometry [66, 67]. In However, the form of γ0 - as a result of Eq. 7 - is Ref. [67] the authors write: “Historically, there have been not unique in the Dirac algebra. Any skew-symmetric γ- many attempts to produce the appropriate mathemati- matrix squares to 1 and is equally useable as symplec- cal formalism for modeling the nature of physical space, tic unit matrix and− can equally well represent the time such as Euclid’s geometry, Descartes’ system of Cartesian direction. The use of the other skew-symmetric matri- coordinates, the Argand plane, Hamilton’s quaternions, ces (γ10,γ14,γ7,γ8,γ9) implies a permutation of the vari- Gibbs’ vector system using the dot and cross products. ables in ψ and a permutation of the indices of the other We illustrate however, that Clifford’s geometric algebra matrices. There are six different equally legitimate sys- (GA) provides the most elegant description of physical tems [48]. If we consider interactions between spinors, space.” If spacetime emerges from dynamics then there we have to take the possibility into account that not all is deeper reason for this elegance. Geometric algebra (sub-) systems are described by the same structure defin- (GA) - if introduced by postulating abstract mathemat- ing transformation according to Eq. 7 - though we will ical anticommuting objects - leaves a mark of “artificial not discuss this in detail here. construction” that has few explanatory power. But GA is way more than an elegant way to write down equations: Using Eq. 62 it is easy to show that symplectic trans- if spacetime emerges from dynamics in the form of the formations are structure preserving, i.e. a transforma- Dirac algebra, then we have the missing link, the physical tion of the dynamical variables ψ that does not change the form of the equations, neither it changes the form of γ0. For the matrix of observables S = Σγ0 we expect a similarity transformation. The linear transformation R 29 Given one system is known as γµ then the alternative system is is given as γ0, γ0γ1, γ0γ2, γ0γ3. 30 The convention of QED (gµν = Diag(1, −1, −1, −1)) cannot be represented with real Dirac matrices [57, 65]. ψ˜ = R ψ . (92) 17

Hence which is a Lorentz boost along x. Same transformation, but X = Ex γ4 + Ey γ5 + Ez γ6 + Bx γ7 + By γ8 + Bz γ9: T S = ψψ γ0 ˜ h T i ′ −1 S = ψ˜ψ˜ γ0 X = RXR h iT T ′ = R ψψ R γ0 (93) Ex = Ex h T i T ′ = R ψψ γ0 γ0 R γ0 Bx = Bx − h iT ′ = RS γ0 R γ0 , Ey = Ey cosh(ε)+ Bz sinh (ε) − ′ (98) Ez = Ez cosh(ε) By sinh (ε) ′ − so that the conditions are fulfilled if R is symplectic (ful- By = By cosh(ε) Ez sinh (ε) ′ − fills Eq. 62): Bz = Bz cosh(ε)+ Ey sinh (ε)

T −1 γ0 R γ0 = R − T (94) which is again a Lorentz boost along x, but now it shows R γ0 R = γ0 . the transformation behavior of electromagnetic fields. Then it is easy to show that symplectic transformations Obviously the algebraic structure of the Dirac matrices are structure preserving, i.e. a symplex F transformed in combination with symplectic dynamics results directly by a symplectic similarity transformation R remains a in Lorentz transformations. Indeed the transformations symplex: above are identical to the Lorentz transformations of the Dirac equation31. F˜ = RFR−1 Our approach does not refer to the speed of light, to time dilation or with alike. N.D. Mermin once wrote: F˜T = (R−1)T FT RT ⇒ −1 “Relativity is not a branch of electromagnetism” [69]. = ( γ0 R γ0) (γ0 F γ0) ( γ0 R γ0) (95) − −1 − Our approach goes beyond this statement. We did not = γ0 RFR γ0 ˜ directly introduce spacetime. But we introduced Lorentz = γ0 Fγ0 . transformations in direct combination with the electro- magnetic fields. Lorentz transformations, electrodynam- In combination with Eq. (68) one finds that a transfor- ics, spinors, and the mass-energy relation all together are mation is symplectic if it is possible to find a (constant) facets of the same phenomenon. Space-time is generated Hamiltonian from which the transformation can be de- by transformation operators, which also represent the rived: a transformation is symplectic, if it can be ex- electromagnetic interaction. The conventional approach pressed as the result of a possible evolution in time. This derives LTs from the “constancy of the speed of light”. implies that not all similarity transformations are possi- This is a phenomenological Ansatz which certainly has ble evolutions in time. a legitimization in the history of physics, but is logically As we show in the following, it is legitimate to call misleading: the “constancy” of the speed of light is nei- the structure preserving property of symplectic transfor- ther the cause nor the reason for, but just one aspect mations the principle of (special) relativity since Lorentz of (special) relativity. Relativity as such is nothing but transformations (LTs) of spinors in Minkowski spacetime structure preservation, i.e. it corresponds to the linear are a subset of the possible symplectic transformations canonical transformations of classical mechanics. Singh for two DOFs. It is well-known that the matrix expo- continued the work of Mermin in the (algebraic) direc- nential of a “Hamiltonian matrix” (i.e. a symplex) is a tion [70]. At about the same time Penrose and Rindler symplectic transformation. It remains to be shown which wrote in Ref. ([71]): “The formalism most commonly symplices generate LTs. Consider we transform a matrix used for the mathematical treatment of manifolds and X = γ +Px γ +Py γ +Pz γ using R = exp(γ ε/2), E 0 1 2 3 4 4 their metrics is, of course, the tensor calculus (or such then we obtain after decomposition of the transformed essentially equivalent alternatives as Cartan’s calculus of ′ matrix X into the RDM-coefficients: moving frames). But in the specific case of four dimen-

′ −1 sions and Lorentzian metric there happens to exist - by X = R4 XR4 ′ ′ ′ ′ ′ accident or providence - another formalism which is in X = γ0 + Px γ1 + Py γ2 + Pz γ3 ′ E many ways more appropriate, and that is the formalism = cosh(ε)+ Px sinh (ε) E′ E of 2-spinors.” What Rindler formulates in terms of com- Px = Px cosh(ε)+ sinh (ε) (96) plex 2-spinors, can (more “classically”) be expressed by ′ E Py = Py real 4-spinors as we present it here. In this way it be- ′ Pz = Pz comes more evident that the emergence of spacetime is

With the usual parametrization where ε is the “rapidity” (β = tanh (ε), γ = cosh(ε), βγ = sinh (ε)), one finds 31 Most textbooks on relativistic QM contain inadequately short and/or incomprehensible descriptions of the covariance of the ′ = γ + β γ Px Dirac equation. The best depiction found by the author is given E′ E (97) in Ref. ([68]). P = γ Px + βγ , x E 18 no accident, but the simplest method of pattern recogni- associate the fk (Eq. 91) as follows: tion of abstract and general symplectic dynamics. Fur- thermore, by the use of real 4-spinors we discover that f0 Energy T ≡ E the electromagnetic fields are intrinsically connected to (f1,f2,f3) P~ Momentum T ≡ the emergent spacetime geometry. This is the reason why (f4,f5,f6) E~ electric field (99) T ≡ the speed of light is related to the geometry of spacetime. (f7,f8,f9) B~ magnetic field This is explained in more detail in Sec. VIII A. ≡ The possibility to derive electrodynamics from rela- tivity alone has been questioned [72]. But in the con- This association is not arbitrary as the transformation text of symplectic dynamics as suggested here there are behavior under rotations and Lorentz boosts of the fk additional restrictions and MWEQs can indeed be de- has to fit to the corresponding physical quantities. How- rived. It is an experimental fact that electromagnetic ever it is not unique. Besides the electromagnetic fields, interactions are structure preserving. No electromag- any second rank tensor would fulfill the formal require- netic process known to the author is able to transform ments. If we would not be interested in electrodynamics a fermion into another. There is pair production and but for instance in hydrodynamics or the motion of rigid particle-antiparticle annihilation, but it is conventional bodies, we would have to use a different scheme of in- wisdom to identify the antiparticle with the particle going terpretation. This ambiguity is unavoidable since our “backwards in time” [68]. The fact that Lorentz trans- starting point was the abstract structure of Hamiltonian formations of the Dirac spinor are indeed linear canonical symplectic dynamics 32. transformations, is long known [62, 73]. But it is rarely However our choice is legitimate if the EMEQ is con- mentioned in discussions on relativity - likely because sistent, meaningful and if it helps in structural analysis relativity is almost exclusively depicted as a theory of as we will show in what follows. Dimensional problems space and time. However - as Edward J. Gillis remarked should not appear as long as we can find physical con- - “relativity is not about spacetime” [74]. stants that allow to translate energy, momentum, electric and magnetic fields into frequencies (the dimension of F is a frequency. For the energy the required factor is ~: A. The Electromechanical Equivalence (EMEQ) /~ = frequency E P~ c/~ = frequency We started our considerations with abstract entities. | | (100) e E~ = frequency It is therefore unavoidable to present an interpretation me c e | | of the terms (i.e. the observables) derived from these B~ = frequency me | | entities. The interpretations themselves can neither be “derived” nor be proven to be “correct”. We can only We assume that in our unit system all these units have ~ show that they are consistent and meaningful and that the numerical value of unity, i.e. me = e = = c = 1. With this nomenclature the eigenvalues iω , iω they have explanatory power. Interpretations make sense ± 1 ± 2 or they don’t - but they do not exclude other interpreta- of F are given as: tions. The transformation properties (96) and (98) suggest ω1 = K1 +2 √K2 the introduction of the electromechanical equivalence ω2 = K1 2 √K2 p − F2 2 ~ 2 ~ 2 ~ 2 (EMEQ), as presented in Refs. ([48],[49]). In those pa- K1 = pTr( )/4= + B E P pers the EMEQ was merely used as a formal tool that al- K = Tr(− F4)/16 KE2/4 − − 2 − 1 lows to obtain a descriptive interpretation of symplectic = ( B~ P~ E~ )2 (E~ B~ )2 (P~ B~ )2 E2 2− ×2 − · − · transformations. Here we argue substantially and present Det(F) = ω1 ω2 = K1 4 K2 a derivation of MWEQs. − In Sec. VI A we have shown that the fundamental so- (101) lution of the EQOM is given by a symplectic transfer Bi-vectors form a so-called even subalgebra of the Dirac 33 matrix. In order to produce constants of motion, the algebra . We have shown that any Clifford algebra Clp,q transfer matrix should represent a strongly stable sys- which is able to represent spacetime has an even number tem, i.e. all of its eigenvalues lie on the unit circle in the complex plane. In this case the symplex F has ex- clusively purely imaginary eigenvalues. We analyzed the structure of F by introducing the RDMs which are a rep- 32 And it tells us that major parts of the laws of physics are the result of pattern recognition in the same sense as the interpreta- resentation of the Clifford algebra Cl3,1. Therefore the list of unit symplices includes the four basic elements tion of acoustic signals in terms of sound and music is. Because, that is what we do here: We analyze the patterns of fundamental γ0 ...γ3 and the six bi-vectors. The other 6 members of variables in Hamiltonian motion. the group of the RDMs are cosymplices. The analysis of 33 We define elements to be even, if they do not change sign when the transformation properties indicates a possible phys- all basic elements γ0,...,γ3 change sign. They are odd other- ical interpretation of the coefficients of the algebra: we wise. I.e. bi-vectors are even, vectors and tri-vectors are odd. 19 of generators N. This holds also for the Dirac algebra. Furthermore we note that there are vector-elements Even dimensional algebras have even subalgebras, which associated with and P~ , but no elements associated di- means that even elements generate exclusively even el- rectly with theE spacetime coordinates. For a represen- ements, whereas odd elements can be used to generate tation of a single elementary particle this makes sense the full algebra: Fermions can generate electromagnetic insofar as the dynamics of a single particle may not re- fields, but photons can not generate fermions. Conse- fer to absolute positions (unless via the bi-vector fields). quently there should be (differential-) equations that al- Spacetime coordinates do not refer to particle properties, low to derive the bi-vector fields E~ and B~ from vectors, but to the relativ position (relation) of particles. Hence but not vice versa. This means, that the description of as long as we refer to a single object based on fundamen- a free particle without external fields should not refer to tal variables, we observe the momentum and energy of a bi-vectors. Hence it is natural to distinguish the vector particle as it behaves in an external electromagnetic field. and bi-vector components and for vanishing bi-vectors As we already mentioned in Sec. (IV), the symplex (E~ = B~ = 0) one finds: S =Σ γ0 is singular, if we use only a single real spinor ψ to define 2 2 T K1 = P~ Σ= ψ ψ γ0 , (104) K = 0E − h i 2 (102) where the angles imply some (unspecified) sort of aver- 2 ~ 2 ω1 = ω2 = √K1 = P , S E − age. Since the eigenvalues of the symplex yield the p “frequency” or “mass” m = 2 P~ 2 of the structure which are the relativistic invariants of matter-fields. At which we identify with a particle,E we− need at least N 4 p the end of Sec. IV we argued that a constant measure- linear independent spinors to obtain a massive fermion.≥ ment reference can be constructed as a quadratic form. In the classical Dirac theory, the electron spinor is a linear 2 ~ 2 Here we have an example, since √K1 = m = P : combination of two complex spinors, i.e. also requires 4 The mass is a constant of motion and has theE physical− p real spinors. From this point of view inertial mass might meaning and unit of frequency. This is an important be interpreted statistically - as a property that can only result and a significant step towards quantum mechan- be derived using averaging - either over samples or in ics: It is the combination of the Planck-Einstein rela- time. The idea that mass might be a statistical phe- tion E = ~ ω and Einstein’s E = mc2. It follows that nomenon is neither new nor extraordinarily exotic, see the “time” parameter as we introduced it in Eqs. (12), for instance Ref. ([75, 76]) and references therein. But corresponds to the eigentime or proper time. The eigen- here we do not refer to entropy but rather to dynamical frequency that defines the scale of the eigentime is the or algebraic properties of real spinors. mass and hence a massive particle is described by an inertial frame of reference in combination with an oscil- lator of constant frequency (i.e. a clock). This is the B. The Lorentz Force deeper meaning of Einstein’s clock attached to all iner- tial frames. The 4-momentum of a particle (fermion) is defined by For vanishing vector components ( =0= P~) we have vectors. The vector components of S are associated with E the 4-momentum: 2 2 K1 = B~ E~ P = γ + Px γ + Py γ + Pz γ = γ + P~ ~γ, (105) − 2 (103) 0 1 2 3 0 K2 = (E~ B~ ) , E E − · where is the energy and P~ the momentum. Accordingly E which are the well-known relativistic invariants of the the fields are associated with the bi-vectors of the “force” electromagnetic field. It follows that the frequencies are matrix F. Then Eq. (19) is (in appropriate units) the only real-valued for E~ B~ 0 and B~ 2 E~ 2 0. Fur- Lorentz force equations · ≥ − ≥ thermore, the eigenfrequencies of a pure electromagnetic dP q = P˙ = (F P P F) , (106) wave vanish since K1 = K2 = 0. This implies that we dτ 2 m − can not transform by whatever means into a system of a q where τ is the proper time and is a relative scaling pure e.m.-wave: The ideal electromagnetic wave has no 2m factor. In the lab frame time dt = γ dτ EQ. 106 yields inertial system, no eigenfrequency and no intrinsic phase (setting c = 1): advance. An analysis of the transformations generated dE q ~ ~ by the bi-vectors yields that the elements γ4,γ5 and γ6 dτ = m P E ~ are responsible for Lorentz boosts, while γ7,γ8 and γ9 are dP = q E~ + P~ B~ generators of (spatial) rotations. This corresponds to our dτ m E × dE ~ physical intuition since (charged) particles are accelerated γ dt = q γ ~v E  ~ (107) by electric fields but their “trajectories” are bended (i.e. γ dP = q mγ E~ + m γ ~v B~ dt m × rotated) in magnetic fields. It is therefore no surprise dE ~ dt = q ~v E  that - using the EMEQ - the derivation of the Lorentz ~ dP = q E~ + ~v B~ . force is straightforward (see Sec. VIIIB). dt ×   20

