1

ON THE ORIGIN OF

C. Darwin Hepburn

Within the context of empirically proven special- relativistic classical , a logical necessity for is demonstrated: Quantum mechanics is the of self-frame effects.

21 June 2001

[email protected] 2

ON THE ORIGIN OF QUANTUM MECHANICS

C. Darwin Hepburn

I INTRODUCTION ...... 4 II PREMISES OF QUANTUM MECHANICS...... 4 II.A Contrast...... 4 II.B Completeness ...... 5 II.C Four-Dimensional Spacetime...... 6 II.D Linearity Of Infinitesimal Functionality In Spacetime...... 7 II.E Symmetry...... 8 II.F Linearity Of Inertial-Frame Spacetime Transformations...... 8 II.G Covariance With Respect To Inertial-Frame Spacetime Transformations ...... 9 II.H Invariance Of The Speed Of ...... 9 II.I Boundary Conditions...... 10 II.J Physical Measurables Represented As Real Numbers ...... 11 II.K Source ...... 11 II.L Second Law Of Thermodynamics ...... 11 II.M Integrity Of Particles...... 12 II.N Rest Mass ...... 12 II.O General Deductive Logic In The Form Of Mathematics ...... 12 III THE GAMMA-DANCE ...... 12 III.A Massless Particle Gamma-Dance...... 13 III.B Massive Particle Gamma-Dance ...... 16 III.C Secondary Gamma-Dances ...... 18 III.D Antimatter Implication of the Gamma-Dance ...... 19 IVDERIVATION OF PLANCK’S CONSTANT...... 21 VHEISENBERG ...... 22 VIWAVE-PARTICLE DUALITY...... 24 VI.AThe Of A Particle...... 24 VI.BIntegrity Of Particle At Source And Detection Points...... 25 VI.CDouble Slit Example...... 26 VI.DPhoton Emission From Energy Transitions In Hydrogen...... 27 VIIAMPLITUDE, PROBABILITY, AND MEASURABLE QUANTITY...... 28 VIIIWAVEFUNCTION COLLAPSE...... 28 IXBELL’S THEOREM, ENTANGLEMENTS, AND THE GAMMA-DANCE...... 31 XEINSTEIN-BOHR DIALOGUES...... 31 3

XIOTHER QUANTUM PROPERTIES...... 32 XIIPOSTDICTION AND PREDICTION...... 33 XIIIINADEQUACIES OF THE FORMULATION...... 34 XIVSUMMARY...... 34 XVACKNOWLEDGEMENTS...... 35 4

ON THE ORIGIN OF QUANTUM MECHANICS

C. Darwin Hepburn

The universe is as simple as it can be, and still be.1

Schwinger’s Maxim: “Keep your theories as close to experiment as possible.” 2

Feynman’s Trick: “What I cannot create, I do not understand.” 3

Within the context of empirically proven special- relativistic classical physics, a logical necessity for quantum mechanics is demonstrated: Quantum mechanics is the special relativity of self-frame effects.

I INTRODUCTION

It is known that the Feynman path integral method4, 5 serves as a general basis for both classical and quantum mechanics6. Inherent in the method is the direct postulate of the path integral7, with a along each path. It has been found that this path integral form, including the phase, can be derived from more primitive postulates. The derivation traces direct lineage from the reasoning of Huygens8, Einstein9, Dirac10, and Feynman11, and provides a deeper physical picture of the origin of quantum mechanics, as a necessary concomitant to any classical system that has measurable special relativistic spacetime.

II PREMISES OF QUANTUM MECHANICS

Deduction from the following set of primitive postulates will be shown to produce all of the Feynman Path Integral structure (and therefore, in consequence, quantum theory12):

(A) Contrast (B) Completeness (C) Four-dimensional spacetime (D) Linearity of infinitesimal functionality in spacetime (E) Symmetry (F) Linearity of inertial-frame spacetime transformations (G) Covariance with respect to inertial-frame spacetime transformations (H) Invariance of (I) Boundary conditions (J) Physical measurables represented as real numbers (K) Source (L) Second Law of Thermodynamics (M) Integrity of particles (N) Rest mass (O) General deductive logic in the form of mathematics

Each premise will be discussed below, and incorporated into the theory in subsequent sections.

II.A Contrast

Contrast, as perceptibility of structure, is required for any system of thought or for physical measurability, and therefore for any mathematics or physical theory. Absence of contrast is synonymous with physical indistinguishability or conceptual meaninglessness13. Contrast is the ancient yin-yang14, the dits and dahs of Morse code, the ones and zeros of the digital computer. Contrast is Sheffer’s refinement15 5 of the Russell-Whitehead postulates for mathematics in Principia Mathematica16, formulated in the notation qp| , meaning “q is distinguishable from p ”. And, contrast is embodied in Dirac’s bra and ket notation ba : If there is no distinction between states b and a , then the braket17 ba is unity; to the extent of contrast between b and a , ba approaches zero, diminishing to zero identically if b has complete contrast from (is independent of) a . Contrast will be used here in Dirac’s notation.

II.B Completeness

Completeness is the inclusion of all elements of a system. In physical theory it generally takes the form of an expansion for the identity , over projections onto all possible states of the system18. In Dirac notation, therefore, completeness becomes: (II-1) 1 = ∑ µµ µ where the sum is over all states µ of the system. Using the Einstein summation convention19 (as will be used throughout unless otherwise stated), this becomes: (II-2) 1 = µµ≡Ξ.

The completeness operator Ξ , applied to a general state Ψ , serves as a projection operator20 of Ψ onto the states µ : (II-3) Ψ = µµΨ .

If Ξ is applied to itself, however, instead of to some Ψ , then Ξ links every element in the complete set of states to every other element in that complete set. There is a common operation to accomplish precisely this sort of linkage: To link every element (point) of the set of points between 0 and x to every element of another copy of the same set, we simply perform a standard integration of x over dx from 0 to x : (II-4) x x 2 xdx = . ∫ 2 0

The division of x 2 by 2 , while usually viewed as providing the area under the line fx()= x instead of the area of the x by x square, can also be viewed as removing redundancies in a set-to-set linkage. That x 2 is, 2 provides the antisymmetrical product of xx⊗ plus half of the points along the diagonal (the line fx()= x). A point along the diagonal corresponds to the µ th element interacting with the µ th element, so redundancies are removed by retaining only half of them. An extension of this interpretation applies to the handling of redundancies in integrations over x 2 , x 3 …, and introduces a variant type of integration for such higher powers of x : To acquire all cancellations of redundancies in multiple linkages, integration will be performed successively for each new factor of x . That is, the integration over x 2 , instead of being x x 3 xdx2 = , is accomplished by ∫ 3 0 (II-5) xx′′ ′ x ′′ xxx′′′′′233 xdxdx′′=== dx , ∫∫ ∫ 2233!⋅ 00 0 and similarly for higher powers of x . This variant of standard integration can be called successive linkage integration, or simply linkage integration. In supplanting x with µµ, and integrating from 0 to 6

µµ, we abstract µµ to be an independent variable internally, amounting to successive inclusion, one-by-one, of each spacetime point into the sum. Only at the end is µµ accorded its other properties, as a projection onto a complete set of points. The standard operation of integration, performed successively for each power of the completeness operator, therefore, automatically provides us with exactly all of the linkage properties we need. Accordingly, and here restricting the Einstein summation convention to apply only within a given ketbra, successive applications of the completeness operator are given by: (II-6) µµ ()µµ2 µµνν≡ µµd () µµ = ∫ 2 0 µµ ()µµ233() µµ() µµ µµννρρ≡ d ()µµ == ∫ 2233!⋅ 0 !

µµ n −1 ()µµ ()µµn µµννρρ""ntimes λλ≡ d ()µµ = ∫ ()nn− 1! ! 0

Thus, the general form for n successive applications of the completeness operator, Ξn , is (II-7) ()µµ n Ξ≡n µµννρρ""ntimes λλ= . n !

Because the completeness operator Ξ1 = µµ is unity (a constant), it is also equal to any other expression with the value unity. If applied to a system where the number of states approaches infinity, the summation over all states can be taken to approach infinity, and the summation becomes an integral. If the states are points in spacetime (the summation goes to an integral whether the region in continuous spacetime is finite or infinite), that integral becomes an integral over spacetime: (II-8) 1()= Ξ1 ==µµ ∫ η xdV.

The weighting factor η()x apportions the relative significance of the included spacetime points, and by definition serves to normalize the volume integral to unity. If η were a finite non-zero constant rather than a function of spacetime coordinates, then its volume integral (over infinite spacetime) necessarily would be infinite, since the volume integral ∫ dV over infinite spacetime is infinite. Since the integral equals unity, then, if spacetime is infinite, η must be a function η()x of the coordinates. If spacetime is finite (meaning that the volume integral ∫ dV is finite), then η can be either constant or a function of the coordinates. To preclude a superfluous declaration of the extent of spacetime, therefore, η is used as η()x , a function of coordinates.

II.C Four-Dimensional Spacetime

21 Four-dimensional spacetime is characterized via the metric gµν , which appears in the formula for the spacetime interval22 (the four-dimensional spacetime generalization of the Pythagorean Theorem): (II-9) 2 µ ν ds= gµν dx dx . 7

In flat spacetime and Cartesian coordinates (Minkowski space), and using standard sign conventions for space and time (the signs on space and time must be opposite, but can be interchanged without physical effect), (II-10) 10 0 0   0100−    gµν =   , 00− 10     00 0− 1 so that the spacetime interval is (II-11) ds22222= dt−−− dx dy dz .

The associated invariant differential volume element23, 24 is (II-12)

dV= gµν dt dxdydz .

where ggµν ≡ is the determinant of gµν ; the Minkowski space value for g is −1 . Because in any physical space, flat or curved, and using any consistent coordinates, space and time have opposite signs in the metric, the general natural value of (II-13) dV= g dt dxdydz is purely imaginary. To derive quantum mechanics from primitive postulates, it is simplest to leave the spacetime volume element in this natural imaginary form, recognizing that spacetime 4-volume is, in general, purely imaginary. It has been customary25, 26, 27, 28, 29 to multiply g by –1 to force the volume element to be purely real, but such a multiplication is an arbitrary or ad hoc insertion, and its inclusion would mandate an additional primitive premise. The inclusion of this minus sign, while having no physical consequences in a wide class of calculations (classical relativity), innocuously has served as one of the historical obstacles to understanding the relation of quantum mechanics to the rest of physical theory. It will be seen below that retaining the naturally imaginary volume helps in understanding the origin of path phase, and hence the origin of quantum phenomena. In Minkowski space the 4-volume element is, therefore, with gi= −1 = , and dtdx dydz≡ d4 x , (II-14) dV= i d4 x .

II.D Linearity Of Infinitesimal Functionality In Spacetime

The postulate of linearity of infinitesimal functionality in spacetime is the premise that a function η()x is a function of x by being a function of one or more functions ΦΨ(),(),xx", and their first derivatives: (II-15) ηη()xxxxx= ((),Φ∂ΦΨ∂Ψµµ ();(), ();)" .

The functions ΦΨ,," will sometimes be called fields. By this postulate any physically relevant properties of functions, including higher order derivatives, always can be derived from knowledge of the functions and their first derivatives, coupled with the content of the rest of the primitive postulates. 8

II.E Symmetry

Symmetry means invariance in some aspect with respect to change in another aspect. Symmetry will be invoked with regard to the completeness expansion. The completeness expansion is a constant, normalized to unity. It is therefore equal to any other value of unity. The completeness expansion µµ discussed above in integral form contains the weighting or normalizing factor η()x . Its properties must be decided by symmetry, as is shown by the following argument:

1. The completeness expansion equals 1 , a constant or invariant. 2. The integral in the completeness expansion is over all spacetime; the totality of spacetime is also an invariant (in Minkowski spacetime). That is, ∫ dV is an invariant. 3. Since η()x is the only other factor in the equation 1()= ∫ η xdV, then η()x also must be an invariant.

For η()x consistently to be both a function of spacetime and an invariant, it must be an extremum: (II-16) ∂η = 0 .

This can be expanded, by including the postulate of linearity of infinitesimal functionality in spacetime (Premise D): (II-17)

∂ηη()xxxxx= ∂Φ() (), ∂ΦΨµµ ();(), ∂Ψ ();" = 0.

In this format, η()x has precisely the properties of the classical Lagrangian density, L ()x 30. Accordingly, all of the relations which have been derived from the Lagrangian density, such as Euler-Lagrange31 classical equations of , apply to η()x . Especially, any change in the configuration of the component fields ΦΨ(),(),xx", that does not change η()x , is a symmetry of η()x . An exemplary class of such possible symmetries is the set of gauge transformations, from which can be derived electromagnetism and the nuclear forces. Henceforth, L ()x will be used instead of η()x ; the preceding discussion establishes the logical necessity for the presence of the Lagrangian density based on the primitive propositions, rather than introducing it as a premise itself (the latter having been done historically32).