(Note that Eq. 106 allows to add arbitrary multiples of of a Taylor series) by algebraic expressions that respect P to F without an effect on P˙ .) the appropriate transformation properties. According to Eq. (99) the field matrix is given by F IX. ELECTRODYNAMICS FROM = Ex γ4 +Ey γ5 +Ez γ6 +Bx γ7 +By γ8 +Bz γ9 . (110) SYMPLECTIC SPACETIME If we write the fields as a Taylor series, the first term must have the following form in order to be linear in X The electromagnetic terms of the symplex F are bi- and to yield a bi-vector: vectors. It has been pointed out in Sec. VIIIA that bi-vectors form an even subalgebra which implies that F = F0 + ( F) X X( F)+ , (111) they are generated by the product of two vectors. Up to D − D ··· where F is an appropriate derivative taken at X = 0 now only a single type of 4-vector has been introduced, D namely the energy-momentum 4-vector of a single system and must be a vector. Alternative forms of the linear (“particle”) of interest. Using a pair of 4-vectors one can term either include an “axial” vector (cosymplex) V or a construct a bi-vector by the use of the commutator of two pseudoscalar, both having a vanishing expectation value: vectors. However, in order to have a non-zero bivector, V = V γ + V γ + V γ + V γ these two vectors must be different. This could either be t 10 x 11 y 12 z 13 (112) F = (V X + XV)/2 the energy-momentum 4-vector of a second system or a 4- 1 vector representing something different than energy and Since we can exclude the appearance of cosymplices in momentum. Firstly we can argue that a second particle structure preserving interactions, the first order term has did - in contrast to the bi-vectors - not (yet) appear in the form of Eq. 111. our considerations and therefore it is unavoidable to in- In the following we analyze first and second order ex- troduce a new 4-vector. And secondly, the EMEQ intro- pressions that can be constructed with respect to the duces momenta without the corresponding coordinates. transformation properties. We skip constant terms for Hence we write this 4-vector as the moment and analyze the relations of the partial derivatives. Linear terms can exclusively be expressed X = γ t + ~γ ~x, (108) 0 · by the commutator of two vector quantities, i.e. X and another (constant) 4-vector J = ρ0 γ0 + ~j0 ~γ: and express the fields E~ and B~ - which were originally · 4π 1 understood as functions of the eigentime τ - by ~x and t: F = (γ E~ + γ γ B~ ) ~γ = (X J J X) , (113) 1 0 14 0 · 3 2 − E~ (τ) E~ (~x, t) → (109) so that we obtain B~ (τ) B~ (~x, t) . → E~ = 4π (~j t ρ ~x) 3 0 0 (114) This means that we extend the dynamics of (single) ob- ~ 4π −~ B = 3 (~x j0) servables to the dynamics of “fields”, depending on the × new vector type parameters t and ~x. For now we keep and find that these linear terms fulfill MWEQs: open the question, how these vectors are related to ( , P~ ). E ~ E~ = 4πρ But we can’t resist to cite Einstein here: “Spacetime does ∇ · 0 not claim existence on its own, but only as a structural ~ B~ = 0 ∇ · (115) quality of the field” [77]. ~ E~ + ∂tB~ = 0 ∇× There have been several attempts to “derive” MWEQs ~ B~ ∂tE~ = 4π~j0 in the past [78–83]. None of these succeeded in find- ∇× − ing general acceptance. Most textbooks treat just the We continue with a second order term 34: Lorentz covariance of MWEQs. A derivation of MWEQs 1 from the covariance condition alone is considered im- F = (XJXJ JXJX) , (116) 2 4 possible [72]. But our ansatz is based on an algebraic − framework with additional (i.e. symplectic) constraints. and it also fulfills MWEQs: Symplectic constraints are well-known in different con- ~ ~ text and led (for instance) to the non-squeezing theorem, E = 4 πρ1 ∇ · i.e. to the metapher of the “symplectic egg” [84, 85]. ~ B~ = 0 ∇ · (117) We have shown that Lorentz transformations as well as ~ E~ + ∂tB~ = 0 the action of the Lorentz force are symplectic. The ap- ∇× ~ B~ ∂tE~ = 4 π~j1 , pearance of cosymplices as generators is excluded for the ∇× − structure-preserving electromagnetic theory. In order to keep track of the transformation proper- ties with respect to symplectic transformations the bi- 34 In first order the two partials of the induction law vanish sepa- ∇×~ ~ ~ vectors fields can only be expressed (for instance in form rately, i.e E =0= ∂tB. 21 where Now we return to Eq. 111 and find that F must be a vector and therefore has (in “first order”)D the form ~2 2 ~ ρ1 = j0 t 3 tρ0 + 4 (j0 ~x) ρ0 − − ·2 2 (118) ~j1 = 4 (~j0 ~x ρ0 t)~j0 + (ρ ~j ) ~x. 1 → ← · − 0 − 0 F = ∂ F F∂ =4 π J , (126) D 2 − Both orders fulfill the continuity equation:   which is nothing but a definition of the vector current ∂t ρ + ~ ~j =0 . (119) ∇ · J = ργ + j γ + j γ + j γ . (127) So far this is not a derivation of MWEQs, but it shows 0 x 1 y 2 z 3 that the transformation properties of bi-vectors and the Written explicitly in components, Eq. 126 is given by construction of spacetime imply MWEQs. There is a method to express these ideas mathematically by the use ~ E~ = 4 πρ of differential operators. Since the aim is the construction ∇ · (128) ~ B~ ∂tE~ = 4 π~j of spacetime, the appropriate generalized derivative must ∇× − by of vector-type, i.e. we define the covariant derivative We define a vector-potential A = φγ0 +Ax γ1 +Ay γ2 + 35 by Az γ3 which has (to first order) the form :

∂ ∂t γ0 + ∂x γ1 + ∂y γ2 + ∂z γ3 . (120) 1 ≡− A1 = (X F0 F0 X) , (129) The non-abelian nature of matrix multiplication requires 4 − to distinguish differential operators acting to the right where F0 is the constant term of Eq. 111. It is easily → → ← and to the left, i.e. we have ∂ as defined in Eq. 120, ∂ verified that F = 1 ∂ A A ∂ and hence the elec- ← 0 2 0 − 0 and ∂ which is written to the right of the operand (thus tromagnetic (bi-vector-) fields of the general symplex F indicating the order of the matrix multiplication) so that is given by ← F∂ ∂t F γ0 + ∂x F γ1 + ∂y F γ2 + ∂z F γ3 1 → ← → ≡ − (121) F = ∂ A A∂ , (130) ∂ F γ ∂t F + γ ∂x F + γ ∂y F + γ ∂z F . 2 − ≡ − 0 1 2 3   In order to keep the transformation properties of deriva- or explicitly in components: tive expressions transparent, we distinguish between the E~ = ~ φ ∂tA~ commutative −∇ − (131) B~ = ~ A.~ 1 → ← ∇× ∂ A ∂ A A∂ (122) ∧ ≡ 2 − It is well-known that the homogeneous MWEQs are a   direct consequence of Eq. 130: and the anti-commutative → ← →2 ←2 1 → ← ∂ F + F∂ = 1 ∂ A A∂ =0 ∂ A ∂ A + A∂ (123) 2 − · ≡ 2     ~ B~ = 0 (132) ∇ · derivative. Then we find: ~ E~ + ∂tB~ = 0 ∇× → ← 1 ∂ vector vector ∂ bi-vector 2 − ⇒ → ← since the squared operators are scalars and commute with 1 A. Accordingly the continuity equation is a direct con- 2 ∂ vector + vector ∂ scalar = 0 ⇒ sequence of Eq. 126: →  ← 1 2 ∂ bi-vector bi-vector ∂ vector → ← →2 → ← → ← ←2 − ⇒ 1 → ← ∂ J + J∂ = 8π ∂ F ∂ F∂ + ∂ F∂ F∂ 1 − − ∂ bi-vector + bi-vector ∂ axial vector = 0   2 ⇒ ~ ~   = ∂tρ + j =0 . ∇ (133) (124) The validity of the Lorentz gauge follows from Eq. 124 Since (in case of the Dirac algebra) any symplex is the sum of a vector and a bivector, we find (for this special 1 → ← ∂ A + A∂ = ∂t φ + ~ A~ =0 , (134) case): 2 ∇   → ← 1 ∂ symplex + symplex ∂ = ∂ symplex = 0 2 ·   35 The commutative derivative of the trivial first order form A ∝ X (125) vanishes and is therefore useless in this context. 22 and one obtains the wave equation of the vector potential where Q is the “charge”. Especially the relation between A: the density ρ(~x, t) and the “phase space density”ρ ˜(ψ) →2 according to Eq. 32 requires some attention. It is clear 2 ~ 2 (135) 4 π J = ∂ A = (∂t ) A . that the naive assumption − − ∇ In a “current free region” the fields have to fulfill the ρ(~x, t)=˜ρ(ψ(~x, t)) (139) homogeneous wave equation: →2 can not be applied directly. According to Eq. 32 we have ∂ F = (~ 2 ∂2)F =0 . (136) ∇ − t It follows that plane electromagnetic fields “in vacuum” 1= ρ˜(ψ) d4ψ = ρ˜(ψ(~x, t)) √g d3x , (140) 2 ~ 2 hold ω = k so that the fields have no “eigenfrequency” Z Z and no dispersion. where g is the appropriate Gramian determinant. We Note that Eqs. (132) and (133) are of the same form. assume in the following that the normalization has been The continuity equation (133) describes charge as some- adjusted accordingly. thing with the properties of a substance in spacetime. If spacetime would be fundamental, we could be sure In summary: We have shown that Lorentz transforma- that the wave function ψ(~x, t) was well-defined and single tions can indeed be understood as symplectic transforma- valued for any coordinate of Minkowski spacetime. How- tions. The generators of these transformations are elec- ever, if spacetime emerges from dynamics, then space- tromagnetic fields. Since electrodynamics is a structure time coordinates might be functions of the phase space preserving interaction, the description of electromagnetic position t(ψ), ~x(ψ). This includes the possibility that a) fields requires that exclusively structure preserving terms different phase space positions are mapped to the same appear. With these “additional conditions” we were able spacetime-position and b) that space and time coordi- to derive MWEQs if we interpret the bi-vectors of the nates are not unique, i.e. particles appear “instanta- matrix F as functions of vector-type spacetime coordi- neously” at different “locations” of spacetime. Finally nates of Cl (R). The bi-vector fields in F fulfill a wave 3,1 this might indeed be the reason why ψ has to be inter- equation that describes the propagation of electromag- preted as a “probablity density”. netic waves, i.e. of light. One finds that electromagnetic waves “in vacuum” fulfill the relation ω2 = ~k2. From ∂ω the constant group velocity ∂k = 1 we obtain the con- B. The Wave Equations stancy of the speed of light. Therefore we did not derive MWEQs from special relativity, but both theories are the result of the construction of spacetime from symplectic According to Eq. 135 the vector potential fulfills a wave dynamics. equation - and also the (electromagnetic) fields can be de- Above equations are not restricted to 3 space dimen- scribed by waves. Solutions of wave equations are usually sions. However the connections between commutators analyzed with the help of the Fourier transformation, i.e. and anti-commutators are way more complex in 10 or 12 the solutions can be written as superpositions of plane dimensions. We had to consider additional “channels”: waves In 3-dimensional space, the symplex F is composed exclu- ~ φ(~x, t) = φ˜(~k,ω) ei(k·~x−ωt) d3k sively of components of the vector and bi-vector type. In ~ (141) φ˜(~k,ω) = φ(~x, t) e−i(k·~x−ωt) d3x , higher-dimensional space (10- or 12-dimensional space- R F time), the general symplex consists (besides vectors R and bi-vectors) of n-vectors with n [5, 6, 9, 10,... ] (see where we skipped normalization constants for simplicity. App. C 1). Accordingly we had to∈ include higher-order One important feature of the Fourier transform is the ~ terms into the Hamiltonian and likely we would have replacement of nabla operator with the wave vector ~ ∇ trouble with the stability of the described objects. ik and of the time derivative with the frequency ∂t iω. The Fourier transformed MWEQs are algebraic→ conditions− for wave functions (we skip the tilde as it is A. The Density usually clear from the context, if we refer to the fields or their Fourier transform): From the combination of Eq. 109 and Eq. 135 it follows that also the (current-) density has also to be constructed i~k B~ = 0 · as a function of space and time: i~k E~ iωB~ = 0 × − i~k E~ = 4πρ (142) ρ = ρ(~x, t) · ~ ~ (137) i~k B~ + iωE~ = 4π~j j = j(~x, t) . × and it is clear that we have to normalize the density by The second of Eq. 142 implies that all (single) solutions 3 ρ(~x, t) d x = const = Q, (138) also fulfill E~ B~ = 0. The homogeneous parts of Eq. 142 Z · 23

are structurally an exact copy of the expressions that we This means that the constant K2 in Eq. 101 vanishes for obtain from Eq. 101 with the condition K2 = 0: every plane wave solution since all single terms forming K2 vanish separately: B~ P~ E~ = 0 E − × ~ ~ ~ 2 ~ ~ 2 ~ ~ 2 P~ B~ = 0 (143) K2 = ( B P E) (E B) (P B) · E − × − · − · E~ B~ = 0 B~ P~ E~ ω B~ ~k E~ =0 E − × → − × · P~ B~ ~k B~ =0 · → · In Ref. [49] we made use of the condition P~ B~ = 0 E~ B~ = 0 · and E~ B~ = 0 which have to be fulfilled in order to block- · (145) diagonalize· the matrix F. But block-diagonalization It is easy to show that “in vacuum” we also have ~ ~ ~ 2 2 (“decoupling”) did not require the third term B P E E~ B~ = 0 so that also K1 = 0 and there are no to vanish. However, if the corresponding MWEQE − would× eigenfrequencies− of electromagnetic waves. Furthermore not yield zero, then we had to deal with a non-zero they do not generate inertial frames of reference. but divergence-free magnetic current density 4π~jm = As we projected the bivector elements (fields) of the ∂t B~ + ~ E~ - which has not been found to date. There general symplex F into spacetime, we need to do the is another∇× more abstract reason, why we may restrict the same with the phase space variables ψ and thus obtain solutions to the case of K2 = 0: Only with this condition the “wave function” or spinor ψ(~x, t). A normalization being fulfilled the symplex F is a symplectic similarity might be written as transformation of the pure time direction as described by Eq. (70) 36. The third scalar product P~ E~ (and ψ(~x, t)T ψ(~x, t) d3x =1 . (146) · the corresponding field equation ~ E~ ) is in the gen- Z ∇ · eral case non-zero and we emphasize that this is another The phase space density function ρ (and optionally the correspondence to the symplectic decoupling formalism Gramian, see Eq. 140) might be “included” into the (Ref. [49]). It is also remarkable, that the so-called du- spinor function, such that ality rotations 37 of the electromagnetic field do not fit into this approach [48]: Electric and magnetic fields can ψ˜ = √ρ ψ . (147) not be exchanged or mixed. Though both are bi-vector fields (forming a “second rank” tensor), the special role Since the electric current must be a vector, the simplest of the time coordinate breaks the suspected symmetry Ansatz for the current density of a “particle” described between E~ and B~ . Furthermore the only generator that by ψ is 38 might be used for such a rotation is the pseudo-scalar, i.e. ¯ a cosymplex. Insofar the symplectic foundation of elec- ρ = ψγ0 ψ −¯ trodynamics has a higher explanatory power than the jx = ψγ1 ψ ¯ “conventional” formalism. jy = ψγ2 ψ (148) ¯ If we summarize (and extend) the stability conditions jz = ψγ3 ψ K 0 and K 2 √K (Ref. [49]) by the condition 2 ≥ 1 ≥ 2 K2 = 0, then it is appropriate (or even mathematically inevitable) to use this striking structural similarity and The EQOM of ψ(~x, t) must result in a current density (Eq. 148) that fulfills the continuity equation. The sim- to postulate the equivalence of ~k and P~ (ω and ). That E plest possible solution is known to be the Dirac equation. is - we claim that P~ ~k ( ω) and hence that the ∝ E ∝ At first sight Eq. (16) does not appear to be similar to the momentum and energy equal via the Fourier transform Dirac equation. But with the help of the Fourier trans- the spatial and time derivatives, respectively. There is form, the equivalence becomes obvious. The eigenvalues still an optional proportionality factor. But for the same of F for a “particle” in a field-free region are given by reason that we did not refer to the speed of light, we also i 2 P~ 2 = im. Together with Eq. (144) we can skip the proportionality factor ~, since it has no physical ±now flipE − the “account”± of Eq. (16) by the replacement of significance, but depends on the choice of units [88–90]: p operators and eigenvalues (and vice versa):

= ω = i ∂t dψ E ( γ + Px γ + Py γ + Pz γ ) ψ = ψ˙ = P~ = ~k = i ~ (144) E 0 1 2 3 dτ − ∇ (i ∂t γ0 i ∂x γ1 i ∂y γ2 i ∂z γ3) ψ = i m ψ − − − ± (∂t γ0 ∂x γ1 ∂y γ2 ∂z γ3) ψ = m ψ . − − − ± (149)

36 For K2 > 0 there are two different frequencies while for K2 = 0 the eigenfrequencies are identical as in case of γ0. 37 Concerning the electro-magnetic duality rotation see for instance 38 This definition looks quite similar to the definition of the mo- the short theoretical review article of J.A. Mignaco [86] and the mentum by the EMEQ: Mass and charge density are proportional latest experimental results [87]. in case of “point particles”. 24