II.F Linearity Of Inertial-Frame Spacetime Transformations

Special relativity was derived from several postulates; they are listed here in the separate sections I.F, I.G, and I.H. It is understood as a definition that inertial reference frames have zero acceleration, and therefore in general can differ from each other in time, position and . The transformation of space and time from one inertial reference frame to another is assumed to be a linear function of space and time: (II-18) tfxt==+(',')αβ x ' t ' xgxt==+(',')ρσ x ' t ' where α , β , ρ , σ are constants (independent of x ' and t ' ). This linearity allows any point to be taken as the origin of spacetime. 9

II.G Covariance With Respect To Inertial-Frame Spacetime Transformations

Physical laws operate the same way in all inertial frames. One can pour coffee the same way today and tomorrow, in the city or in the country, standing still relative to an observer or moving at constant velocity on a very smooth train relative to that observer. Non-inertial frames are excluded here.

II.H Invariance Of The Speed Of Light

The speed of light, c , is the same, independently of the inertial frames of source and observer. Linearity of spacetime transformations (Premise F), special covariance (Premise G), and the invariance of the speed of light (Premise H), are well known as premises for special relativity33. One of the logical consequences of these premises is the set of Lorentz Transformations34: (II-19) 100v t ' t  rel            vx100'   x   rel      = γ    , y         0010 y '       z       0001 z ' where

• laboratory coordinates are unprimed and rocket coordinates are primed • space and time units are used such that c = 1

• vrel is the relative velocity of the rocket, along the positive x -axis, as viewed from the

laboratory, in the same units as c , so that in general 01≤≤vrel . 1 • γ = (II-20) 2 1 − vrel

One of the known but under-appreciated consequences of this transformation arises when the rocket is considered to be a itself (or any other massless particle, like a gluon or the hypothetical graviton). Time in the lab frame becomes (II-21) tvx''+ ttvx==+rel γ()′′ 2 rel 1 − vrel so that for the photon in its own frame, x '0= , and we have (II-22) t tt'1= − v2 = . rel γ

This is the familiar formula for time dilation, but for the photon, vrel = 1, with the consequence that t '0= , literally. It is this fact that has been under-appreciated in physical theory; it will be shown below to be a key ingredient in understanding quantum mechanics as a logical necessity in any classical relativistic universe. Time dilation (or Lorentz contraction) is complete for a massless particle like the photon and is not complete for a massive particle like an electron. Therefore, any deductions involving time dilation (or Lorentz contraction) can be expected to be different in some way (discussed in Section III.B) for massless and massive particles. Or, equivalently, considering distance instead of time, distance in the lab frame is

(II-23) ′′ xvtx=+γ ()rel . 10

In the frame of the rocket, if we have an object of length x ′ , measured at t′ = 0 (with careful attention given to the meaning of simultaneity in the context of a non-zero length35), then (II-24) xx= γ ′ , or (II-25) ′ 2 xx= 1 − vrel .

′ If the rocket is a photon, vrel = 1, so x = 0 . That is, a non-zero distance x to be traveled by the photon in the lab frame is seen by the photon in its own frame as exactly zero. If we look at just time, then in the photon frame, literally, zero time elapses however far the photon travels. If we look at just distance, then in the photon frame, literally, there is zero distance (in its direction of travel) for the photon to travel to reach its destination. This zero-time or zero-distance separation can be referred to as direct Lorentz contact, or dLc. When the photon starts its journey at its source, it already has arrived at its target. Therefore at its embarkation, it already is sensing all the conditions along the path, including conditions at its terminus.36 This is true even if, in the lab frame, conditions anywhere along the path, including the terminus, are changed ahead of the photon while the photon is in flight. This is part of the understanding for what Feynman referred to as the photon “sniffing out, instantly, all possible paths”37. It is part of what makes it possible, in the famous two-slit photon interference experiment (Section X), for one photon going through one slit to know instantly whether the other slit has just been opened or closed . This realization explains why quantum mechanics was found to be consistent with the results of the Aspect experiments38 (Section IX). The full derivation will be given in conjunction with the development of the completeness operator expansion, in Section III, entitled The Gamma-Dance. That odd relativistic disjunction (designated above as direct Lorentz contact) between lengths of space and time in the rocket (self-frame) and in the lab frames will be demonstrated to require, for full consistency, all the properties of quantum mechanics.

II.I Boundary Conditions

Boundary conditions are required, in general, when laws of nature are applied to individual situations to allow experimental measurements39. Part of every experiment is the specification of initial states, like “initially, the were in a -up state before entering the apparatus”, or of spatial boundaries, like “the two ends of the guitar string are almost fixed in position”. The general law of nature will remain the same over different experiments, but the boundary conditions can be altered over those different experiments, while still testing the same law of nature. Therefore it might be said that the boundary conditions are not present in specific form in a basic law of nature, but are present uniquely in every experiment testing the law, and should be acknowledged as present. Or, it could be stated that the existence of boundary conditions in every physical measurement is itself a law of nature, and accordingly must either be taken as a premise or derived from other premises. Here, it is taken as a primitive premise. The spacetime domain of all points accessible for any paths between a source point a and a destination point b , will be referred to as the realm of a and b , with regard to the particular particle that is considered for travel from a to b . For example, consider the 2-slit experiment (invented by in the early 1800s and championed by Feynman)40 using , with the experiment performed in a rectangular box with a at one end and a particular target point b at the other end. The realm includes the entire spacetime interior of the box except for those points occupied by the actual 2-slit barrier. The realm includes all time points contained between the time of event a and the time of event b , in addition to the empty space points inside the box. The realm is not limited to a straight-line locus between a and b ; rather, it includes all spacetime points accessible for all possible trajectories given any number of turns in the trajectories, commensurately with Huygen’s Principle41. If the barrier were opaque to visual photons, yet transparent to radio , then the realm would differ from the visual photon realm. The radio wave realm would include the spacetime points occupied by the visual 2-slit barrier as well as the rest 11 of the box’s interior, meaning that there would be no radio barrier, and the experiment would not be a radio wave 2-slit experiment. If the box itself were transparent to another particle type P, then the realm with respect to P would be larger than the box. The realm with respect to any particle is all spacetime within boundaries that block passage of that particle. Equivalently, the realm can be defined as the total contiguous spacetime region within which any two points can be in direct Lorentz contact. For example, the realm of cosmic background photons is the entire universe (except for objects opaque to specific ) from the time of recombination to the present. It is through the concept of realm that boundary conditions become intrinsic to the process of physical interaction, as will be developed starting in Section III.

II.J Physical Measurables Represented As Real Numbers

Mathematical theory (with both positive and negative numbers) includes use of complex numbers to provide closure under fractional-root operations, like the square-root operation. In addition, the volume of spacetime is itself inherently complex42. Yet, all physical measurements are made in terms of real numbers. Therefore, general theory that leads to physical measurables must include provision for conversion from complex numbers to real numbers, as an intrinsic part of the theoretical structure. A common technique for this is the process of taking the square root of the complex square (SRCS): (II-26) aib+=()() aibaiba + − =+22 b≥ 0 where a and b can be positive, zero, or negative, but aib+ is always positive or zero. Classical probability theory is understood to have been founded on assumptions that excluded both negative and complex numbers. The necessity for a general probability theory that allows at least complex numbers appears to have been encountered first in attempts to interpret43 elements in the empirically based Schrödinger Equation. Yet, a complex-number probability theory also can be viewed as a generalization of (real positive-number) probability theory that had been overlooked by previous mathematicians44. If it is maintained that any probability must be represented by a , then this general probability theory must include a provision for deduction of a real-number probability from a complex expression. The SRCS technique then arises as perhaps the simplest such method. It does not yet, however, carry any implication for summing over complex possibilities before performing the SRCS. This will be addressed, not in these premises, but in the deductions of Section III (The Gamma-Dance). The explicit premise, without reference to probability, is that all physical measurables must be represented in terms of real numbers. Therefore, because some of the internal mathematical modeling (like the modeling of spacetime volume) uses complex numbers, a method like SRCS for conversion from complex to real is required for logical consistency and meaning.

II.K Source

To address the motion of an object from a starting point to a final point, we need the concept of the object coming into existence at the starting point. This concept will be referred to as the source of the object, and lacking a derivation of it from more primitive premises, it is taken as a fundamental premise.

II.L Second Law Of Thermodynamics

Physical processes run from higher energy to lower energy, or equivalently, from lower to higher entropy45. It appears to be this that propels a particle like a photon to leave its source (Premise K) and proceed to a target, rather than just to stay at its source. Its derivation from more primitive premises is not known, and therefore it is taken itself as a primitive premise. Without this premise, it appears that sources would not function as active beginning points (a source would be just another point in the realm), and therefore no measurable consequences logically could be predicted to arise from contrast46. 12

II.M Integrity Of Particles

From one point in spacetime to the next, an electron is recognizably an electron. We do not know fully why energy coalesces to form rest matter in the form of any particle: lepton, quark, gauge boson, or possibly Higgs boson. From imposing gauge symmetry47 on a classical Lagrangian, we can derive the classical field properties of massless gauge bosons, like the photon and the gluons. By further imposing spontaneous symmetry breaking48 on gauge symmetry, we can attribute rest masses to the weak gauge bosons, re-assign a zero mass to the photon, and get correspondence with experiment. Yet, we have not succeeded in deriving fully satisfactorily the need for the localization of the field in the form that allows any one of these to be called a particle. For the masses of fundamental fermions (leptons and quarks) we do not yet have a viable theory49. It is reasonable to assume that we may in the future comprehend particularization in terms of more primitive concepts (such an understanding might be expected to be part of the derivation of the mass spectrum of the fundamental fermions). For now, the concept of the basic existence and persistence of coalesced classically-detectable particles (instead of mere fields) is taken as a primitive premise, and will be referred to as the integrity of particles.

II.N Rest Mass

Some fundamental particles, when stationary with respect to an inertial coordinate system, posses the property of rest mass, that is, a non-zero residual of energy, or mass in its self-frame. Why non-zero rest masses occur for some particles is not understood fully. Rest mass does emerge naturally in special relativity50 as the Pythagorean invariant hypotenuse of the energy-momentum 4-volume: (II-27)

pEpppµ = (,−−−123 , , ) such that (II-28) µ 22222 ppµ = E−−−≡ pxyz p p m

or in 2-dimensional spacetime, with pp23==0 , (II-29) 222 mEp= − x .

So, we see a logical necessity for rest mass, given that experiments demonstrate that pµ acts as a 4-vector, and further characterization of rest mass from special relativity will be made in Section III.B, but we do not yet understand it sufficiently to derive rest-mass spectra for all fundamental particles. Although spontaneous symmetry breaking51 in the GWS electroweak theory52 produces finite masses for the W ± and Z 0 particles, and zero mass for the photon, it produces no predictive power whatsoever for lepton and quark masses. Therefore, the idea of rest mass has not yet been deduced entirely from the other premises, and must for now be invoked as an independent premise. It is considered plausible that it may eventually be understood at a deeper level.

II.O General Deductive Logic In The Form Of Mathematics

All of the standard rules of deductive mathematical analysis53 are assumed as default. These include use of equality; substitution; operations of addition, subtraction, multiplication, division; calculus, etc.

III THE GAMMA-DANCE

The primitive postulates will be combined to find what structure naturally emerges from the idea of contrast when joined first with completeness and then with the rest of the postulates. All that is being done is development of deductive logic (Premise O) from explicit premises. And, detailed steps will be shown, to minimize ambiguity of meaning. 13

Starting with the primitive postulate of contrast (Premise A), between spacetime (Premise C) points a and b expressed in Dirac’s braket54 form, (III-1) ba , we have a symbol for the meaning that a and b are located at measurably distinct points in spacetime. This distinctness in a connected spacetime implies the possibility of one or more trajectories or paths between a and b . Let a and b be defined relative to an inertial reference frame F. The special relativity postulates (Premises F, G and H), through their implication of time dilation and Lorentz contraction, then require that

(1) An object α moving at the speed of light c = 1 (relative to any other frame, including F ) sees a and b as identical points in spacetime (direct Lorentz contact, or dLc), while an observer in any other frame such as F, who sees α as moving at velocity c , sees b as distinct from a . Object α is

55 necessarily massless (mα = 0 ), again from special relativity .

(2) An object β moving at velocity vcrel < relative to frame F and along the line from a to b sees a and b as distinct points, yet closer together than would an observer ξ moving at velocity u relative to

F , colinearly with vrel , where uv< 0 ), again from special relativity56.