Apart from “missing” unit imaginary, which is “hidden” Multiple spinors are required to form the representa- in the definition of the Dirac matrices γµ, Eq. 149 is tion of a massive particle, more precisely: A massive ob- the Dirac equation. A unitary matrix which transforms ject is composed of ν 4 linearly independent spinors. between RDMs and the conventional Form of the Dirac This raises questions about≥ the 4-dimensional Γ-space - matrices (˜γµ) is explicitly given in App. E 5. However the “phase space” or “spinor space”, for instance: How - unitary transformations do not change the signs of the do we construct and represent these spinors? In a “clas- 2 metric tensor. If we desire to have γ0 = 1, we still need to sical“ phase space we have two general methods to repre- multiply the transformed matrices by the unit imaginary, sent an ensemble: The first is given by a classical density so that: distribution ρ(ψ) which describes the number of phase space points per unit volume and the second is sampling, ( i ∂t γ˜ + i ∂x γ˜ + i ∂y γ˜ + i ∂z γ˜ ) ψ = m ψ . (150) − 0 1 2 3 ± i.e. the representation of the space space by ν “samples” that could for instance be represented by the columns of C. Lightlike Spinors and massive Multispinors a 4 ν-matrix Ψ. The single spinors are the columns of this× matrix and the Σ-matrix is given by 39

So far we described spinors as a single list of funda- 1 T T Σ= ΨΨ . (153) mental variables of the form ψ = (q1,p1, q2,p2) i.e. as ν a 4-dim. column vector and the matrix of second mo- T ments Σ as Σ = ψ ψ or explicitly Σij = ψi ψj . In this Now consider that the density distribution ρ(ψ) has an case the matrix Σ as well as the matrix S = Σ γ0 have internal symmetry. In the simplest case it might have a a vanishing determinant and the eigenfrequencies (101) point-symmetric of the form: are equally zero. The matrix S is a symplex with the ρ(ψ)= ρ( ψ) (154) RDM-coefficients being second order monomials of the Γ − space coordinate ψ. In order to apply the EMEQ and to distinguish these productions from the “external” forces In this case we might replace ρ by F, we use different names (i.e. lower case letters and 1 instead of [62]: U ρ (ρ(ψ)+ ρ( ψ)) . (155) E → 2 − 1 2 2 2 2 2 (q1 + p1 + q2 + p2) U ∝ 1 2 2 2 2 More generally, if the symmetry properties of the 4- px ( q + p + q p ) ∝ 2 − 1 1 2 − 2 dimensional phase space can be expressed by a matrix py (q q p p ) ∝ 1 2 − 1 2 η, then: pz (q p + q p ) ∝ 1 1 2 2 ex (q q p p ) ∝ 1 2 − 1 2 1 ey ( q1 p2 p1 q2) (151) ρ (ρ(ψ)+ ρ(η ψ)) , (156) ∝ 1− 2 −2 2 2 → 2 ez (q p + q p ) ∝ 2 1 − 1 2 − 2 bx (q1 q2 + p1 p2) or - if there are N symmetries: ∝ 1 2 2 2 2 by (q + p q p ) ∝ 2 1 1 − 2 − 2 1 bz (p2 q1 p1 q2) ρ (ρ(ψ)+ ρ(ηk ψ)) . (157) ∝ − → N k X For dimensional reasons - Eqs. 151 represent actions - we Instead of plugging the symmetry into the density, it stay with proportionality. Evidently Eqs. 151 defines a could also be expressed by the spinor, i.e. by replacing R4 R10 bilinear mapping , i.e. for every single “point” the single spinor ψ by a sampling matrix of spinors: in 4-D Γ space there→ is a 3+1-dimensional structure in- cluding electromagnetic fields. This mapping has the fol- ψ (ψ, η1ψ,...,ηN ψ) . (158) lowing general structural properties with respect to every → single phase space point: In this form, it is still possible to multiply the matrix of second moments by the density without any restriction ~p2 = ~e2 = ~b2 = 2 U to ψ. This is a mixed form that allows to represent the 0 = ~e2 ~b2 − phase space symmetry by the spinor and still write spinor 2 = 1 (~e2 +~b2) U 2 and density as a product. In this case we compute the ~p = ~e ~b matrix of second moments according to U × (152) 3 = ~p (~e ~b) mU 2 2· ×~p2 =0 2n T ∝ U − Σ= dψ ρ(ψ)ΨΨ . (159) ~p ~e = ~e ~b = ~p ~b =0 · · · Z

Since we made no other assumptions about the spinor, 39 1 it follows that single spinors are lightlike, i.e. have mass The root of the normalization factor ν could equally well be zero and the vectors ~e, ~b and ~p form a trihedron. included into the definition of the spinor. 25

The expectation value of an operator O is then expressed two matrices must also be skew-symmetric. In case of by a 2 n = 4-dimensional spinor phase space, we can select some of the 15 traceless RDMs for the Rk. In this case O = dψ2n ρ(ψ) Tr(ΨT Ψ) . (160) one needs to select 3 out of 6 skew-symmetric matrices. h i Z The 3 selected skew-symmetric matrices must pairwise anticommute. There are (up to an orthogonal transfor- The highest degree of symmetry is given, if all single mation and up to a signed permutation) only two sets spinors (column vectors) that form the multispinor Ψ are that fulfill this conditions: pairwise orthogonal. In this case the Σ-matrix is propor- tional to a unit matrix 1 Ψs = 2 (ψ,γ7 ψ,γ8 ψ,γ9 ψ) 1 (167) T 2 2 2 2 Ψc = 2 (ψ,γ0 ψ,γ10 ψ,γ14 ψ) . ΨΨ = (q1 + p1 + q2 + p2) 1 , (161) and describes - according to the EMEQ - a massive par- The two multispinors are then formally orthogonal ma- ticle in its rest frame. If the single spinors of Ψ (i.e. trices, i.e. the columns of Ψ) - are ψk, then such a system can be T 2 2 2 2 constructed from a single spinor ψ by the use of linear Σ = Ψs,c Ψs,c = (q1 + p1 + q2 + p2) 1 2 2 2 2 (168) orthogonal operators Rk according to S = (q1 + p1 + q2 + p2) γ0

ψk = Rk ψ They can also be interpreted as geometric objects, i.e. T ψj ψk = ψj ψk (162) as representations of a “phase space unit cell” and/or · T T = ψ Rj Rk ψ =0 . as a certain symmetry pattern of a phase space distribu- tion 40. However the possibility for such a ψ-independent T The orthogonality then requires that the product Rj Rk definition of an orthogonal multispinor does not work must be skew-symmetric: in arbitrary even dimensional spaces: The next consid- ered Clifford-algebra Cl has a spinor of size 2 n = 32. T T T T 9,1 (Rj Rk) = Rk Rj = Rj Rk Hence we would need 31 orthogonal skew-symmetric and T −T (163) R Rj + R Rk =0 . ⇒ k j pairwise anticommuting matrices. If such a system of matrices would exist, then it would be the basis of a 31- Now it is known, that the γ-matrices are all orthogonal. dimensional Clifford algebra, which is obviously impossi- If we replace Rk by γk, then we have the following con- ble. Therefore in 9+1 and higher dimensions we cannot dition: construct an orthogonal massive multispinor in the de- T T scribed way simply by phase space symmetries. Neither γ γj = γ γk . (164) k − j a Hurwitz- nor a Kustaanheimo-Stiefel transformation (see App. A) exist for 32 or more variables. And there- If both operators γj and γk are symmetric or both skewsymmetric, then the condition is equivalent to the fore the wave function can not be written in a general requirement that they anticommute: way as a product of density and 32-component spinor: Only in 2 and 4 dimensions the spinor space is separable γk γj + γj γk =0 . (165) from the phase space in the way described above. Hence the condition expressed by Eq. 85 excludes all but the 4- If one is symmetric and the other one is skew-symmetric, dimensional phase space to be the simplest possible basis then they have to commute: of “objects” of a physical world. If our arguments are co- gent, then the corresponding space of observables of any γk γj γj γk =0 . (166) − physical world must have 3 + 1 dimensions. What is remarkable here is the fact that orthogonal op- We argued that the multispinor approach forces the phase space density ρ(ψ) to have a specific symmetry. erators which are independent of ψ exist only in even- dimensional vector spaces. In odd dimensional spaces Then we should also have a method to suppress a sym- metry. This is technically difficult with a positive definite such operators are impossible: Though it is always pos- density ρ. If the (multi-) spinors are multiplied with ρ, sible to define a linear orthogonal operator R such that √ then the option to have a negative ρ is mathematically (R x) x = 0, only in even dimensional spaces the ro- √ not easily expressible. It is preferable to write the density tational· operators R can be defined in such a way that as the square of a function φ: they are independent of x, i.e. which rotate any spinor T in such a way that ψ R ψ = 0. One such example is γ0. 2 Without loss of generality we can restrict the first op- ρ = φ (ψ) , (169) erator to equal the identity matrix R0 = 1. Then all Rk, k [1, 2,..., 2 n 1] have to be skew-symmetric ∈ − to make the product ψ ψk = ψ Rk ψ (k = 1, 2, 3) equal T T T 40 to zero. In this case the product ψj ψk = ψ Rj Rk ψ Note that the 4 matrices 1, γ7, γ8,y9 as well as 1, γ0, γ10, γ14 for j = k must vanish and therefore the product of the are representations of the quaternions 1, i, j, k. 6 26 such that it is possible to express asymmetry without the In this context we have to take the effect of certain use of negative densities. If we then write RDMs into account, which are used to model charge con- jugation (γ14), parity conjugation (γ0) and time reversal ˜ 1 φ = 2 (φ(ψ)+ φ(X ψ)) γ10: ˜2 1 2 2 ρ = φ = 4 (φ (ψ)+ φ (X ψ)+2 φ(ψ)φ(X ψ)) . F = γ + P~ ~γ + γ E~ ~γ + γ γ B~ ~γ (170) E 0 · 0 · 14 0 · then the density vanishes, where φ(ψ) = φ(X ψ), i.e. γ0 F γ0 = γ0 P~ ~γ γ0 E~ ~γ + γ14 γ0 B~ ~γ − E − · − · · where φ is purely skew-symmetric with respect− to the γ F γ = γ P~ ~γ + γ E~ ~γ γ γ B~ ~γ 10 10 E 0 − · 0 · − 14 0 · transformation X. γ F γ = γ + P~ ~γ γ E~ ~γ γ γ B~ ~γ 14 14 E 0 · − 0 · − 14 0 · (173) D. Deformation of the Phase Space Unit Cell The expectation values of an operator F using the mul- tispinor Ψc can also be written as Now consider a deformed phase space ellipsoid. We 3 give weights to the phase space lattice as defined by the T 1 T F = Ψc γ0 F Ψc = 4 ψk γ0 F ψk multispinors in Eq. 167 h i k=0 1 T T T = 4 (ψ γ0 F ψ + ψPγ0 γ0 F γ0 ψ T T T T Ψs = (aψ,bγ7 ψ,cγ8 ψ,dγ9 ψ) + ψ γ10 γ0 F γ10 ψ + ψ γ14 γ0 F γ14 ψ) (174) (171) 1 T T Ψc = (aψ,bγ ψ,cγ ψ,dγ ψ) , 0 10 14 = 4 (ψ γ0 F ψ ψ γ0 γ0 F γ0 ψ T − T + ψ γ0 γ10 F γ10 ψ + ψ γ0 γ14 F γ14 ψ) T 1 Then the corresponding symplex ΨcΨ γ0 has the follow- c = 4 F γ0 F γ0 + γ10 F γ10 + γ14 F γ14 ing form: h − i If we compare this to Eq. 173, then the result is clearly T T 2 2 2 2 Ψc,sΨc,sγ0 = ψ ψ Diag(a ,b ,c , d ) γ0 equivalent to a projection resulting in F = . = ψT ψ (a2 + b2 + c2 + d2) h i E 2 2 2 2 E + ( a + b + c d ) Px (172) −2 2 2 −2 + (a b + c d ) Ez F. Duality between Operators and Observables 2 − 2 2 − 2 + (a + b c d ) By) /4 − − It is well-known that observables have a dual role in This corresponds to a decoupled oscillator
matrix as de- both classical as well as in quantum mechanics. Eq. 19 scribed in Ref. [49]. Any symplex with purely imaginary describes the change of the second moments of an en- eigenvalues is symplectically similar to this type of sym- semble of spinors due to the force matrix F. In classi- plex, which represents a deformed phase space lattice. cal point mechanics we would have written “external” Note that the energy is represented by the squared diag- forces, since self-interaction is almost always divergent in onal of the phase space lattice while the mass is repre- theories that assume space-time to be fundamental. In sented by a function of the volume. In the next section our approach the situation is different and in order to we show how the phase space lattice is related to CPT- account for the dynamics of fundamental variables, self- Transformations. interaction should be included. This means that the force matrix F is the sum of self-forces Fs and “external” forces Fx. If no external forces are present, the spinor ensemble E. CPT-Transformations nevertheless “oscillates” with the frequency determined by the eigenvalues of Fs. Since both, the self-force and The use of multispinors imprints a certain phase space the S-matrix of the system are symplices, it is natural symmetry. But this phase space symmetry is correlated to write the functional dependence of the self-force as a with (or can be mapped to) specific “real world” symme- matrix-function of S: tries. F S If we consider the difference between the multispinors s = f( ) . (175) Ψs and Ψc. The first thing we may note is that all single Since only odd powers of a symplex are again a symplex, spinors (columns) of Ψs are constructed by symplectic the Taylor series of f may contain odd powers only. It is transformations. In a sufficiently ergodic system, they obvious that the self-force and the S-matrix will commute could - at least in principle be interpreted as a single so that the S-matrix either depends explicitly on time phase space trajectory at different times, while Ψc is com- (which implies external influences) or it can assumed to posed of (partially) disjunct components. It is one of the be static: remarkable features of the 4-dim. phase space that there are these disjunct areas. A system with the phase space S˙ = Fs S S Fs =0 . (176) position ψ at t = 0 can by no means be moved symplecti- − cally to position γ14 ψ and vice versa. The 4-dim. phase Without “external” influences the matrix of second mo- space is split into two symplectically disjunct regions. ments remains static though the spinor itself oscillates - 27 in accordance we the onto-logic of time. Insofar we have The commutator of the two extra cosymplices is a sym- to take care when interpreting Eq. 106. “Classically” the plex and hence yields a non-zero field value λ, which force matrix F contains only the external forces. But acts only between the two additional time-like dimen- here we use the sum of external and the self-forces. In sions. Considerations like these are the background for the absence of external interactions the oscillation is ex- the claim that time might indeed be “multidimensional” clusively due to self-interaction. - but it remains to be shown that we would be able to notice.

X. ANYTHING ELSE? B. Higher-Dimensional Spaces A. Once More: Cosymplices If spacetime is an emergent phenomenon, then the Much of what has been derived above is a consequence question arises whether other (high-dimensional) space- of the distinction between symplices and cosymplices (or times may not emerge “in parallel”. We have given a “Hamiltonian” and “Skew-Hamiltonian”) matrices and number of algebraic arguments why a 3 + 1 dimensional their expectation values. One central argument was that space-time is a very special case which can not be re- the generators of the considered Clifford algebras have placed easily by other dimensions. Specifically we believe to be symplices since only simplices have non-vanishing to have shown that it is not so much a question of the expectation values. Only symplices are generators of sym- properties of a “real” external world (which is “indepen- plectic transformations and hence only symplices repre- dent” of the observer) which we experience and inves- sent forces with measurable effects. One might argue tigate. We should always keep in mind that our argu- that some entity might exist though its expectation value mentation is founded simply on the possibility in princi- and its obvious consequences remain zero. Maybe it has ple of observation. If our argumentation is sound, then hidden or indirect effects? For instance the algebra of the possibility in principle for the existence of completely 2 n 2 n = 8 8 (co-) symplices represents a Clifford different universes with different dimensionality and dif- × × algebra Cl3,3 that has - compared to the Dirac algebra - ferent laws of nature is much smaller then considered two additional cosymplices as time-like generators. Let elsewhere [91]. the generators ζµ with µ [0,..., 5] (with the real Pauli But let’s ignore the above given arguments for 3 + 1 ∈ matrices ην defined in App. (C 2) be defined by: dimensions for a moment: Indeed the strongest evidence for a 3+1-dimensional spacetime that we have at hand, ζ = η η η is based on light and electrodynamics. Chemistry is al- 0 3 ⊗ 3 ⊗ 0 ζ1 = η3 η3 η1 most exclusively based on electromagnetic interactions of ζ = η ⊗ η ⊗ η electrons: we experience the world through electronic in- 2 3 ⊗ 1 ⊗ 2 (177) ζ3 = η3 η2 η2 teraction. Our world is the world of electrons. What we ζ = η ⊗ η ⊗ η see, is light emitted, reflected (or absorbed) by electrons. 4 1 ⊗ 0 ⊗ 2 ζ5 = η2 η0 η2 . If we touch a solid object, then electrons are “touching” ⊗ ⊗ electrons. We can experimentally investigate weak and The first four elements correspond to (a variant of) strong forces, gravitation and so on - but whatever we the Dirac algebra, and the last two represent “hid- “see” with our own eyes, is to almost any degree of ap- den” dimensions corresponding to additional energy con- proximation based of the electromagnetic interaction of tributions ε1 and ε2, but are cosymplices, if ζ0 is electrons. the symplectic unit matrix. All ζµ are pairwise anti- Assume that a 9+1-dimensional systems exist in par- commuting and the metric tensor is given by gµν = allel by the properties of other (high-dimensional) fields diag( 1, 1, 1, 1, 1, 1). The commutator table of the - then it still remains questionable, if and how observers − − − first four generators then is (as in case of the Dirac ma- would interpret these 9 + 1-dimensional objects. Cer- trices) the electromagnetic field tensor [48]. But since tainly we can not expect that 9 + 1-dimensional entities (according to Eq. 29) the commutator of a symplex and a are just “3 times of the same”, since the algebraic fea- cosymplex is a cosymplex, the mixed terms between sym- tures (for instance of rotations) are different: An essential plices and cosymplices have vanishing expectation values: feature of 3 dimensions is that the spatial rotations do not commute. In more than 3 dimensions we can form 0 Ex Ey Ez 0 0 commuting rotators, say γ1 γ2 and γ3 γ4. Hence in 9 di- Ex 0 Bx By 0 0 − − mensions we sort out 3 sets of 3 non-commuting rotators.  Ey Bx 0 Bz 0 0  [ζµ, ζν ] = − − (178) Consider we use h i  Ez Bx By 0 00   − −   0 0 0 00 λ  1) γ2 γ3, γ3 γ1, γ1 γ2  0 0 0 0 λ 0  2) γ γ , γ γ , γ γ (179)  −  5 6 6 4 4 5   3) γ8 γ9, γ9 γ7, γ7 γ8 Hence the two additional elements are (in average) “de- coupled” from the first four (and all other symplices). We then find that any two rotators from different groups 28