The massless particle and massive particle cases will be discussed separately in the next two sections.

III.A Massless Particle Gamma-Dance

For a massless particle (using as an example a photon) going from a to b , starting with the primitive postulate of contrast (Premise A) in Dirac braket form, (III-2) ba and replacing the identity operator (that by default is present between the bra and the ket) by the completeness expansion (Premise B) in the index µ (see Section II.B), (III-3) bababababa≡ 1 = Ξ = µµ≡ ∑ µµ . µ

The meaning now is that a photon going from a directly to b is the same as a photon going from a to one of the µ -points and then to b , plus going from a to another of the µ -points and then to b , and similarly for all possible µ -points. The µ -points represent literally all spacetime points ()1, 2, 3, " accessible to the photon within the realm (see definition in Section II.I) of a and b . Let (III-4) µµ=++11 22 33 +"

The photon, in its self-frame, sees a and 1 as in direct Lorentz contact (dLc – see definition in Section II.H), and also 1 and b as in dLc. It also sees a and 2 in dLc, 2 and b in dLc, and similarly for all µ - points. At the next higher application of µµ, or µµνν (as developed in Section II.B), the photon will also see 1 and 2 in dLc, and 1 and 3 in dLc, and 2 and 3 in dLc, etc. Because of the dLc, there is no overt distinction between positive and negative time intervals between these sets of points; all time intervals for the photon in its self-frame are zero. Or, equivalently, the photon sees each trajectory leg as having zero distance. Thus it is that all spacetime points accessible by sequential combinations of straight-line paths between a and b , and between a and any µ and b , and between a and any µ and any ν and b , and … (that is, in the realm of a and b ) are immediately touched, through dLc, as soon as the photon leaves the source at a . This is precisely the underlying physical picture that requires the uniquely 14 quantum effects in the behavior of massless particles in spacetime. It is required by classical special relativity. It can be pictured as the photon “dancing” with all points in its realm of spacetime, and to all infinite orders of permutations of points in its realm of spacetime, all instantly as seen from the photon’s self frame, in going from point a to point b . Since the direct Lorentz contact is being caused by the γ - factor in special relativity, (III-5) 1 γ = , 2 1 − vrel this “dance” will be called the gamma-dance, or γ -dance. It is the central feature in the origin of quantum mechanics57. The logical consequences of this γ -dance will be derived below. In the process, the completeness operator µµ will be inserted repeatedly for successively higher orders of the γ -dance, each order being equivalent to the original contrast postulate, ba . The normalized sum of all orders (an infinite number of them) will constitute the full physical γ -dance58. Accordingly, repeating from (III-2) the process of inserting the projection operator, and noting that there now are two brakets in the center of which the completeness expansion can replace the identity operator: (III-6) ba= bµµ a = ba1 µµ = baµµ1

If we consider incorporating these logical possibilities by summing over them, (III-7) 21bababa=+µµ µµ 1 =+baba11µµ µµ = 21baµµ .

That is, summing reduces to the same expression from which we started: (III-8) ba== b1 µµ a b µµ a .

Therefore, because we are expanding each time over all spacetime points, ba is independent of the placement of the second completeness expansion. That is, ba is independent of the existence of two possibilities of ordering. In consequence, ba can be written with two completeness expansions (keeping track of redundancies but not of ordering) simply as: (III-9) ba= b1 µµ a = baννµµ = baΞ 2

()µµ2 = ba 2

1 using in the last step Eqn (II-6). (Recall that the factor of 2 arose in the integration over two completeness expansions, meaning that it arose to remove redundancies in linkages between points, rather than from ordering of the completeness operators.) By similar means it can be shown that the ordering independence carries over into the insertion of n completeness expansions with n ! linkage redundancies, so that 15

(III-10) ba= bλλ" σσννµµ a = baΞ n ()µµ n = ba n ! where the completeness expansion has been inserted n times in the first line, and in the last step the formula for Ξn is used from Section II.B. Note that for n = 0 and n = 1, this general formula consistently produces: (III-11) ()µµ 0 nbab==0: aba = 0! ()µµ1 nbab==1: aba =µµ 1!

If we sum all such n th order completeness expansions for n values from 0 to N − 1 , then since each composite braket still equals ba , we must divide by N : (III-12) 11NN−−11()µµnn() µµ ba==()∑∑ b a() b a NnNn00!!

Taking the limit as N →∞, which implies the limit as (1)N −→∞, and representing the now infinite

N in the denominator as N∞ , contrast with the inclusion of completeness becomes: (III-13) 1 ∞ ()µµn   ba=   b∑ a . Nn∞ 0 !

Recognizing the simple Maclaurin series expansion (III-14) ∞ ()µµn ∑ = e µµ, 0 n ! we have (III-15) 1   µµ ba=   be a . N∞

Using results from Sections II.B, II.C, II.D, and II.E, (III-16) 4 µµ==∫∫η()xdVLL () xdV = i ∫ () xd x and the original structure for contrast becomes: (III-17) 1 4   ixdx∫ L () ba=   be a. N∞

To be consistent with the presence of µµ between b and a , it is recognized that the structure baµµ is a representation of the sum over paths from a to µ to b , for all points µ in the 16 system. The completeness operator itself, µµ, is a sum over all points in the system. Therefore, for a set of discrete paths, (III-18)  baµµ=  µµ ∑∑  paths: ab→→µµ  where the summation over µ is shown explicitly on the right-hand-side. For a continuum of paths Xtxyz(, , , ), representing the differential for a path by D (instead of the differential d for a simple variable), and reverting to the Einstein summation convention over µ , (III-18) becomes (III-19) baµµ= ∫ µµ d(paths from a→→ µ b ) b ≡ ∫ µµDX(t,x,y,z) a b ≡ ∫ µµDX(x). a

Since this is the form for only a single completeness operator inserted between b and a , inserting the full γ -dance (from above), (III-20) b 1 4 ba=   ei(x)dx∫ L X(x) N  ∫ D ∞ a which displays the form of a Feynman path integral, with Planck’s Constant equal to unity (see Section IV). Because both quantum mechanics and have been shown previously59 to be derivable from the Feynman path integral, the derivation of this structure here from primitive premises constitutes a derivation of quantum mechanics from primitive premises. This demonstrates the logical necessity of quantum mechanics, given the primitive and purely classical concepts listed as premises. The foregoing derivation, it is stressed, has nowhere presumed any of the standard uniquely quantum concepts, yet it has produced the heart of quantum mechanics. Henceforth quantum physics thrives as a logically necessary, as well as empirically necessary, part of the model of the structure of the physical world. The derivation so far has assumed that the particle going from a to b is massless, so that direct Lorentz contact occurs, and powers of µµcould be constructed via simple integration. Massive particle motion will be addressed in the next section.

III.B Massive Particle Gamma-Dance

A particle with non-zero rest mass necessarily travels at a speed vrel less than the speed of light: vcrel <=1 . This derives from the principles of special relativity (premises F, G and H), as in standard texts60, via the relations (for motion in time and one space dimension): (III-21) mEp222= − (III-22) pmv= γ rel (III-23) Em= γ where 17

• m =rest mass of a particle (Premise N), a natural invariant in special relativity

• vrel =velocity of particle (as measured from a given frame) in units where the velocity of light is equal to 1 • E =energy of a particle (as measured from the same given frame) • p =momentum of a particle (as measured from the same given frame); pp≡ x from (II-29) 1 • γ = (III-24) 2 1 − vrel

. Dividing (III-22) by (III-23) provides that (III-25) p = v , E rel for all particles, massive or massless. From (III-21), if m = 0 , we must have Ep= , and therefore (III-

25) requires that vrel = 1, which in our units is c , the speed of light. Again from (III-21), if m > 0 , then for p ≥ 0 , we necessarily have that pE< , and therefore from (III-25) that vrel < 1 . Therefore special relativity implies that massless particles travel at light speed, and that massive particles travel necessarily at less than light speed. Thus, a massive particle cannot participate in the γ -dance in exactly the same way, using direct Lorentz contact, as did the massless particle. To simplify, at first we will use hypothetical scalar (spin- zero) bosons for both the massless and the massive cases. Although the photon is the most commonly discussed massless particle, and the electron the most commonly discussed massive particle, their respective characters as a vector boson and a spin-12fermion introduce extraneous concepts with regard to the basic understanding of the γ -dance. Let the massless scalar boson be particleG and the massive scalar boson be particle H , in discussions that follow. Particle G travels at velocity c , and thus experiences dLc, expressible either as complete time dilation (duration of any photon travel, as measured in the particle-G frame, is exactly zero), or complete Lorentz contraction (any distance to be traveled by particle G , as measured in G ’s frame, is exactly zero).

Particle H , with m > 0 , travels at vcrel < , and therefore cannot experience simple dLc. In the way that the dLc for G allows (requires) the instantaneous mutual contact of all spacetime points in the realm, with all orders of redundancy (this is the definition of the γ -dance), which implies directly a quantum nature for particle G , then particle H lacks the simple dLc. That is, the γ -dance derivation in Section III-A (for massless particles) requires dLc to produce the MacLaurin expansion of the completeness operators which, in turn, produces the complex exponential which is a central feature of the Feynman Path Integral (FPI). And, of course, the presence of the FPI implies the presence of quantum properties61. The lack of an obvious dLc for massive particles would appear to prevent derivation of the FPI for massive particles (like H ), and hence to preclude quantum phenomena for them. Yet, overwhelming experimental data (including the central critical features of the whole of the solid state industry, for example) demonstrate to us that massive particles do indeed behave in accordance with quantum mechanics. We are therefore obliged to try to derive, from premises, a logically necessary mechanism, possibly similar to the G -particle γ -dance, which will imply H s quantum mechanical nature (that is, the FPI for massive particles). Foreknowledge of the experimental property of the quantum nature for a massive particle is taken as a motivation, postdictively, to search for a derivation of that quantum nature from first principles. But, the derivation itself must produce, not postulate, that quantum nature, if we are to have a true understanding of it62. We only understand features that we can derive directly and exclusively from self-evident premises, and that understanding is only as deep as the premises are simple and the deduction direct. The Lagrangian for a massless scalar boson G , required from Premise D (linearity of infinitesimal functionality in spacetime) and from the invariance (Premise E: symmetry, and (II-7)) of the which contains the Lagrangian, is 18

(III-26) µ L = ∂∂µGG.

Particle G , in its γ -dance, fills its realm instantly as viewed from its frame (Section III-A) with paths back and forth, resulting in the FPI and therefore quantum behavior of G . For a massive scalar boson H , the Lagrangian is, from the same principles63, (III-27) µ 22 L = ∂∂−µHHmH.

Particle H has the curious mass term, mH22, which prevents H from traveling at velocity c . Yet, if we consider only the kinetic term for H s motion, (III-28) µ L = ∂∂µHH then it has the formalism of a massless particle. The effect of the mass term cannot be incorporated into the physical behavior of H at a detector until subluminal speed (vcrel < ) allows it, from special relativity. We can refer to the region already identified for massless particles as the realm, as the luminal realm; and that portion of it within the envelope expanding at vcrel < as the subluminal realm. There intrinsically is no known reason to preclude the massless kinetic term from performing the massless particle γ -dance in the luminal realm, exactly as a massless particle would. Therefore, proceeding minimalistically, we allow it. The effect of the mass term is factored in, as the subluminal realm expands. Thus the natural motion of a massive particle appears to be

(1) the kinetic term acting as a massless particle throughout its realm (luminal realm), by performing the γ -dance, including direct Lorentz contact, and thereby acquiring quantum attributes

(2) the mass term expanding from the source at vcrel < (thereby defining its subluminal realm), and meshing with the quantum attributes of the kinetic term within its expanding front (within its subluminal realm).

The full set of quantum attributes, of which the central elements are discussed in subsequent sections, then occur similarly for both massless and massive particles, within respectively the luminal and subluminal realms. According to this picture, there will be the full γ -dance (quantum) effects for massive particles, but with the proviso of awaiting the subluminally-expanding front for the incorporation of the massive particle. This will apply equally to bosons and fermions that are massive. Phrased differently, an electron can “sniff out” its luminal realm in the 2-slit experiment, but detection awaits vcrel < travel time. The “kinetic” part of an electron acts like a photon with respect to the γ -dance except for spin effects, including all of the uniqueness of the self-frame. The full massive electron meshes with these effects, but its self-frame is subluminal. Thus the electron has three relevant frames of reference: the massless self-frame, the massive self-frame, and the standard lab frame, which may or may not coincide with the massive self-frame. An implication of the massive self-frame (compared to other lab frames) regarding antimatter will be discussed in Section III-D, and other empirical consequences will be covered in Section VIII. The above picture is proposed as a deeper model for the fundamental idea of motion of a massive particle, or in general, of motion of any massive object.