commute 41. On the level of observables we derive from of constants of motion. We have shown that this the sim- Eq. 21 that if two observables commute they are decou- plest mathematical model that allows us to do so is given pled. Presumably observers that are socialized in their by the Hamiltonian formalism. We found that the osten- perception to adapt to a 3+1 dimensional world would sible necessity of the use of complex numbers in quantum not interprete the 9 + 1 dimensional entity correctly. In- mechanics can be translated into the “classical” finding stead of “seeing” a 9 + 1-dimensional object “parallel” to that dynamical variables come as pairs. The Hamiltonian our “electronic” 3+1-dimensional spacetime, an observer ansatz resulted in an algebra of symplices and cosymplies. might see 3 objects located in 3 + 1 dimensions. The Reasonable assumptions as for instance a proper distinc- ontologically motivated idea that our spacetime dimen- tion of measurable and unmeasurable quantities guided sionality must be a fundamental and a unique property us to Lie algebras that are isomorphic to Clifford algebras of the world (and may not depend on the type of the Cl(N 1, 1). The most fundamental of these algebras is interaction), might have guided physics into the wrong the Dirac− algebra represented by real matrices. 42 direction after all . Looking back we can identify pure variables with the Tab. II summarizes the theoretical number of (co-) components of the Dirac spinor, i.e. with the compo- symplices that are to be expected in dynamically gen- nents of the quantum mechanical wavefunction. If we erated spacetimes based on Clifford algebras. Obviously include a density function, then the spinors can be in- all algebraically possible spacetimes beyond 3 + 1 dimen- terpreted as “probability amplitudes”. We argued that sions include high-order symplices, i.e. penta-, hexa- the separation into a multispinor and a pure phase space and deca-vectors. However we would like to add a re- density function only works, if 2N = (2N)2 holds, i.e. mark here: If one would argue that only vectors and if the dimension of the Clifford algebra and spinor alge- bi-vectors have non-negligible effects, then we find that bra are the same. This was the last missing argument to the N-dimensional symplex-algebra ClN−1,1 is composed show that Hamiltonian dynamics has special properties of N vectors, N 1 boosts and (N 1)(N 2)/2 rota- in 4-dimensional phase spaces. tions. With N =2− n this sums up to− an effective− number eff One goal of our gedankenexperiment was to demon- of symplices νs given by strate the emergence of a Minkowski geometry (i.e. spacetime geometry) through symplectic dynamics. eff (N−1)(N−2) N (N+1) νs = N + N 1+ 2 = 2 Spacetime can indeed be a mere interpretation of dynam- 2 n (2 n+1)− (180) = 2 , ics if the Lie algebraic construction of geometry based on symplices is isomorphic to the representation of a Clif- just as if the phase space dimension would be N = 2 n ford algebra. Most (if not all) arguments that we used N and not 2 . Since only the generators are observables, in the derivation are related to specific (algebraic) sym- one might find arguments for an effective dimensional metries. Apparently these symmetries are less obvious reduction even of higher-dimensional dynamical systems. in the conventional form of the Dirac matrices. Further- The effective phase space dimension neff = 2 n is then more the use of complex numbers in the conventional given by [48]: form of the Dirac theory wrongly suggests that we are in a non-classical domain. We have shown that some of eff 1 1 the apparent differences between quantum and classical n = 2 ν + . (181) eff s mechanics are of ontological nature and loose much of r 4 − 2 their significance in the light of the ontology of existence in time. Non-classicality is in this scheme neither con- XI. SUMMARY AND CONCLUSION nected to ~ or the unit imaginary but to the fact that all quantities of the fundamental level vary at all times and Based on the identification of time with change, we in- to the consequences thereof. troduced fundamental variables, which are defined exclu- The idea that the elements of Clifford algebras are re- sively by the property of variation in time. Measurability lated to Minkowski spacetime is well-known and has been requires reference to invariant constant rulers. Since pure described by D. Hestenes and others in various publi- constants are by definition not available on a fundamen- cations on spacetime algebra [66, 93]: “The Dirac ma- tal level they have to emerge from dynamics in the form trices are no more and no less than matrix representa- tions of an orthonormal frame of spacetime vectors and thereby they characterize spacetime geometry”. How- ever (to our knowledge) Hestenes never discussed why 41 Dynamically emerging spacetimes have their own properties, Minkowski spacetime should have these properties. Ac- which are determined by the structure of the corresponding sym- cording to our interpretation Clifford algebras are the plectic Clifford algebra. One can not simply add another dimen- optimal mathematical representation of spacetime be- sion as in arbitrary dimensional Euclidean spaces. Insofar phys- ical spaces are much more restricted than mathematical spaces. cause spacetime emerges by a pattern of special sym- 42 There are however suggestions how to derive particle physics and metry properties which can be expressed by the isomor- even dark matter from a triplet algebra of dimension 212 = 4096 phism of the Hamiltonian dynamical structure to Clif- based on a 64-dimensional phase space [92]. ford algebras. The methods of GA are fascinating, but 29 the formal elegance of GA remains unexplained unless we duced by the isomorphism of certain classical quantities understand that both space-time and electromagnetism (energy, momentum, electric and magnetic field compo- emerge from the structure of the Dirac algebra. It is this nents) with the abstract observables of the 3+1 dimen- connection that can explain why GA is the (only) appro- sional Clifford algebra. The fact that we can derive the priate mathematical tool to represent spacetime [94]. But Lorentz force equations from it, is neither a physical this connection unfolds its full explanatory power only in nor a mathematical “proof” of this interpretation, but combination with fact that the Dirac algebra is also the it is a proof that our interpretation is compatible with algebra of symplectic coupling. This can be shown in this fundamental dynamical law. It appears to be re- the simplest and most obvious way by the use of the real markable to the author that a single 3-dimensional or- Dirac algebra. bit is mathematically isomorphic to the average envelope We also claimed that these insights are arguments for of a 2-dimensional ensemble. In this sense we join the the (apparent) dimensionality of spacetime. This might view of Mark Van Raamsdonk: “Everything around us - appear unacceptable to physicists who are committed to the whole three-dimensional physical world - is an illu- a realistic ontology. However, already the question why sion born from information encoded elsewhere, on a two- space-time should be 3+1 dimensional, implies the possi- dimensional chip” [95]. Most authors that analyzed the bility in principle of a mechanism which could explain the structure of the symplectic group Sp(4, R) interpreted dimensionality of space-time. If this question is accepted this space to be 2+1 dimensional. However - as we ar- as legitimate, then it appears to the author that the only gued above - the space of observables that corresponds logically possible answer must be related to the structure to Sp(4, R) is a 3 + 1 dimensional Minkowski spacetime. of interaction as suggested by Einstein. We have given Mathematically it is evident that this view implies re- a mathematically simple and sound quasi-classical expla- strictions on the 3 + 1 dimensional space - for instance nation. Our derivation of relativity does not require the the “quantization” of the angular momentum. principles of relativity, neither the constancy of the speed But even if we could present a more rigorous deriva- of light nor the principle that the laws of physics must be tion of the (fundamental) relativistic equation of motion the same in all inertial frames. The presented approach of quantum mechanics - the Dirac equation - it would not allows to derive both principles from the conventional automatically imply a “derivation” of quantum mechan- classical concept of canonical transformations. The lat- ics as a whole. Though it is rarely explicitly mentioned, ter is a result of the fact that Lorentz transformations are it is known (though not well-known) that the equations structure preserving (i.e. symplectic) transformations of of motion used in quantum mechanics are (taken as such) the spinor part of the wavefunction. The former is a classical [15]. With respect to this question, our presen- result of our derivation of Maxwell’s equations. 3 cen- tation is not new. What is new (to the knowledge of the tral arguments were given in preparation of the deriva- author) is the algebraic connection between the classi- tion of the Lorentz force and electrodynamics in form cal (co-)symplex algebra with the Clifford algebra of the of MWEQs: The first argument is based on the differ- Dirac matrices. ence between even and odd elements of the algebra, the We started with the classical Hamilton function, and second on the fact that the expectation values of cosym- were guided to interpret the fundamental variables as the plices must vanish in symplectic dynamics for symmetry quantum mechanical wave function in momentum space. reasons and the last one on the transformation proper- Mathematically the alleged antagonism between classical ties. The wave equations follow from MWEQs and the and quantum mechanics seems to evaporate in the light comparison of the Fourier transformed MWEQs with the of what we described above. At least we could show that structure of decoupling then lead us to the identification it is not the mathematical structure of the equations of of momentum (energy) with space (time) derivatives and motion that accounts for the interpretational difficulties hence to the Dirac equation. with quantum mechanics. It is often claimed that the way in which probabilities On the basis of the presented ontology of time we found appear in quantum mechanics is special and unusual in an explanation for why the quantum mechanical wave- some unspecified way. This alienation is often related to function has some features that seem to be mysterious the “complex probability amplitudes”. But if we rein- or “non-classical”: Impossible to be directly measured terpret the formalism of quantum mechanics by its close and without a well-defined dimensional unit and physi- relation to the matrix of second moments, then its form cal meaning. However these are exactly the properties of and postulates are quite familiar. fundamental variables that we derived from the ontology We do not claim that our “derivation” is rigorous in a of time and a proper logic of measurement. Supposing strict mathematical sense, nor that it is complete. And our ontology is “correct”, it is impossible to know what it can not be: our abstract ansatz may only include ab- the wave function is. We think that we gave an expla- stract objects and becomes a physical theory only with nation why this is so: The question has no meaningful an appropriate interpretation. Interpretations can not physical answer - it can not. The consequences of the be proven, they can only be adequate and consistent or presented ontology are certainly difficult to accept: On not. the fundamental level, the world is apparently very dif- Our central “interpretation” is the EMEQ which is in- ferent from our every day experience: loosely speaking it 30 is “flat”. The apparent 3+1 dimensional spacetime is “in transformation [96]: reality” a four-dimensional phase space. However there 1 are interpretations of quantum mechanics (and general Ψhw = 2 (ψ, γ8 ψ,γ9 ψ, γ7 ψ) q− p q − p relativity) which are way more esoteric than this. 1 − 1 − 2 − 2 The picture that we painted is a generalized, ab- 1 p1 q1 p2 q2 (A1) = 2  −  , stracted review of algebraic methods for a low- q2 p2 q1 p1 p q p −q dimensional Γ-phase-space of classical statistical mechan-  2 − 2 1 1  ics based on arbitrary continuously varying fundamental   or (with a flipped sign of the last column) to the so-called variables. This is a generalization and abstraction be- Kustaanheimo-Stiefel (KS-) transformation [97]: cause we did not use a-priori assumptions on the num- ber of space-phase points making up a “something”, a 1 Ψks = (ψ, γ ψ,γ ψ,γ ψ) . (A2) piece of matter, nor did we make any a-priori assump- 2 − 8 9 7 tions about the type of interaction other than arguments based on symmetry considerations. It is the onto-logic of The matrices (167) are also used in the Euler-Rodrigues time that guided us to the idea that observables are mo- formulation of attitudes and represent left- and right- ments of fundamental variables and the phase space co- multiplicative isoclinic mappings [98]. ordinates themselves are not directly accessible. Here we Since the KS-transformation is designed to map the discussed almost exclusively the second moments. How- Kepler- (or Coulomb-) problem to the harmonic oscil- lator, it is not unreasonable to speculate that the ap- ever it might be worth considering the role of 4-th order 1 and higher moments in more detail for both, the static pearance of the r -coulomb-potential of massive (charged) case of “quantum systems” in their eigenstates as well as particles can also be explained in this way. It has been the potential role that odd moments might play in quan- shown elsewhere that non-bijective quadratic transforma- tum jumps. There are strong arguments to assume that tions allow to map the hydrogen atom to the harmonic quantization cannot be finally understood as a linear the- oscillator [99]. ory. It might be enlightening to investigate the physical and algebraic meaning of higher order moments. We only Appendix B: Arbitrary constant Hamiltonians shortly touched this area with Eq. 151 and Eq. 152 with respect to a single phase space points. Following this logic we found strong arguments for Can the Hamiltonian equivalently be of higher order? the dimensionality of energy-momentum-space. Since We presumed that the Hamiltonian is a constant function the energy-momentum space is conjugate to space-time, of the fundamental variables ψ, i.e. a constant of motion, these arguments directly concern the dimensionality of we may ask if our derivation holds also for higher order the latter. Along the way we found an interpretation functions. Given the Hamiltonian has the general form that describes fundamental laws of relativistic electrody- of Eq. 15: namics and quantum mechanics. We described the bridge (ψ) = + εT ψ + 1 ψT A ψ + 1 B ψ ψ ψ + leading to covariance matrices and statistical moments, 0 2 3 ijk i j k H + H1 C ψ ψ ψ ψ + ..., i.e. the bridge to probability theory. Significant theories 4 ijkl i j k l (B1) have been built on less solid grounds. Maybe the above where all coefficients ε , A , B , C are constants. is also a contribution to the question of whether, and in i ij ijk ijkl Then we still assume that not only the matrix A but what sense, quantum mechanics deserves to be regarded also the higher order “tensors” are symmetric in all in- as the final theoretical framework. dices. The term 0 is a constant anyway and does not contribute and againH we choose the zeropoint of ψ such that the linear term vanishes. Then the time derivative ACKNOWLEDGMENTS of this Hamiltonian reads:

R ˙ 1 ˙ ˙ Mathematica has been used for part of the symbolic = 2 (ψ A ψ + ψ A ψ)+ H 1 ˙ ˙ ˙ calculations. Additional software has been written in + 3 (Bijk ψi ψj ψk + Bijk ψi ψj ψk + Bijk ψi ψj ψk)+ “C” and compiled with the GNU c -C++ compilers on + ... different Linux distributions. XFig 3.2.4 has been used (B2) to generate the figures, different versions of LATEXand Due to the symmetry in all indices, this symplifies to: GNU c -emacs for editing and layout. ˙ = (ψk Aki + Bjki ψj ψk + ... ) ψ˙i H (B3) = ( ψ )i ψ˙i ∇ H Appendix A: Hurwitz and Kustaanheimo-Stiefel with the gradient given by Matrices

( ψ )i = (Aik + Bijk ψj + Cijlk ψj ψl + ... ) ψk ∇ H With a minor modification (without influence on ob- γ ( ψ ) = F(ψ) ψ 0 ∇ H servables) the matrix Ψs is proportional to the Hurwitz (B4) 31 where F(ψ) is the product of γ0 with a symmetric matrix k 1 2 3 4 5 6 7 8 9 10 11 12 ˜ A - and therefore a symplex: s 0 2 5 9 14 20 27 35 44 54 65 77 (−1)s + + - - + + - - + + - - A˜ ik = Aik + Bijk ψj + Cijlk ψj ψl + ..., (B5) c 1 4 8 13 19 26 34 43 53 64 76 89 so that the solution still is: (−1)c - + + - - + + - - + + -