III.C Secondary Gamma-Dances

Superimposed on the γ -dance of a particle (massless or massive) going from a lab-frame source a to a lab-frame detector b , there are additional γ -dances for all secondary particles generated by that 19 original particle throughout its motion. For example, if a particle is charged, then in addition to the particle’s own γ -dance, there is a secondary gamma dance of photons throughout the realm, with the source and the target being the original particle. If gravitons later prove to exist, then all particles (defined as having energy, whether massless or massive) will act as sources and targets of graviton γ -dances, meaning that all particles will have at least one secondary γ -dance occurring. Particles that participate in the weak interaction act as sources and targets of γ -dances of the weak gauge bosons, and particles that participate in the strong interaction are sources and targets for γ -dances of the gluons. Therefore, an electron will have its own massive-particle γ -dance, and will generate secondary γ -dances of photons, W ± and Z 0 , and possibly gravitons. And, a quark will have its own massive-particle γ -dance, and will generate secondary γ -dances of photons, W ± and Z 0 , eight gluons, and possibly gravitons. A photon will have its own γ -dance, and possibly a secondary one of gravitons. Similar breakdowns can be made for all fundamental particles (including quarks, leptons, gauge bosons, and Higgs bosons if they exist). The secondary γ -dances will start and end on the primary particle at all instants of time. There will thus be infinitely many criss-crossing paths of these multiple γ -dances, and the full set of them is part of the definition of the vacuum, and intrinsically part of the definition of the primary particle in relation to that vacuum. The method of computation of physical effects in the γ -dance picture remains the extant method encompassed by . The physical necessity of various known computational infinities is obvious in the γ -dance picture. The γ -dance picture, while supporting (not changing) the method of quantum field theory computation, gives a deeper insight into why it must be so, illustrating the virtual- particle-filled structure of the vacuum (a consequence of special relativity) as a necessary ingredient in all quantum phenomena, including elementary (or so-called “non-relativistic”) quantum mechanics.

III.D Antimatter Implication of the Gamma-Dance

In the Stückelberg-Feynman interpretation64 of particle motion, which is itself a direct derivation from special relativity through Dirac’s work65, a particle traveling backward in time is an antiparticle. It has been supposed historically that this interpretation implies that matter and antimatter should occur in equal proportion. Yet, the physical evidence is that matter ubiquitously predominates in common (low energy) experience, and that only in high-energy situations do we see approximately equal proportion in the presence of matter and antimatter. This has been viewed as an enigma, and various methods have been devised for accommodating such evidence in theory. One of the methods, involving CP violation in the weak interaction coupled with general CPT conservation66 does imply an asymmetry between matter and antimatter, and can be expected to pertain, in concert with the model discussed below; but does not clearly resolve the question (predict proportions) for general circumstances.

In the massive particle γ -dance, the outward expansion at velocity vcrel < of the massive component of particle H (see Section III.B), produces an asymmetry as viewed from either the lab frame or from H s self-frame, between paths proceeding forward in time and paths receding (proceeding backward) in time. The paths proceeding forward are intrinsically longer than the paths proceeding backward, because by the time (measured in either the self-frame of the massive particle or in the lab frame) the return path is traversed, the massive particle has moved a finite distance toward the target. This asymmetry (as measured from the self-frame) is present only with massive particles (vcrel < ); for massless particles, the γ -dance, as viewed from the massless particle self-frame, is instantaneous, so the outward and return paths are of equal length (zero). In the lab frame, the γ -dance return path is shorter for both massless and massive particles. It is suggested that because the inequity in the lab frame, between the lengths of the outgoing massless particle paths and the returning massless particle paths, constitutes a preponderance of time- forward paths over time-backward paths, it therefore corresponds to a preponderance of matter over antimatter. In a universe with zero relative motion, there would be only matter. In a universe with only massless particles, and therefore with all particles moving at light speed, there would be equal amounts of matter and antimatter. For complete symmetry in this picture, in a backward-time universe with zero relative motion, there hypothetically would be only antimatter, but such a declaration is meaningless 20 because there is no associated measurable67. In particle experiments with massive particles at near-light speed, the numbers of particles and antiparticles are approximately equal. In experiments with massive particles at speeds much less than light speed, the zero-motion case is approximated, and essentially all measurable massive particles are matter instead of antimatter.

If the path length in the lab frame is L0 , then the Lorentz-shifted path length as seen from the rocket (massive particle) frame is (III-29) 2 LL= 0 1 − vrel so that the ratio of the difference between the uncontracted and contracted lengths, to the uncontracted length, is (III-30) ∆−L LL− 11γ 0 2 ==11−−vrel = 1− = LL00 γγ

It is postulated that, in an experiment with a characteristic lab-frame particle velocity of vrel , this ratio is a statistical measure of the likelihood of detecting antimatter. Further and more specifically, it is suggested that this ratio, ∆LL0 , is directly proportional to the probability of detecting antimatter (Pm ): (III-31) ∆L = αPm , L0 where α is a constant to be determined. If we define Pm as the corresponding probability to detect matter, and require that the two probabilities sum to unity, (III-32) PPm +=m 1 . Therefore, (III-33) 11 PP= 11− = − + 1− v2 . m m αα rel

If we then impose the condition, as a generalization from experience, that at lightspeed (vc==1) the ratio of the two probabilities is unity, or (III-34)  P 11−−v 2  mrel   == 1 Pm  11 vrel =1 11− + −v 2 rel  αα vrel =1 and solve for α , we get α = 2 . Therefore, the probabilities (or proportions) are (III-35) ∆L 111 Pv==11−−2 =  1−  m ()rel   22L0 2γ (III-36) ∆L 111 Pv= 1111− =+− 2 =+  m ()rel   . 22L0 2γ 21

These are graphed in Figure 1, with Pm for antimatter and Pm for matter:

GAMMA-DANCE PROPORTIONS Antimatter & Matter

1.0 0.9 0.8 Matter 0.7 0.6 0.5 0.4 Proportion 0.3

0.2 Antimatter 0.1 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Particle Speed (relative to lightspeed)

Figure 1: γ -Dance Proportions For Matter And Antimatter

For the specific values of zero, half-lightspeed, and lightspeed velocity, these formulae provide:

v rel Pm Pm Pm Pm 0 0 0 1 1 2 .134 .067 .933 1 1 1 1 2 2

Table 1: γ -Dance Proportions For Matter And Antimatter

It may be of value to compare these with empirical data, along with CP-violation factors. The significance of these results lies in the emergence of antimatter as an intrinsic consequence of the same mechanism that produces quantum mechanics itself, namely self-frame effects in special relativity (the γ -dance). The formalism naturally produces the prevalence of matter over antimatter in common (low velocity) experience, the equal production of matter and antimatter in high-energy processes like pair production, and predicts the proportion for intermediate energy values.

IV DERIVATION OF PLANCK’S CONSTANT

From the form of the expression (IV-1) b 1 4 ba=   eixdx∫ L () X(x) , N  ∫ D ∞ a 22

the phase is i times the classical action. Implicit in that phase is the presence of a denominator of value unity:

(IV-2) 4 ixdx∫ L () ixdx()4 = ∫ L 1

If the gamma-dance derivation had produced a phase denominator of zero (as some researchers have supposed would obtain in a purely classical universe), then the phase would be infinite, and there would be no structure whatsoever of physical law derivable from this path integral. Indeed, the Lagrangian structure for classical mechanics would not emerge naturally, if this phase denominator were zero. This implies that even the natural derivation of classical mechanics requires that this phase denominator be non-zero. Since in common successful usage in the Feynman path integral this same phase appears as (IV-3) 4 ixdx∫ L () , # we may identify directly that # = 1 from the gamma-dance derivation. Since many calculations of quantum effects are carried out in “natural units”68 with # = 1 and c = 1 , the γ -dance derivation includes implicitly the derivation of both the existence, and the correct value, of Planck’s Constant, # , in natural units. Conversion of # into other units proceeds as in the standard literature69. Logical consistency, therefore, requires that # exist as a non-zero value, and specifically with the natural-units value of unity, even for classical physics, as well as for quantum physics.

V HEISENBERG UNCERTAINTY PRINCIPLE

With any wave, classical or otherwise, there always is an associated uncertainty principle. Since the γ -dance70 produced a phase along a path from source to detector, implying a wave, it should therefore be expected that an uncertainty principle should arise. The relationships are standard in quantum mechanics literature, but will be presented here briefly, in one of the several possible ways71, for thoroughness72. Letting any wave ψ()x be a superposition73 of plane waves eikx of phase φ = kx and wavelength 2π ∂φ λ = k , and with k = ∂x , define ∆k as the extent of the domain of variation of the parameter k . In order that the wave ψ()x be restricted to a certain region ∆x , it is necessary that constructive interference among these different waves occurs only in that region, and that they interfere destructively kx∆ 2π everywhere else. The number of waves, each of wavelength k , contained in ∆x is 2π . In order that the various plane waves forming ψ()x may interfere destructively at the limits of the interval ∆x , this number must change at least by one unit when k runs over its domain of variation ∆k :

∆∆xk ≥ 1 , 2π or (V-1) ∆∆xk ≥2 π .

SCl 4 The phase produced by the γ -dance was φ = , where Sxdx= () , the classical action (III- # Cl ∫ L 17). This phase, in the derivation of the γ -dance, operated between the initial point a and the final point 23

Sba(), b . Changes in the end point x will cause changes74, in general, in Cl . The change in phase per b # unit displacement of the end point is, from the definition of the wave number k ,

(V-2) ∂φ 1 ∂Sba(), k ==Cl . ∂∂xxbb#

b 3 Yet, from the γ -dance derivation, and with ∫ L ()xdx≡ L, a (V-3)

b tb 4 SxdxLdt==() . Cl ∫∫L ata

Since the action was an invariant75 in the γ -dance, (V-4)

δδSSxxSxCl==0[ Cl + ][]− Cl .

Then, (V-5)

tb Sxx[]+=δδδ Lxxxxtdt$$ + ,, + Cl ∫ () ta t b ∂∂LL =++Lxxt()$$,, δδ x x dt ∫ ∂∂xx$ ta t b ∂∂LL =+Sx[] δδ x$ + x dt Cl ∫  ∂∂xx$  ta

Integrating by parts, (V-6) t t b ∂∂∂LdLLb δδSx= −− δ x   dt. Cl ∂xdtxx$$∫  ∂ ∂ ta  ta

76 If we change the end point xb , (V-7) δS ∂L  Cl =  $  δ xxb ∂  xx= b

∂L δS But, classically the momentum is defined as p = . Thus, p = Cl . Therefore, using Eqn. (V- 2), ∂x$ δ xb (V-8) 1 ∂Sba(), p k ==Cl . ##∂xb 24

∆p Therefore, ∆k = . Inserting this into Eqn. (V-1), we have # (V-9) p ∆∆x ≥2 π , ()# or (V-10) ∆∆xp ≥ h, which is a standard version of Heisenberg’s Uncertainty Principle. ∂L Since the relation p = emerges naturally from classical Lagrangian theory (which itself is ∂x$ implied by the results of the γ -dance), and includes Premise D (linearity of infinitesimal functionality in spacetime), all of the inputs required come from classical premises. Therefore, the above development constitutes a derivation of Heisenberg’s Uncertainty Principle from classical premises. This implies, it is emphasized, that the consistency of classical physics requires Heisenberg’s Uncertainty Principle.

VI WAVE-PARTICLE DUALITY

Wave-particle duality occurs as a consequence of the γ -dance and the integrity of a particle undergoing any process. The γ -dance was derived in Section III, and its implication of a wave nature for particles is discussed below, including the effects of particle integrity, followed by examples of the γ - dance picture in 2-slit interference and hydrogen-atom electron transitions.