ψ˙ = γ ( ψ ) (B6) 0 ∇ H TABLE I. Signs for products of k anti-commuting (co-) sym- We look at the second moments: plices according to Eq. (C3) and (C5). The “plus” signs cor- respond to symplices: Products of 2, 5, 6, 9, 10, ... symplices d T ˙ T ˙ T are again symplices. dτ (ψ ψ ) = ψ ψ + ψ ψ T T T = γ0 ( ψ ) ψ + ψ ( ψ ) γ0 (B7) = F(ψ∇) ψH ψT + ψ ψT F∇T (Hψ) If s and c are even (odd), respectively, then the prod- Multiplication from the right with γ0 and replacing ucts are (co-) symplices. Tab. I lists the resulting signs T F (ψ) with γ0 F(ψ) γ0 again yields: for k =2 ... 10. Clifford algebras with more than 4 gen- erators include penta- and hexavectors as “observables”. d T F T T F 2 dτ (ψ ψ γ0) = (ψ) (ψ ψ γ0) + (ψ ψ γ0) (ψ) γ0 , Furthermore one may conclude from Tab. I: In 9 + 1- (B8) dimensional spacetime, the pseudoscalar is the product so that with γ2 = 1 and S ψψT γ we have the Lax 0 − ≡ 0 of all 10 generators and therefore a symplex. This allows pair: (in principle) for a scalar field. Since spatial rotators are ˙ combinations of two spatial generators, in 9 + 1 dimen- S = F(ψ) S S F(ψ) . (B9) 9 − sional spacetime there are 2 = 36 spatial rotations. A Note however, that the step from Eq. B8 to Eq. B9 is in derivation of the corresponding MWEQs for such higher- the general case only valid for single spinors. dimensional spaces (if possible
or not), lies beyond the scope of this paper, but if all bivector boosts have a cor- responding electric field component and all bivector ro- Appendix C: (Co-) Symplices for higher-dimensional tations a corresponding magnetic field component, then Spacetimes we should expect 9 “electric” and 36 “magnetic” field components in 9 + 1 dimensional spacetime. 1. Which k-Vectors are (Co-) Symplices

Given that we have a set of N pairwise anticommuting 2. The real Pauli matrices and the structure of (co-) symplices Si (Ci), which can be regarded as gen- (co-) symplices for n DOFs erators of real Clifford algebras, then the question if the k-products (i.e. bivectors, trivectors etc.) are symplices For one DOF and an appropriate treatment of the 2 2- or cosymplices, depends on a sign and can be calculated blocks of the general case we introduce the real Pauli× as follows: matrices (RPMs) according to T T T T T (S1 S2 S3 ... Sk) = Sk Sk−1 Sk−2 ... S1 k−1 = ( 1) γ0 Sk Sk−1 Sk−2 ... S1 γ0 0 1 1 0 − (C1) η0 = η1 = 1 0 ! 0 1 ! The number of permutations that is required to reverse − − (C6) 0 1 1 0 the order of k matrices is k (k 1)/2, so that η2 = η3 = 1 = − 1 0 0 1 T s ! ! (S S S ... Sk) = ( 1) γ (S S S ... Sk) γ , 1 2 3 − 0 1 2 3 0 (C2) η is the symplectic unit matrix (i.e. corresponds to γ of with s given by: 0 0 the general case). η0, η1 and η2 are symplices. If we con- k2 + k 2 sider general 2 n 2 n Hamiltonian matrix F composed s = k 1+ k (k 1)/2= − . (C3) of n2 2 2 blocks× A , then γ has the form given in − − 2 ij 0 Eq. (9).× The 2 n 2 n-symplex F that fulfills Eq. 20 has × If we consider k cosymplices Ci instead, we obtain k more the form sign reversals, so that D A A ... A (C C C ... C )T = ( 1)c γ (C C C ... C ) γ , 1 12 13 1n 1 2 3 k 0 1 2 3 k 0 A˜ D A A − (C4)  12 2 23 ... 2n  − ˜ ˜ with c given by: F = A13 A23 D3 ... A3n (C7)  − −   . . .  k2 +3 k 2  . . .  c =2 k 1+ k (k 1)/2= − . (C5)    A˜ n A˜ n A˜ n ... Dn  − − 2  − 1 − 2 − 3    32 where the 2 2 matrices Dk on the diagonal must be so that 2 2-symplices× and can therefore be written as T × A (η0 A η0) = (a12 a21 a11 a22) 1 = Det(A) 1 . k k k − − Dk = d0 η0 + d1 η1 + d2 η2 . (C8) (C19) A matrix A is symplectic, if A A˜ = 1. But we can say If the blocks A above the diagonal have the general ij in any case, that A A˜ is a co-symplex: form 3 A A˜ = (C + S) (C S) − (C20) A = ak ηk , (C9) = C2 S2 + SC CS k=0 − − X then Eq. 20 fixes the form of the corresponding blocks since squares of (co-) symplices as well as the commutator of symplex and cosymplex are cosymplices. below the diagonal A˜ ij to A 2 n 2 n-cosymplex C has according to Eq. 27 the ˜ T form × Aij = η0 Aij η0 − 0 1 2 3 T = η0 (a η0 + a η1 + a η2 + a η3) η0 ij ij ij ij E1 B12 B13 ... B1n − 0 1 2 3 = η0 ( a η0 + a η1 + a η2 + a η3) η0 ˜ − ij ij ij ij  B12 E2 B23 ... B2n  = a0 η + a1 η + a2 η a3 η ˜ ˜ ij 0 ij 1 ij 2 − ij 3 C = B13 B23 E3 ... B3n (C21) (C10)    . . .  From this one can count that each subblock A has two  . . .  ij   antisymmetric and two symmetric coefficients. We count  B˜ B˜ B˜ ... E   1n 2n 3n n  n(n 1)/2 sublocks Aij . Together with n diagonal sub-   − where the 2 2 matrices Ek on the diagonal must be blocks with each having one antisymmetric coefficients, × we should have 2 2-cosymplices and are hence proportional to the unit matrix× νa = n (n 1)+ n = n2 (C11) s − Ek = ek 1 . (C22) antisymmetric symplices. The number of independent a a a antisymmetic parameters is ν = νs + νc = n (2 n 1) = 2 − 2 n n. Therefore we have 3. Which k-Vectors of Cl −1 1 are (Anti-) − N , Symmetric νa = n (2 n 1) n2 = n2 n (C12) c − − − antisymmetric cosymplices. Since we have as many Possible dimensionalities for emergent spacetimes are cosymplices as we have antisymmetric matrix elements given by Eq. 84. Since all but one generator of ClN−1,1 a s νc = ν , the number of symmetic cosymplices νc is: are symmetric, the analysis of how many k-vectors are (anti-) symmetric, is reasonably simple. The total num- s a 2 2 2 νc = νc νc =2 n n (n n)= n (C13) N − − − − ber of k-vectors is given by k , the number µk of k- s The number νs of symmetric symplices is: vectors generated only with spatial elements, hence is N−1 s a 2 2 . For k-vectors that are generated exclusively νs = νs νs =2 n (2 n + 1)/2 n = n + n . (C14) k − − from spatial basis vectors, one finds: A˜ ij is called the symplectic conjugate of Aij . If S and T T T T T C are the symplex and cosymplex-part of a matrix A = (S1 S2 S3 ... Sk) = Sk Sk−1 Sk−2 ... S1 C + S, then the symplectic conjugate is = Sk Sk−1 Sk−2 ... S1 (C23) a A˜ = γ AT γ = C S . (C15) = ( 1) S1 S2 S3 ... Sk , − 0 0 − − As well-known one can quickly derive that where first step is possible as all S are spatial basis vec- tors and hence symmetric. The second step reflects the ˜ ˜ ˜ AB = B A . (C16) number of permutations that are required to reverse the order of k anticommuting elements: If A is written “classically” as a = k(k 1)/2 . (C24) a a − A = 11 12 (C17) a21 a22 ! If one of the symplices equals γ0 (i.e. is anti-symmetric), then we have T then η0 A η0 is given by: T T T T T (S1 S2 S3 ... Sk) = Sk Sk−1 Sk−2 ... S1 T a22 a12 = Sk Sk−1 Sk−2 ... S1 (C25) η0 A η0 = , (C18) − − a+1 a21 a11 ! = ( 1) S1 S2 S3 ... Sk − − 33

N=p+q 4=3+1 10=9+1 12=11+1 scaling factor γ to transform between the eigentime τ 2N 16 1032 4096 and the time of an observer t - which is again a similar- ity to the use of the KST in celestial mechanics, where a n 2 16 32 s a Sundman transformation µk = µk + µk dt k=0 1=1+0 1 =1+0 1=1+0 c = f(q, p) (D2) k=1 4=3+1 10 =9+1 12=11+1 s dτ k=2 6=3+3 45 =9+36 66=11+55 s is used [39, 40]. Despite these remarkable similarities, k=3 4=3+1 120=36+84 220=55+165 c there are also significant differences, since in contrast to k=4 1=0+1 210=126+84 495=330+165 c the KST, we do not increase the number of variables and k=5 - 252=126+126 792=462+330 s we do not transform coordinates to coordinates. The k=6 - 210=126+84 924=462+462 s “spinor” ψ was introduced as two canonical pairs (i.e. 2 k=7 - 120=84+36 792=462+330 c coordinates and the 2 canonical momenta) and is used k=8 - 45=9+36 495=165+330 c in Eq. (D1) to parameterize a 4-momentum vector, while k=9 - 10=1+9 220=55+165 s the KST uses 4 “fictious” coordinates to parameterize 3 cartesian coordinates. k=10 - 1=1+0 66=55+11 s k=11 - - 12=11+1 c k=12 - - 1=0+1 c Appendix E: Non-Symplectic Transformations s 2 νs = n + n 6 272 1056 s a 2 νc = νs = n 4 256 1024 1. Simple LC-Circuit a 2 νc = n − n 2 240 992 2 µs = 2 n + n 10 528 2080 The key concept of symplectic transformations was µa = 2 n2 − n 6 496 2016 mentioned to be the structure preservation. In the follow- ing we exemplify the meaning of structure preservation s a TABLE II. Numbers µk = µk + µk of k-vectors for by giving examples for non-symplectic transformations. N-dimensional dynamically emergent spacetime Clifford- s a Consider a simple LC-circuit as shown in Fig. 1. Start- algebras ClN−1,1. µk (µk) is the number of (anti-) symmetric k-vectors. The 0-vector is identified as the unit matrix which y 2 I I is (of course) symmetric. νx = n + n is the number of (anti-) L C symmmetric (co-) symplices, were y = (a),s denotes (anti-) symmetry and x =(c),s denotes (co-) symplices. LC U FIG. 1. Simple LC- circuit.