VI.A The Wavelength Of A Particle

The γ -dance produced a phase (III-17), and a k (V-10) such that (VI-1) 12∂Sba(, ) p π k ==Cl ≡ . ##∂xb λ

The wavelength emerging from the γ -dance is therefore, using h = 2π# , (VI-2) h λ = p where p is the classical momentum magnitude: (VI-3) ∂L ∂S p ==Cl . ∂∂xx$ b

Immediately from (VI-2), if p = 0 (implying the rest frame of the object), then λ = ∞ , and from mEp222= − (III-21), we have that the entire energy of the object is in its rest mass, so the object necessarily will be recognized as a particle, not a wave. That is, in its own rest frame, an object always is a particle, not a wave. For massive particles (m > 0 ), this is intuitively reasonable. Again for massive objects, if p > 0 (implying non-rest frame) in addition to m > 0 , then (III-21) indicates that the energy is greater than the rest mass energy. That excess is well known as kinetic energy, is present in any frame other than the rest frame, and through (VI-2) requires a wave nature for the thusly-viewed non-stationary massive particle. 25

For massless particles, in order to have any structure at all, the energy must be non-zero: E ≠ 0 . Therefore, given that m = 0 , it must be that p ≠ 0 , again using (III-21). Yet, if p ≠ 0 inherently, the idea of a simple rest frame for a massless object breaks down, from special relativity. From any lab frame, however, an observer can see the massless object moving at speed c ≡ 1 (in our units), and from that lab frame can speak of a frame moving along with the massless object, and term that the massless object’s self- frame. In that frame, of course, an observer would see direct Lorentz contact in the direction of travel (see Section II-H). The available momentum-energy information allowed by special relativity from such a self- h h frame is only that p = λ and Ep= . Therefore, Eh==λ ν , using the standard classical wave relation that λν = u , where u is the velocity of the wave, which in the case of a massless object is c (see discussion following (III-25)). Therefore, from the γ -dance a massless object inherently is a wave, given that it always is observed from a non-self frame. Yet, having defined, as above and externally, a self-frame for the massless object, within that self-frame the massless object will indeed register itself as a particle; its lack of rest mass, or indeed energy, in that self-frame is counter-balanced by the direct Lorentz contact phenomenon. The missing concept for full consistency is provided by integrity (Premise M) introduced in Section II.L. The implications of integrity, in terms of a self-frame for a massless particle, will be developed in the next section.

VI.B Integrity Of Particle At Source And Detection Points

In the physical transition of any particle from point to point in spacetime (propagation), we can only recognize the particle as a physical object because it maintains its integrity, that is, its recognizable identifying properties. Without integrity of physical objects there would be no structure at all in the cosmos (there would be no measurable universe). Given that there is structure, and that all observable structure can be synthesized from the building blocks of particle-wave objects, we ask how the particle integrity is achieved, based on simple premises. We have observed in Nature so far, with certainty, two types of particles: fundamental fermions and gauge bosons. Current empirically correlated theories suggest, for consistency, at least another class, of Higgs bosons; that class may prove to consist of a single element or multiple elements. Speculations, not yet empirically correlated, include other classes of particles. In the current discussion, the emphasis will be on the gauge bosons and fundamental fermions. Imposition of gauge symmetry in the context of the Lagrangian has been demonstrated to require the existence of gauge bosons as exchange particles between fundamental fermions, in successful theories of the Standard Model for the electroweak and the strong interactions. The gauge symmetries require the gauge boson particles to exist, meaning to continue to exist from point to point in spacetime. Without such continuity we couldn’t detect them. Therefore it consistently can be phrased that gauge symmetry in the Lagrangian is at least partly the cause of the integrity of gauge bosons. The spontaneous of the electroweak theory gives the W ± and Z 0 non-zero masses which are retained for finite durations, and the zero mass of the photon. Unbroken gauge symmetry in the quantum chromodynamics theory gives zero masses to the eight gluons. Therefore, symmetry produces or at least contributes to the existence and integrity of the photon, the W ± and Z 0 , and the eight gluons. These symmetries operate within the context of the Lagrangian, with symmetry properties (the action is an invariant), and that Lagrangian has been demonstrated to emerge as a natural part of the γ -dance formalism. (We have not yet produced theoretically, in clarity, a necessary presence of fundamental fermions in the Lagrangian). During propagation of a massless particle (for example, a photon), the γ -dance process, with direct Lorentz contact, requires that the detection point be identical (in the self-frame of the photon) with the emission or starting point. Therefore, the process (for the photon, it is gauge symmetry) that produced the particle and maintains its integrity, still applies (the end point equals the starting point), so the integrity is maintained from the source to the end point of the transition. This is in spite of the fact that, in the γ - dance process, the photon touched every point in the entire system allowed by the boundary conditions of the system (that is, in its realm), as a spread-out wave. Therefore, the γ -dance and symmetry together produce both particle-like integrity at the transition termini (because of direct Lorentz contact in the self- frame), and wave-like adaptation to the boundary conditions of the complete environment in the lab frame. This constitutes (is) wave-particle duality, and it is, through the direct Lorentz contact, a relativistic effect. 26

The γ -dance of a massive particle, from the discussion of Section III.B, has elements of the massless particle γ -dance and the sub-luminally expanding effect of the mass term in the Lagrangian, together producing the phase (and therefore wave-like) structure in the Feynman Path Integral. Because in the massive particle’s self-frame the momentum is by definition zero (Section VI.A), all of its energy in that frame is in its rest mass, and it intrinsically is purely a particle. From any other frame it is seen to have an excess of energy over its rest-mass energy, as a contribution from non-zero momentum, and therefore from (VI-2) must be wave-like, generating multiple-path wave effects using boundary conditions. At its source and detection, its self-frame state is altered, thereby requiring recognition of its particulate nature. There is not yet any known derivation that equivalently accounts for the presence in the Lagrangian of the fundamental fermions (leptons and quarks) based on primitive premises, although their mass terms are accommodated by symmetry in the Lagrangian. Accordingly, a full understanding of their γ -dance cannot be given here. It is speculated, because leptons and quarks do exhibit integrity, that symmetry requirements will eventually emerge that produce these fermions, including their mass spectra, and that their γ -dance effects will be similar to those of other massive particles (the massive gauge bosons), save for the difference in spin (which in itself has long been understood).

VI.C Double Slit Example

In a standard quantum process, like double-slit interference (see Fig. 2 in Section VIII) using either photons or electrons, the classic wave-particle duality is demonstrated. It is of use to visualize the process in the γ -dance picture, discussed here for the photon. The well-known phenomena to be coordinated in the 2-slit interferometer are:

• When both slits are open, interference fringes are detected on the screen, indicating wave behavior. • When only one slit is open, the screen pattern is that of a classical particle. • With both slits open but only one particle is being sent through the apparatus at a time, each individual particle registers on the screen integrally and distinctly as the same particle that entered at the source. That is, the particle may be said to maintain its integrity at the source and registration points, whatever happens in between. Yet, after many individual particle hits have accumulated on the screen, the pattern is the same interference pattern that was seen with both slits open and many particles at once going through. Therefore each individual particle appears simultaneously to be behaving as a particle and as a wave. The distribution of end points indicates wave behavior, while each end point itself indicates an integrity-maintaining particle hit.

In the γ -dance for the photon, in the lab frame, there is in the region around the source point and infinitesimally close to it, an isotropic wave emanating from the source. This wave is subjected to boundary conditions (including the obstructing barrier, the two slits, the outer limits of the realm, and the detector). But, as seen from the photon frame, there is an infinite set of points in immediate equal contact, because of the complete Lorentz contraction for the massless photon in any direction of travel. It appears that any single direction of travel (start of a path) in the lab frame is picked, randomly and isotropically, and then the photon, in its frame, negotiates the path to the target instantly. In the lab frame, the isotropic symmetry at the beginning is sculpted by the boundary conditions into varying densities of trajectories in the γ - dance. The photon just touches all end points at once and one registers (detection occurs). When the photon registers, then the same physics that caused it to have integrity at the source point also cause it to have integrity at the detection point, because in the photon frame the source and detection points are identical (due to the Lorentz contraction). The cause of the photon’s integrity in the first place is the cause of its continued integrity at the end point. What has happened in between to generate the wavelike placement of the photon on the final screen, is the γ -dance, which was shown in Section III to have phases along multiple paths, and hence to be wavelike. In more detail, the photon’s integrity comes from its creation to maintain gauge symmetry; that same gauge symmetry is maintained in the photon frame at the source and the registration points, because they constitute the same point. In the lab frame, the source and registration points are not the same. This discrepancy, which comes directly from the Lorentz transformation of special relativity, is real, physical, and logically necessary for consistent photon behavior. It is the heart of the intuitive disjunction in 27 visualizing quantum effects, and it is actually a relativistic effect, even in so-called “non-relativistic quantum mechanics”. The photon, unaware of the γ -dance in its frame, leaves the source and arrives at the detector instantly, with the paths in the lab frame modulated by boundary conditions. In the lab frame, the uncertainty principle (Section IV) associated with the wave nature generated from the γ -dance renders the position in the immediate locale of the source point indeterminate; this feedback causes the isotropic direction of departure from the source point. And, in turn, the randomized choice of path results. On this random choice of path is superposed, again by the γ -dance, the mandated by the realm’s geometry. When the photon hits the detector it is still the same photon that left the source, because it has no reason in its frame not to be; it still “thinks” it’s in the same place. In the lab the observers have to wait for the finite light travel time to detect the registration (hitting of the screen). If the boundary conditions in the lab frame are changed downstream of the photon while the photon is traveling, then the photon in its frame “sees” the new configuration instantly and reacts to it instantly, because the 100% Lorentz contraction makes it see everything around it instantly. The lab observers see the altering boundary conditions as set out in time along the lab-frame multiple paths of the photon. Accordingly, the γ -dance picture directly implies wave-particle duality, and includes in-flight adaptation to lab-frame changes to boundary conditions. Quantum probability will be addressed in Section VII.

VI.D Photon Emission From Electron Energy Transitions In Hydrogen

In the case of emission of photons from electrons in atoms, the well-known phenomena to be coordinated are:

• Transitions occur from higher energy to lower energy. • Electron energy levels are determined from wave considerations. • Emitted photons carry only the quantized energy equal to the difference in energy levels. • The emitted photons exhibit all electromagnetic wave phenomena, yet are quantized and maintain integrity until explicit interactions occur.

The electrons, in their γ -dance, generate phases along multiple paths. Those paths are sculpted by the electromagnetic attraction between electron and atom (simple hydrogen case of electron and proton), so that the waves must follow periodic boundary conditions around the proton, leaving their energies quantized. From the Second Law of Thermodynamics (Premise L), a higher energy electron tries to lower its energy. If the electron’s γ -dance detects a lower-energy configuration, it will try to make the transition to it, still subject to the atom’s boundary conditions and the integrity conditions for the electron. If the simplest transition mechanism, for removing energy, is an electromagnetic emission of a photon (possible because the electron is charged), the electron will try to emit a photon. At that point, the photon will initiate its own γ -dance, with its own integrity maintenance process. The result is the electron in a lower-energy state and an emitted discrete-energy photon. The electron has maintained its integrity through a transition between energy levels maintained by wave-like periodic boundary conditions. The emitted photon has originated with particle-like discrete energy and maintains its particle-like integrity, yet exhibits wavelike properties if measured downstream. The γ -dance, plus the Second Law of Thermodynamics, plus the symmetries that create and maintain particle integrity, plus the boundary conditions of the atom, plus the electromagnetic properties, require thereby the extensively-observed photon emissions from electrons in atoms. Each step in this picture can be reduced to the primitive and intuitively obvious premises. 28

VII , PROBABILITY, AND MEASURABLE QUANTITY

The path integral resulting from the γ -dance is complex, and includes all possible paths allowed by the geometry of the boundary conditions. The volume of spacetime, id4 x , is a component of the γ - dance path integral, and that volume is explicitly imaginary. Therefore the exponential

4 eixdx∫ L () is inherently complex, rendering the result complex in general. Note again, to emphasize, that the γ -dance inherently and exhaustively sums all possible paths (consistent with the boundary conditions) with their respective phases intact. All experiments measure real quantities. That is, measurables are represented as real numbers (Premise J). For comparison of a theoretical calculation with experimental results, therefore, it must be ensured that the calculated result is real, not complex or imaginary. The simplest way to accomplish this is to take the absolute square. Any theory in which spacetime volume intrinsically is part of the calculation requires that an absolute square be taken before a measurable can be constructed from the theory, unless there is cancellation of the sign of the determinant of the metric (this exception is realized in classical relativity applications). In the physical γ -dance, all paths that can be traversed instantly in the photon frame are included in the sum. That is, the dividing criterion between what is included or not included in the summation of the γ -dance is the possibility of complete Lorentz contraction (or equivalently, complete time dilation) along the path (this is called direct Lorentz contact). The result produces a measurable. In the theoretical model of the γ -dance, we must accordingly sum the complex of all paths that can give direct Lorentz contact between the source and registration points. Any apparent path that lacks the possibility of direct Lorentz contact is not to be included in the γ -dance. The resulting total complex amplitude is what must be absolute-squared to get a measurable. Any paths that lack direct Lorentz contact (like separate one-slit observations in the 2-slit experiment) can only be combined to form measurables after absolute squaring. This is the origin of the standard quantum rule for summing amplitudes and then absolute squaring. It is clear that through the phenomenon of Lorentz contraction in the γ -dance, this summing of amplitudes is actually a relativistic effect, even for non-relativistic quantum mechanics. Since this process is the heart of all quantum mechanics, it emerges that quantum mechanics is one of the consequences of special relativity. In conjunction with the preceding calculations, quantum mechanics is the special relativity of self-frame effects. This can carry many implications. One of the strongest is the possibility for extension of the γ -dance process from special relativity to general relativity, to explore the consequences77. This has not yet been carried out.

VIII WAVEFUNCTION COLLAPSE

The γ -dance provides a different view of the process of classical measurement of quantities calculated from quantum theory. The old view envisioned pictures like an infinitely burgeoning array of simultaneously valid and real universes from each point in our universe (many-worlds interpretation78), or of interactions with the environment to decohere79. In the γ -dance picture, there is only one universe needed, and the exact mechanism for decohering interaction is demonstrated. The oddity of apparent collapse occurs because there exists, in the self-frame of a massless particle (example: photon), direct Lorentz contact (dLc) between the source point and any other point in the realm. This dLc is not seen in the lab frame (any non-self frame of the particle), so there appears to be a disconnect. That disconnect, however, is an already known feature (proven to be consistent with experiment and theory) from special relativity. Therefore, it can be phrased as a Galilean disconnect or inconsistency, but a special-relativistic consistency. In any lab frame, an observer has to await light-travel-time (LLT) to know for sure what happened in any dLc-mediated event80 event, no matter how convoluted or simple the strategy for detection 29 is set up to be. In the self-frame of the particle, however, the transition from source to detector is instantaneous, literally, physically, and fully consistently with special relativity. In the self-frame of the massless particle, there is no wavefunction to collapse. There is just the exact identity of all points in its direction of travel, with the source point. The detection point is identical with the source point81. The choice of a direction in which to propagate is isotropically random at the source (because of the uncertainty principle, derived from the γ -dance in Section IV). Given that choice, the transition is trivial, with the entire set of the particle’s attributes transferring to the detector, because of the identity of the detection and the source points. In more detail, each possible path and end point (detector) in time and space is sensed as a proximate point instantly in the self-frame of the particle while still at the source, and randomly a direction from the source is picked. This randomness in source- departure direction is weighted in the lab frame by the density of paths throughout the realm. The paths (more easily imagined in the lab frame than in the self-frame because of dLc), symmetric about the source in the infinitesimal region about the source, spread throughout the realm subject to all boundary conditions (Premise I). Those boundary conditions cause the path density to vary throughout the realm. Simultaneously, each path acquires from the γ -dance an intrinsic phase (again, more easily visualized in the lab frame), and the detector point is arrived at instantly in the self-frame of the particle. The detection, using the wave effects from the γ -dance, registers in a particulate sense because the integrity of the particle at the source is still in play at the detector (in the self-frame). That is, in the process of the source and detector points being proximate dimensionless points, they are indeed literally identical points. In any lab-frame (called here the lab-frame), the observer must wait for LTT from the source to a detector. Until the registry in that detector, the observer has no information whatsoever on the position or state otherwise of the particle. Any such information would constitute a detection, and would end the direct Lorentz contact in the self-frame. The γ -dance provides a phase along each path, that phase being an i action formed of the spacetime integral of the Lagrangian for the particle’s classical field, multiplied by # (see Section III.A). The necessary presence of this phase along each path, as viewed from the lab frame, gives the lab’s interpretation (including measurement) of the particle as a wave. In the self-frame, the particle remains intrinsically a particle, including at source and detector, with the intermediate points identical to them. Since the particle registers at the source and detector as a particle in the self-frame, it must do so also in the lab frame. Hence we arrive at a separation of the causes of wave-particle duality: In the self-frame, the particle always is a particle. In the lab-frame, the observer sees selection of the detection point as wavelike, but the actual detection as particle-like. In the γ -dance picture, therefore, there is only consideration of known special relativistic effects, and other primitive classical concepts, with derivation of their detailed consequences. Those consequences include wave-particle duality and other features of quantum mechanics, but nowhere require infinitely many universes or other such exotic devices that themselves would require extreme revamping of much of the rest of physical theory. That is, the γ -dance incorporates quantum effects into physical theory without the need for exotics. The γ -dance picture indeed provides a physical means of decoherence, through the mechanism of direct Lorentz contact and its implications. The γ -dance picture also suggests that alterations in the realm between the particle’s departure from the source and arrival at the detector (as seen from the lab frame) must affect the result, whether the changes are upstream or downstream on the particle’s imagined path from source to detector. It is required that such changes be within the realm of the process, meaning equal to or after the time of source departure and before or equal to the time of detection, and within the space limits defined by the boundary conditions, as viewed from a lab frame (most easily, the lab frame in which the detector is at rest). This is because the γ -dance necessitates total back-and- forth (positive time and negative time) passages of the massless particle between the source and detector events in the spacetime of the lab frame. This is compared with the instantaneous dLc in the particle’s self- frame. It is emphasized that the wavefunction of Schrödinger’s Equation is a mathematical approximation to the physical reality that is measured experimentally82. The γ -dance provides a deeper mathematical mechanism from which Schrödinger’s Equation can be derived as an approximation (the γ -dance derives the Feynman path integral, from which Schrödinger’s Equation has been derived83). 30

As an archetypal example, Young’s 2-slit experiment (Fig. 2) illustrates both the standard argument for the collapse of the Schrödinger wavefunction, and the γ -dance resolution of that argument:

1

Source 2

Detector Barrier

Figure 2: Young’s 2-Slit Experiment

The common argument for collapse of the wavefunction is that if slit-2 is closed, then “instantly”, the interference pattern at the detector reverts to the particle pattern for the source→ slit-1 → detector path. In the γ -dance picture, careful accounting is made of events in the self-frame of the particle and in any (the) lab frame. In the self-frame, if slit-2 is closed at any time during the experiment, then the particle “knows it” instantly, because the particle is in dLc with all points in its realm. Here it is assumed the experiment began with slit-1 and slit-2 open, and the realm as a spacetime volume: In time, the realm lies inclusively between the time of leaving the source and the time of arriving at the detector, as viewed from any frame. In space, the realm is a box with source at one end, and detector at the other end, and with the barrier with the two slits in the middle of the box, as viewed from the detector lab fame, with consistent Lorentz transformations to any other frames. Therefore, in the self-frame, closing slit-2 has an instant effect at the detector point, because all of the points along the path are the same. In the lab frame (choose the easiest lab frame, that in which the box, source, and target are stationary), the time for a photon between leaving the source and arriving at the detector will be the light travel time along its particular path. The γ -dance will be performed using all paths that the photon can access. If slit-2 is closed before light could have reached it classically, the detector will register a particle distribution, because all paths must go through only slit-1. If slit-2 is closed after light could have reached it classically, it still should affect the detector, because the back-and-forth γ -dance is continuing until the detection event. If slit-2 is open throughout, then the detector will register the interference of waves, while still detecting each photon as a particle. But, in all cases, any observer at the detector must await light travel time before he knows, by any means, the state of slit-2. Any technician who had instructions to open or close slit-2 at any particular time cannot be known with certainty to have followed instructions, until he can send at light speed a separate signal, in proof, to the observer at the detector. So, in the lab frame, there is neither instantaneous nor superluminal knowledge from events a distance away. So, in the self-frame, there is instantaneous but not superluminal knowledge of events throughout the realm (because light travel time is zero owing to dLc), but in the lab frame the non-zero light travel time is required for all signal transmission. It is this special-relativistic consistency, which is a Galilean disconnect between the self-frame time and the lab-frame time, that gives rise to both the γ -dance itself and the historical confusion over wavefunction collapse. The Schrödinger Equation is outwardly nonrelativistic, so that, although it tracks marvelously many quantum features, it fails to account quite consistently for this special relativistic effect, the Galilean disconnect between the self and lab frames. It is expected that the arguments applied here to the 2-slit interference experiment will map to all other experiments of alleged wavefunction collapse, such as the Stern-Gerlach experiment, revealing the same γ -dance solution. 31

For massive particles, as described the Section III.B, the free-particle term in the Lagrangian produces all the features of a particle γ -dance, while the mass term in the Lagrangian catches up at its mass velocity, which is sub-luminal. So, the particle itself will react instantly to changes anywhere in its realm, but the lab-frame measurement of that self-frame reaction must await light signal time. All instances of wavefunction collapse (as in the double slit and the Stern-Gerlach experiments) should be found to proceed, comprehensibly, in accordance with this picture84.

IX BELL’S THEOREM, ENTANGLEMENTS, AND THE GAMMA-DANCE

Quantum mechanics violates Bell’s Theorem85 because it is not a local hidden-variable theory. In fact, terms like “hidden-variable” or “non-local” are unnecessary, merely malapropos, in the γ -dance theory. Quantum mechanics indeed displays apparent, but not real, “non-local” behavior in the lab frame and is a trivially local theory in the massless-particle self-frame (see Section III.B for massive particle effects). There is no “hidden variable”; the agent causing the apparent disjunction is the Lorentz transformation of special relativity, which provides direct Lorentz contact (dLc) for massless particles and for the massless (kinetic) terms of massive particles. The direct Lorentz contact in turn is part of what mandates the γ -dance. The γ -dance, operating throughout its realm, includes among its effects all of the properties collectively described as ; all paths of the subject particle within its realm are entangled. The sense in which entangled states86 appear to decohere instantaneously (undergo wavefunction collapse) is precisely the sense caused by the entire entanglement resolving to a single point in the subject particle’s self-frame (direct Lorentz contact), yet extending over light-travel-time paths as viewed from any non-self-frame (lab frame). This renders the subject particle’s motion entirely (trivially) local in its self- frame, and apparently disjunctive in the lab frame. The Lorentz transformation between the two frames bridges the effects in the self-frame and the lab frame in this phenomenon of direct Lorentz contact. The Aspect experiments87, 88, 89 proved experimentally the reality of the entanglements of quantum mechanics as viewed from the lab frame, with both time-constant and time-varying analyzers. In so doing, the experiments also demonstrated the expected (post-dictive) effects of the direct Lorentz contact of the γ -dance picture, especially strikingly in the case of time-varying analyzers. In none of the Aspect experiments could certainty be established90 in the correlation of the measurements on entangled states, until allowing for light-travel-time (in the frame of the detectors, i.e. in the lab frame) between the two or more detectors. It is therefore suggested that the γ -dance is the physical resolution, via special relativity, of the quantum enigma of entanglement, wavefunction collapse, Bell’s Theorem, and the Aspect experiments.