Hence, depending on whether a is even or odd, we count µs,a = N−1 symmetric and µa,s = N N−1 ing up naive, we take a Hamiltonian description using k k k k − k skewsymmetric matrices or vice versa. The result for the the total system energy, i.e. we associate the potential       C 2 simplest Clifford algebras is given in Tab. II. energy with the energy stored in the capacitor 2 U and L 2 the kinectic energy with 2 IL: C L C L Appendix D: The Kustaanheimo-Stiefel = U 2 + I2 = U 2 + I2 , (E1) Transformation H 2 2 L 2 2 C

and assume that U is the canonical coordinate and IC is During the “derivation” of the Lorentz force equation the canonical momentum. Then the EQOM are: in Sec. VIIIB, we associated elements of the matrix of second moments (or S =Σ γ ) with energy and momen- ∂H 0 U˙ = = L IC ∂IC tum of a “particle”, i.e. we assumed that: ∂H (E2) I˙C = = CU − ∂U − ¯ 2 2 2 2 = ψγ0ψ ψ0 + ψ1 + ψ2 + ψ3 Quite obviously these equations fail to describe the cor- E − ∝ 2 2 2 2 Px = ψγ¯ ψ ψ + ψ + ψ ψ 1 ∝− 0 1 2 − 3 rect relations for capacitors and inductors, which are: Py = ψγ¯ 2ψ 2 (ψ0 ψ2 ψ1 ψ3) (D1) ∝ − ˙ Pz = ψγ¯ ψ 2 (ψ ψ + ψ ψ ) C U = IC 3 ∝ 0 1 2 3 L I˙L = U = L I˙C (E3) − Eq. (D1) is (up to factor) practically identical to the regularization transformation of Kustaanheimo and We note that the mistake is a wrong scaling of the vari- Stiefel [97], (KST). At the same time we introduced a ables. We recall that the product of the coordinate and 34 the corresponding conjugate momentum should have the its derivative: dimension of an action, while voltage times current re- ˙ sults in a quantity with the dimension of power. Thus φ = F φ we introduce a scaling factor for the current and write: 0 0 1/C1 1/C1 − −  0 0 01/C2  (E8) C L = φ . = U 2 + P 2 , (E4) 1/L1 0 0 0 2   H 2 2 a  1/L 1/L 0 0   2 − 2    so that the canonical momentum is now P = a IC and The matrix F represents the structural properties of the the EQOM are: LC-circuit. The matrix is not a symplex and can not be transformed into a symplex by any symplectic trans- ˙ ∂H L L U = ∂P = a2 P = a IC formation. This does not mean that the dynamics of the ˙ ∂H P = ∂U = CU (E5) system can not be derived from a Hamiltonian - it simply ˙ − C − IC = a U means that the transformation matrix which is required − to map the system to a Hamiltonian system, is not struc- The comparison with Eq. E3 then yields a = L C. But ture preserving but structure defining. note that the scaling changes the product pi qi and hence In case of two coupled LC-circuits, it is likewise not is not a symplectic (structure preserving), but a non- sufficient to use scaling factors to obtain the canonical symplectic (structure defining) transformation. momenta and even if it was, the eigen- frequencies of the circuit can not be guessed anymore. The square of the matrix from Eq. E8 is given by: 2. Coupled LC-Circuits L1+L2 1 0 0 − C1 L1 L2 C1 L2 1 1 0 0 If we restrict ourselves to the simplest case (i.e. two ¨  C2 L2 C2 L2  φ = − 1 1 φ . capacitors, two inductors), the structure of the only non- 0 0 C1 L1 C1 L1  − −  trivial way to couple two LC-circuits is shown if Fig. 2.  0 0 1 C1+C2   − C1 L2 − C1 C2 L2  The equations of motion can be derived directly from   (E9) Obviously the state vector φ is not composed of canon- I1 I ical variables. The Lagrangian L is given by I2 3 U L C C 1 L1 2 L2 2 C1 2 C2 2 1 1 2 = I1 + I3 U U1 . (E10) U L 2 2 − 2 − 2 Eq. E8 can be used to replace the currents by the deriva- L U2 2 tives of the voltages:

I1 = C1 U˙ C2 U˙ 1 FIG. 2. Two coupled LC-circuits. − − I3 = C2 U˙ 1 = L1 (C U˙ + C U˙ )2 + L2 C2 U˙ 2 C1 U 2 C2 U 2 , ˙ 2 1 2 1 2 2 1 2 2 1 the drawing using the general relations UC = IC /C for L − − (E11) an ideal capacitor and I˙L = UL/L for the ideal inductor so that for the coordinates q1 = U and q2 = U1, the and Kirchhoff’s rule: canonical momenta are: ˙ ∂L I1 = U/L1 pi = ∂q˙i ˙ ∂L U = (I1 + I3)/C1 p = = L C (C U˙ + C U˙ ) − 1 ∂U˙ 1 1 1 2 1 I˙ = (U U )/L 3 − 1 2 (E6) = L1 C1 I1 (E12) U˙ = I /C −∂L 2 1 3 2 p2 = = L1 C2 (C1 U˙ + C2 U˙ 1)+ L2 C U˙ 1 ∂U˙ 1 2 0 = I1 + I2 + I3 = L1 C2 I1 + L2 C2 I3 U = U1 + U2 − so that the transformation matrix from the state vector The energy sum again yields a Hamiltonian of the diag- φ to the canonical variables ψ is given by: onal form: ψ = T φ C1 2 L1 2 C2 2 L2 2 q 10 0 0 U = U + I1 + U1 + I3 . (E7) 1 H 2 2 2 2 q 01 0 0 U (E13)  2  =    1  We use the voltages and currents of the Hamiltonian to p1 0 0 L1 C1 0 I1    −    define the state vector φ = (U,U1, I1, I3) and we find for  p   0 0 L C L C   I   2   − 1 2 2 2   3        35

The transformed matrix F˜ = T F T−1 then is: Q can be used to transform S into all possible 6 already skew-symmetric matrices used to build the two types of L1+L2 1 0 0 2 C1 L1 L2 − C1 C2 L2 spinors in Eq. 167. And it concerns the role that the 1 1  0 0 2  “axial 4-vector”-components play in the transformations F˜ C1 C2 L2 C2 L2 = − (E14) between these two spinor-types (or types of phase space  C1 0 0 0   −  symmetry).  0 C2 0 0   −    The Hamiltonian can then be expressed in the canonical 4. Nonsymplectic Transformations within the coordinates as: Dirac Algebra = 1 ψT A ψ H 2 C1 00 0 In the following we present an orthogonal (non- symplectic) transformation that enables to “rotate the  0 C2 0 0  (E15) A = L1+L2 1 , time direction”, i.e. to transform from γ0 (as we used 0 0 2  C1 L1 L2 − C1 C2 L2  it here) to other representations of γ0. We define the 1 1  0 0 2   C1 C2 L2 C L2  (normalized) Hadamard-matrix H4 according to  − 2    ˜ γ2−γ1−y4+γ5 (γ15+γ0)(γ2−γ1) where A = γ0 F. As was shown in Ref. [49], the Hamilto- H4 = 2 = 2 nian formulation (for stable oscillating systems) is always 11 1 1 “similar” to the case of completely decoupled oscillators, 1 1 1 1 = 1  − −  i.e. the non-symplectic transformation maps the struc- 2 1 1 1 1 ture of Fig. 2 to two seperate systems as shown in Fig. 1.  − −  1 1 1 1 (E17) The transformation T is not symplectic and may serve   T − −  as an example for a structure defining transformation in H4 = H4  2 contrast to symplectic structure preserving transforma- H4 = 1 tions. γ0 H4 γ0 = H4 H γ H = γ 4 0 4 − 0 3. The Direction of Time H4 is a symplex and it is antisymplectic. Furthermore we define a “shifter” matrix X according to If the matrix Q used in Eq. 7 is not only non-singular, but orthogonal, then 0001 γ0 + γ2 + γ6 + γ7 1000 QT SQ X = − =   (E18) = diag(λ0 η0, λ1 η0, λ2 η0,..., 0, 0, 0) , (E16) 2 0100   with some real coefficients λ [100]. In this case we would  0010  k   restrict to λk = 1 by the argument that the dynamics of   the system has to be defined by the Hamilton function Some properties of the X-matrix are and not by the (otherwise arbitrary) skew-symmetric ma- T T 1 S γ0 Xγ0 = X X X = trix . However this is true only for closed systems, i.e. T T systems in which the matrix A is a function (of the con- X γ0 X = γ7 X γ7 X = γ0 T − T − (E19) stants of motion) of the variables ψ only. Since we inter- X γ8 X = γ14 X γ14 X = γ8 T T − preted γ0 as the (unit vector in the) direction of time, it X γ9 X = γ10 X γ10 X = γ9 . appears to be wise to return to the possible algebraic and − physical implications of this choice for the parameters λk. Note that X is a symplex and hence could be used as a In Ref. [49] we have demonstrated that a symplectic de- force matrix. In this case we have a 4-th order differential coupling analysis of any oscillatory symplex F leads to a equation of the form: normal form exactly of the form on the right-hand side ′ of Eq. E16. The only difference is that the decoupling ψ = X ψ ′′′′ (E20) transformation is symplectic, while the transformation ψ = ψ ,

Eq. E16 is obviously not symplectic - and not even gen- 4 erally orthogonal. We said that the system evolution since one finds X = 1. The product of these two matri- in time is symplectic. However the fact that the only re- ces R6 = H4 X is an orthogonal matrix that transforms striction for the time-matrix S is its skew-symmetry calls cyclic through all possible basis systems: for a deeper analysis of the possible physical implications 1 1 11 of the non-symplectic transformation by Q. Such a pro- 1 1 1 1 1 gram extends the scope of this paper. It may concern R6 = H4 X =  − −  . (E21) the number, type, and possible interactions and trans- 2 1 1 1 1  − −  formations between the fermions. Recall that the matrix  1 1 1 1   − −    36

The most relevant properties are then it is quickly verified that

T T T R R6 = R6 R = 1 R γ0 R6 = γ7 6 6 6 † T T γ˜µ = i U γµ U , (E27) R6 γ10 R6 = γ8 R6 γ14 R6 = γ9 T − T R6 γ7 R6 = γ14 R6 γ8 R6 = γ0 T − R6 γ9 R6 = γ10 , where µ [0 ... 3] andγ ˜µ are the conventional Dirac (E22) matrices footnoteThe∈ explicit form of the real Dirac ma- so that R6 can be iteratively used to switch through all trices is given for instance in Refs. ([48, 49]). Usingγ ˜µ, possible systems: the other matrices of the Clifford algebra are quickly con- structed. However, since we multiplied by the unit imag- T R6 γ0 R6 = γ7 inary, it is clear that the conventional Dirac algebra is a T 2 2 (R6 ) γ0 (R6) = γ14 rep of Cl1,3. T 3 3 − Appendix F: Graphical Representation of Dirac (R6 ) γ0 (R6) = γ9 T 4 4 − (E23) Matrices (R6 ) γ0 (R6) = γ10 T 5 5 − (R6 ) γ0 (R6) = γ8 T 6 6 Fig. 3 illustrates the geometric interpretation of the (R6 ) γ0 (R6) = γ0 RDMs. 6 Finally one finds that (R6) = 1. But R6 is neither sym- plectic nor is it a symplex. Expressed by the γ-matrices, 4 Vertices (0−Faces): 4 Faces (2−Faces): R6 is given by 6 Edges (1−Faces): 0 0 4 R6 = 1 + γ0 + γ1 γ2 + γ3 γ4 + γ5 + γ6 − − 5 + γ + γ γ γ γ γ γ γ . 4 11 7 8 − 9 − 10 − 11 − 12 − 13 − 14 13 (E24) 6 Another matrix with equivalent properties is given by 9 2 12 2 1 1 10 ˜ T 7 R6 = H4 X . (E25) 8 3 These transformations exemplifies part of the claim (i.e. 3 the use of a specific (though arbitrary) form for γ0) in 1 Volume (3−Face) Eq. (7). FIG. 3. The group structure of the RDMs (i.e. the Clifford algebra Cl3,1) can be represented by a tetrahedron: The ver- tices represent the “basic” symplices γ0,...,γ3, the edges the 5. Transformation to the Conventional Dirac bi-vectors γ4,...,y9, the surfaces the components of the ax- Algebra ial vector γ10,...,γ13 and the volume the pseudoscalar γ14. Another graphical representation has been given by Good- According to the fundamental theorem of the Dirac manson [102]. matrices [101], any set of Dirac matrices is similar to any other set up to a unitary transformation. However a change in the sign of the metric tensor requires in addi- tion a multiplication with the unit imaginary. Let U be the following unitary matrix

1 i i 1 − REFERENCES 1 i 1 1 i U =  − − −  (E26) 2 i 1 1 i  − − −   1 i i 1   − − −   

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