X EINSTEIN-BOHR DIALOGUES

A series of dialogues91, expressing both deep affection and deep difference of viewpoint between Neils Bohr and , spanned the decades from the advent of quantum theory in the early 1900s to the loss of Einstein in 1955. Although approximate and incomplete, a summary of their views will facilitate an analysis of their interchange in terms of the γ -dance. Bohr maintained that, in the absence of satisfactory theoretical understanding, the empirical findings should stand as the structure of any theory, including quantum mechanics. He noted that the empirical facts requiring the existence of quantum mechanics seemed frequently to assign two apparently contradictory qualities to a given phenomenon. He called this aspect of quantum mechanics complementarity, or duality. A prime example is wave-particle duality. He advised that this complementarity and other quantum enigmas should be accepted as part of the essence of quantum mechanics, unexplained in theoretical terms, while mandated experimentally. Einstein maintained that elsewhere in physical theory, it appeared that when we worked hard enough to unravel Nature’s secrets, Nature would reveal itself to us. He produced superb statements like “The incomprehensible thing about the Universe is that it is comprehensible”. He wanted to find some 32 hypothetical “hidden variables”, through which an explicitly rational quantum theory could be modeled, with full development from primitive premises to deduced physical measurables. Both Bohr and Einstein were speaking for legitimate properties of historically good physical theory. But, the full theory remained undiscovered, despite Einstein’s efforts, and all that was left with assurance was the experimental data, stitched together with bits of theory which required, frequently, a dual model of a single phenomenon, among other apparent quirks. Yet this cobbling of data and bits of theory worked so well that it has provided a physical basis for a significant percentage of the entire world economy (solar cells, computer chips, lasers, nuclear power, genetics…). The γ -dance picture (Section III above) demonstrates the logical and physical necessity of two distinct Lorentz frames of reference in viewing the detailed behavior of massless particles (Section III.A), and derivatively, of massive particles (Section III.B). The discussion here will be in terms of massless particles. The two Lorentz frames are literally the self-frame of the particle, and any other (non-self) frame, that is, a lab frame. In the self-frame, a massless particle remains a particle, because it stays at a single point from source to detector, thereby intrinsically and trivially retaining, throughout its existence of an instant, the integrity (Section II.L) as a distinct particle that it had on formation. In its self-frame, it travels zero distance and notices zero elapsed time, even if in a lab frame it originates as a cosmic background radiation photon at recombination and is detected on Earth about 15 billion years later92. In the lab frame, the γ -dance intrinsically produces wavelike behavior, as the paths dance throughout the realm (Section II.I), but the paths focus at two nodes, the source and detector. At each of those two points, in both the lab frame and the self-frame, the propagation is at a terminus, and therefore the self-frame and the lab frame views must produce the same results. Since the particle integrity in the self-frame is still in play, the registry at source and detector in the lab frame must be as a particle. Yet, the profusion of registry at any particular point follows the necessarily wavelike behavior in the γ -danced lab frame. That is, discrete particles are detected in a wavelike distribution pattern. So, in a two-slit experiment, the detector over a variety of points, in time and space, detects the wave’s interference pattern, but comprised of particulate strikes. There is, therefore, a duality or complementarity inherent in the structure; it is the duality of the frames of reference: the self-frame versus any other (lab) frame. This is a fundamental split, and the two views give apparently disparate results, until one notices that they are connected by the Lorentz transformation. So it is that the hidden variables so long sought by Einstein are melded into one factor that was intrinsically a vital part of Einstein’s own special relativity theory: the Lorentz transformation93. In this sense, then, the good reasoning of both of these old friends comes together: The experimental results do indeed display a complementarity, a duality, and there is a connector. There are two critically different frames, the self –frame and the lab frame (synonymous with any non-self frame) , each producing distinct parts of the totality of quantum effects. Yet, the two are linked by the Lorentz transformation. With the Lorentz transformation and the rest of special relativity, theory does indeed bridge the gap (indeed, it both requires the gap and provides the bridge) of quantum complementarity. Einstein also was concerned over the element of chance in quantum phenomena. In the γ -dance picture the element of chance is visible as the equal probability of any direction into the realm, on leaving the source, and the precise timing of departure. The γ -dance resolution, as discussed in Sections VI and X, is that if we start out assuming only the source position and time, as measured in the lab frame, the γ - dance produces the uncertainty principle. This in turn makes any direction equally likely about the source point, and provides, given any energy, a complementary time spread for departure. We are left with an essentially complete theoretical understanding of quantum mechanics94, and with the impossible wish of testing the reactions of these two fine people to the proposed γ -dance picture.

XI OTHER QUANTUM PROPERTIES

Other quantum properties, like spin95 and its associated statistics96 and exclusion principle97, have in turn been derived in the literature from the quantum features already discussed above, and those derivations will not be repeated here. The main function of the γ -dance picture is to derive consistently from simple primitive ideas the more complex concepts (which had seemed mutually incompatible) that have historically been used as the higher-level premises of standard working quantum theory. It is noted that the 33 existence of intrinsic spin98, which is noticeable in non-relativistic as well as relativistic quantum mechanics, was derived by Dirac in the context of his approach to relativistic quantum mechanics. In this, Dirac presaged the necessity of special relativity in deriving even non-relativistic quantum mechanics. It has been shown99 that the Schrödinger Equation can be derived from the path integral. In turn, the standard host of quantum results, including the utility of commutators, can be derived from the Schrödinger Equation, and can be accessed in good quantum mechanics textbooks100. Since the path integral was derived from the γ -dance, all of the Schrödinger Equation solutions also follow from the γ -dance. Quantum field theory101, the successful quantum theory for relativistic energies and for deriving quantum versions of forces from symmetries, is generally derived from premises that include the path integral. It therefore also follows from the basis of the γ -dance picture, given the additional introduction of premises for features like force coupling constants, fermion fields, fermion masses, specific additional symmetries of the Lagrangian and their effects (like non-Abelian gauge symmetries, implying Feynman- Fadeev-Popov ghosts), spontaneous symmetry breaking, and Cabibbo-Kobayashi-Maskawa rotations. The Feynman Path Integral usually is used with the path differential DXtxyz(, , . ) replaced by a field differential DΨ()x " including factors for all fields Ψ()x ", which themselves are functions of spacetime. Future theories that check with experiments will be required to corroborate the experiments that have secured quantum mechanics and quantum field theory, at the current level of detail; minimally at that level at least, it can be expected that they too will be consistent with the γ -dance. Perhaps in that process the γ -dance will assist in probing deeper into natural law.

XII POSTDICTION AND PREDICTION

Phenomenologically, quantum mechanics is extremely well-developed, despite a century of incomplete theoretical consistency. Major industries support many people day after day based on the successful match between empirical quantum effects and the limited theory. Therefore, in suggesting a theoretical framework to unify those quantum effects, it is vastly more likely that, if correct, the theory will postdictively accommodate, rather than predictively expand, the set of experimental data. The γ -dance picture provides a deductive model, from primitive classical premises, that produces the central features of quantum mechanics, from which the other known features already have been shown to be derivable. Those central features are the Feynman path integral, the presence of the Lagrangian within an action integral (in the form of a phase), and the existence and value of Planck’s constant. The immediate consequences include wave-particle duality, Heisenberg’s Uncertainty Principle, and the full lore of so-called wavefunction collapse. Based on these, the standard derivations in the literature, like Feynman’s derivation of the Schrödinger Equation from the path integral102, follow directly. All of these constitute successful postdiction by the γ -dance theory. There is the suggestion in the γ -dance picture that any change of geometry, anywhere in the spacetime of the realm, including upstream from the classically expected position of the particle, as well as downstream, should cause a physical signature in the measurables. Consider alterations made in the geometry of the realm while the particle is “in flight”, that is, between the times (in the lab frame) of departure from the source and arrival at the detector. A downstream effect is a measurable property caused by a change in the geometry of the realm made after the particle leaves the source (in a classical sense, from the point of view of the lab frame), but before the particle (again in a classical sense and in the lab frame) would be expected to have reached the realm geometry in question. An upstream effect is a measurable property caused by a change in the geometry of the realm made after the particle leaves the source, and after the particle (again in the classical sense and in the lab frame) would be expected to have passed the realm geometry in question, but before reaching the detector. It appears that past experimental tests (including the Aspect experiments) have extended, beyond separate experiments on distinct static geometries of the realm, only to downstream effects. Measurement of an upstream effect would have to await, in the lab frame, light-travel-time from the location of the alteration in the realm geometry to the location of the detector, thereby arriving at the detector inherently later than the classical expectation of the particle’s arrival time at the detector. Yet statistically, because of the γ -dance’s wave effects through the 34 uncertainty principle, some registrations of particle arrivals at the detector will occur later than the light- travel-time from source to detector, after the source has been shut off (in the lab frame). Of these late arrivals, it is predicted that those arrivals, after the light-travel-time from an upstream realm alteration to the detector, should show (if the γ -dance picture is physically correct) the interference effect at the detector of the altered realm geometry. It appears that the previous representations of quantum theory can predict this result103, of upstream quantum effects, but that notice has not been focused on it. Thus, the possibility of upstream quantum effects appears to constitute an additional postdiction of, and an increased emphasis (which may constitute a prediction) by, the γ -dance picture. “Apart from the theory, there is the question whether it is possible with the equipment at present available to detect”104 an upstream quantum effect. There also is a derivation from the γ -dance, in Section III.D, of the presence of antimatter in comparison to matter, as a function of relativistic velocity of the observer. This is not known to be entirely postdictive.

XIII INADEQUACIES OF THE FORMULATION

In discussing the γ -dance of massive particles, the massless kinetic term in the Lagrangian for massive particles was allowed to behave as in the massless particle case, with the effect of the mass term expanding subluminally. This separation is necessary experimentally and is justified by special relativity as in the discussion immediately following (III-25). Yet, there is the sense in the argument of some ambiguity remaining. But, this in no way alters the massless particle theory, and it is possible that the suspected ambiguity in the massive case will prove to have been illusory. Fermion masses are not understood in a fundamental sense, and gauge boson masses, as in the electroweak interaction, are only partly understood. Electroweak theory appears to require Higgs boson interaction in generation of fermion masses, to maintain the internal consistency of the theory105, but actual predictive power of fermion masses is missing from the theory. It is an item of future work in physics to understand the origin of fundamental particle masses, and especially of fundamental fermion masses. It is expected that this question is reasonably separate from the fundamental theory of the γ -dance picture. Postdictive theory presents insidious challenges for avoiding underived assertions. Yet, quantum mechanics required almost completely postdictive theory. The problem is discussed in Section XII. The antimatter implication of the γ -dance, discussed in Section III.D, has been discussed only in the context of the γ -dance, without the details of pair production and CPT symmetry. The derived proportions should be viewed as the γ -dance component. There are 15 basic premises of the γ -dance theory. The last of these is a composite of all standard rules of inference in mathematics. Of the others, most are distinctly primitive, like contrast and completeness. The premises of source, integrity, rest mass, and the 2nd Law Of Thermodynamics are somewhat less so, and may eventually be explained in terms of concepts more primitive. To the extent that the premises are distinct and primitive, can the theory be robust.

XIV SUMMARY

From a set of primitive premises embodying only classical concepts, including special relativity, a direct derivation, called the γ -dance, of the Feynman path integral, and producing wave-particle duality and Planck’s constant, has been demonstrated. The Lagrangian and its property of invariance have been shown to be necessary given the same set of premises. The Heisenberg Uncertainty Principle has been shown as necessary again from these classical concepts. Resolution of the questions of quantum entanglements and wavefunction collapse, commensurate with experiments, has been suggested by the theory. Implications with regard to observable proportions of matter and antimatter have been derived106. Despite several remaining open smaller questions mentioned in the preceding section, it is suggested that the process of the γ -dance directly produces quantum phenomena from the basis of special relativity and other concepts in classical mechanics. 35

Henceforth107 classical and quantum mechanics shall cease to exist as unique and mutually incommodious sets of natural law. Only a kind of union of the two, displaying a deeper, at once simpler and more complex108, harmony of logic and measurable knowledge, midwifed by special relativity, will preserve an independent reality. Quantum mechanics is the special relativity of self-frame effects.

XV ACKNOWLEDGEMENTS

Dedicated to the memory of Richard P. Feynman and Lionel P. Hepburn

The author thanks friend and colleague William Kells for useful criticism of an early manuscript of several concepts presented herein. Encouragement has been provided by Charles Kohlhase, Kenrick Vassall, John Paredes, Patty Michelson (in memoriam), Kendall Wicks, Donna Wicks, Richard Turco, William Kells, Lorraine Kells, Mary Jeannet (in memoriam), Donald Jeannet, Anna Jeannet, Glynn Anderson, Randy Nelson, Jason Hull, Richard Damian, Ogden Newton, James Smyth, Brigitte Curt, Stephanie Tsuchida, Leslie Shultz, and Robert Dvorak. This work has been funded independently by the author.

1 Functional simplicity, not trivial simplicity, is the referent. The simplest structure that allows any structure whatsoever appears to be the true physical, or functional, structure. Nature is minimalist. The least action principle, gauge symmetries, and Fermat’s Law give hints in this direction. Absolute nothingness, including the absence of virtual processes, without even spacetime or momentum-energy, constitutes trivial simplicity. It is not that which is implied here; even the physical vacuum, with its virtual processes, is not trivially simple. The measurable complexity is the result of (fractal) iterations of functionally simple structure. 2 Julian Schwinger, in conversation in his office at UCLA with author (while author was a student in Schwinger’s course on source theory), around 1983. 3 Feynman’s blackboard at Caltech, approx. 1978-1988. 4 R. P. Feynman, Space-Time Approach To Non-Relativistic Quantum Mechanics, Rev Mod Phys, 20 (1948), p. 267. 5 Feynman & Hibbs, Quantum Mechanics And Path Integrals, McGraw-Hill, Inc. (1965). 6 Ibidem, pp.29-31. 7 Ibidem, pp.25-39. 8 C. Huygens, Traité de la Lumière (1690), facsimile reprint, London: Dawsons of Pall Mall (1966). 9 A. Einstein, On The Electrodynamics Of Moving Bodies, Annalen der Physik 17 (1905), translated in The Principle Of Relativity, Dover Publ., Inc. (1952), pp. 37-65. 10 P. A. M. Dirac, The Lagrangian In Quantum Mechanics, Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933), reprinted in J. Schwinger, Selected Papers On , Dover Pub. Inc. (1958), pp. 312-320. 11 Ref. 4. 12 Ibidem. 13 Measurability is required for meaning. Lack of measurability implies meaninglessness. This appears to be universal. History is replete with grotesque demonstrations of the cruelty of the opposite, usually promoted in the name of superstitions (including newageisms and all religions) or nationalism or greed by itself. 14 Ancient Chinese tradition of “opposite principles, one negative, dark, and feminine; and the other positive, bright, and masculine” (definition from Random House Dictionary, Coll. Ed., Random House, Inc. (1968)). 15 Russell & Whitehead, Principia Mathematica, Introduction, Cambridge Univ. Press (about 1915). 16 Russell & Whitehead, Principia Mathematica, Cambridge Univ. Press ( about 1915). 17 P. A. M. Dirac, Quantum Mechanics, 4th Ed., Oxford Univ. Press (1967). Definition: braket: Intentional variant spelling of bracket, to indicate a joined Dirac bra and ket, in the form . The term ketbra also can be useful for indicating a joined Dirac ket and bra pair, in the form . 36

18 Ibidem, p. 63. 19 A. Einstein, The Foundation Of The General Theory Of Relativity, Annalen der Physik, 49 (1916); translation in The Principle Of Relativity, Dover Publ., Inc. (1952), pp.122-123. 20 Ref. 17, pp. 38-39. 21 Ref. 19, p.119. 22 Ref. 19, p.119. 23 Ref. 19, p. 129. 24 S. Weinberg, Gravitation And Cosmology, John Wiley & Sons, 1972, p. 99. 25 In his original paper on General Relativity (Ref. 19, pp. 120-130), Einstein derives the 4-D volume invariant using +g , and then introduces − g , “which is always real on account of the hyperbolic character of the spacetime continuum”. He discusses the inability of g to change sign in the spacetime continuum, but gives no further rationale for introducing the negative sign than to keep − g always real. Einstein (The Meaning of Relativity, Princeton Univ. Press (1970), p. 68) derives the 4-D volume invariant as: g′ dx ′ = g dx , with g = g µν , which is intrinsically negative, and uses it consistently with a negative sign inserted. 26 Eddington (The Mathematical Theory of Relativity, Cambridge Univ. Press (1965), p. 109), after having proven that (dV )2 g is an invariant, simply notes that − g is therefore also an invariant, and thereafter always uses ( −g ) whenever the natural formalism calls for g . 27 Dirac (General Theory of Relativity, Wiley-Interscience (1975), p.37) notes that “ g is a negative quantity, so we may form − g ”, without further justification for the negative sign. 28 Feynman (Feynman Lectures on Gravitation, Addison-Wesley Publ. Co. (1995), p. 121) simply identifies − g as the determinant of the linear transformation of the spacetime coordinates. 29 Weinberg (Gravitation &Cosmology, John Wiley & Sons (1972), p.98) defines g as the negative of the determinant of the metric tensor without justification. 30 For a typical contemporary discussion: L. Ryder, Quantum Field Theory, Cambridge Univ.Press (1996), pp.80-82. 31 Ibidem, pp. 83-84. 32 Ref. 7. 33 Ref. 9; and a contemporary version in Taylor and Wheeler, Spacetime Physics, W. H. Freeman and Co. (1992). 34 Ref. 9, p. 47. 35 Taylor & Wheeler (Ref 33), pp. 53-93. 36 T. S. Eliot, The Four Quartets, in Collected Poems 1909-1962, Harcourt Brace & Company, 1968. The entire poem is a symbolic discourse on the synonymity of the ending and the beginning. 37 Feynman, a particle “sniffing out …all possible paths”: Author recalls hearing or reading Feynman making a similar statement, but has not succeeded in tracing the time or place of the reference. 38 Refs. 87, 88 and 89. 39 M. Gell-Mann, The Quark and the Jaguar, W. H. Freeman and Co. (1994), pp. 129-131. 40 Thomas Young, quoted in J. Gribben’s Schrödinger’s Kittens and the Search for Reality (Back Bay Books, 1995), p. 50; Feynman’s quantum mechanical use of the 2-slit experiment: Ref. 5, chapter 1. 41 Ref. 8; Huygen’s Principle: The propogation of a wave can be modeled as spherical waves emanating, at the wave velocity, from every point in the wave front at a given instant. Interference effects will construct the actual wavefront for the next instant, including bending around corners. 42 Section II.C of this report 43 M. Born, Z. Physik 38 (1926) p. 803; W. Heisenberg, Z. Physik 43 (1927) p.172; N. Bohr, Naturwiss. 16 (1928) p. 245; 17 (1929) p. 483 and 18 (1930) p. 73. For thorough discussions: W. Heisenberg, The Physical Principles of Quantum Theory, Univ. of Chicago Press (1930); also Dover Publ. Inc.; N. Bohr, Atomic Theory And The Description Of Nature, Cambridge Univ. Press (1934). 44 once commented in a conversation with the author (at a cast party in about 1978 after one of the famous Cal-Tech musicals in which Feynman performed, and for which the author, then doing interplanetary mission work at the Jet Propulsion Laboratory, was a musician in the orchestra) that 37

“Physicists always have to invent their own mathematics. The mathematicians are interested in something else. I donno what they’re interested in.” The author asked “What about Reimannian geometry?”, and Feynman replied “Oh, that’s an exception.” Yet, it would appear that Feynman was indeed largely right, that the process of trying to model the real physical world forces us to develop particular tools that might escape us based on purely non-empirical reasoning. 45 Landau & Lifshitz, Statistical Physics, 3rd Ed. Part 1, Butterworth/Heinemann Publ. (2000), p. 29. 46 Section II.A of this report. 47 L. Ryder, Quantum Field Theory, Cambridge Univ. Press (1996), Chapters 3 and 7. 48 S. Weinberg, The Quantum Theory Of Fields, Volume II, Cambridge Univ. Press (1996), Chapter 21. 49 Electroweak theory suggests a connection between fundamental fermion masses and spontaneous symmetry breaking, but without any predictive power. For example, Peskin and Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley Publ. Co. (1995), pp. 713-715. 50 A contemporary text: E. Taylor & J. Wheeler, Spacetime Physics, W. H. Freeman and Co. (1992), p. 211. 51 Ref. 48. 52 Ref. 48. 53 G. Arfken, Mathematical Methods For Physicists, Third Ed., Academic Press Inc. (1985). 54 Ref. 17. 55 See proof at beginning of Section III.B of this report. 56 Ibidem. 57 Wheeler referred to a possible “smoking gun” to explain quantum mechanics; it appears that Wheeler’s smoking gun is the γ -dance. In fantasy, one envisions Texan Wheeler in his 37.85-liter hat holding a smoking gun inside a saloon, under a big ornate wild-west sign emblazoned with “γ -Dance”, where barmaids and cowboys are doing a dance that requires each to bump into all of the others on the floor constantly. Was the gun fired to initiate the dancing of the γ , or was it over a devious hidden variable in a game of chance? 58 The γ -dance differs from the Wheeler-Feynman theory of half-retarded/half-advanced waves (J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17 (1945), p. 157), in that, instead of postulating a retarded and an advanced wave that interfere, in the γ -dance the wave itself in the lab frame is deduced from premises, and the instantaneity occurs in the direct Lorentz contact of the massless particle self-frame. Yet, the Wheeler-Feynman theory did help anchor the need for dLc. J. Gribbin’s discussion of the Wheeler- Feynman theory and related theories in Schrödinger’s Kittens and the Search for Reality (Back Bay Books, 1995) is considered excellent. 59 Ref. 4. 60 A. Einstein, The Meaning of Relativity, Princeton Univ. Press, Fifth Ed. (1953), pp.37-47. 61 Ref 4. 62 Ref. 3. 63 Ref. 47, p.82. 64 Stückelberg, Helv. Phys. Acta. 15, 23 (1942); R. P. Feynman, Phys. Rev. 74, 939 (1948). 65 P. A. M. Dirac, Theorie Du Positron, Rapport du 7e Conseil Solvay de Physique, Structure et Proprieté des Noyaux Atomiques, p. 203 (1934). 66 M. Peskin & D. Schroeder, An Introd. To Quantum Field Theory, Addison-Wesley Publ. Co. (1995), Section 20.3. 67 Ref. 13. 68 Ref. 66, p. xix (Notations and Conventions). 69 Particle Physics Booklet, Particle Data Group, Springer (July 2000), p. 4. 70 Section III.A of this report. 71 One common alternative approach is to discuss commutation relations, as in the text by Robinett, Quantum Mechanics, Oxford Univ. Press, 1997, pp. 280-282. 72 And at least to minimize what M. Gell-Mann refers to as “flapdoodle” (The Quark and the Jaguar, W. H. Freeman and Co.,1994, pp. 167-176. 73 Paraphrased from Messiah, Quantum Mechanics, Vol. I, North-Holland Publ. Co., Amsterdam, 1968, p. 130. 74 Paraphrased from Ref 5, p. 45. 75 Ibidem, p. 27. 38

76 Ibidem, p. 28. 77 It may be that a search for gravity that has a specifically quantum nature is a slightly misdirected effort: The general-relativistic γ -dance may generate a new form of mechanics that is inherently different from the known quantum mechanics. 78 H. Everett, III, Relative State Formulation In Quantum Mechanics, Re. Mod. Phys. 29 (1957), pp. 454- 462; Y. Ben-Dov, Everett’s Theory And The ‘Many Worlds’ Interpretation, Amer. J. Phys. 58 (1990), pp. 829-832. 79 G. Greenstein & A. Zajonc, The Quantum Challenge, Jones and Bartlett Publ. (1997), pp. 166-171. 80 The term “dLc-mediated event” is used pointedly in place of the popular term “nonlocal event”. The latter is an unfortunate malapropism seeming to imply superluminal signal velocity, and is completely unnecessary in the γ -dance picture; what is really occurring is direct Lorent contact (Section III). Gell- Mann’s objection (Ref. 72, p.173) to the use of the label “nonlocal” is upheld. 81 Ref. 36. 82 Ref.2. 83 Feynman and Hibbs, Quantum Mechanics and Path Integrals, pp. 76-84, McGraw-Hill, Inc., 1965. 84 Rumors of the inherent demise of reasoning, the failure of logic, the incomprehensibility of the real world, and a host of other intellectual armageddonisms, based on the phenomenon of wavefunction collapse, were perhaps premature. 85 J. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics, vol. 1, pp. 195-200 (1964); “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys., vol. 38, pp. 447-452 (1966); A. Greenstein & A. Zajonc, The Quantum Challenge, pp. 117-124, Jones & Bartlett Publ. (1997). 86 Greenstein & Zajonc, ibidem, Chapter 6. 87 A. Aspect, P. Grangier and G. Roger, Experimental Tests Of Realistic Local Theories Via Bell’s Theorem, Phys. Rev. Lett., 47 (1981), pp. 460-463. 88 A. Aspect, P. Grangier and G. Roger, Experimental Realization Of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation Of Bell’s Inequalities, Phys. Rev. Lett., 49 (1982), pp.91-94. 89 A. Aspect, J. Dalibard and G. Roger, Experimental Test Of Bell’s Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49 (1982), pp. 1804-1808. 90 It is one thing for an associate experimenter to promise to perform a measurement or alter a setting at a particular pre-synchronized and parallel-transported time, but it is a profoundly different thing to verify that the promised deed has indeed been done. The latter cannot be done more rapidly than in light travel time. A cornerstone in the consistency of human rationality should not be left to rest on whether or not the concept of a research associate in a thought experiment or a real research associate in a physical laboratory kept its word. 91 Greenstein & Zajonc, Ref. 85, Chapter 4 and references therein. 92 E. Kolb and M. Turner, The Pocket Cosmology, Review of Particle Properties, The European Physics Journal C, Vol. 15, No. 1-4, 2000, p 125 93 Discovered originally by Lorentz (H. Lorentz, Versuch Einer Theorie Der Elektrischen Und Optischen Erscheinungen In Bewegten Körpen, Leiden (1895) §§ 89-92; translated in The Principle Of Relativity, Dover (1952), pp. 1-7) and later, independently, by Einstein (Ref. 9). 94 Sections II through IX of this report, and Ref. 4. 95 P. A. M. Dirac, The Principles Of Quantum Mechanics, Fourth Edition, Oxford Univ. Press (1967), pp.263-269. 96 W. Pauli, The Connection Between Spin And Statistics, 58 (1940), p. 716; reprinted in Schwinger, Ref. 10, pp.372-378. 97 Ref 95, p. 211. 98 Ref. 95. 99 Ref. 4. 100 A. Messiah, Quantum Mechanics, Vols I & II, John Wiley & Sons (1966); or an elementary text: Robinett, Quantum Mechanics, Oxford Univ. Press (1997). 101 Refs. 47, 48, and 49. 102 Ref 4. 39

103 It is possible that previous theories are compatible with this result, but that the possibility of upstream quantum effects was not as obvious owing to the structure of those theories, with the consequence that corresponding experiments would not have been tried. Future consideration on this point may prove useful. 104 The wording is borrowed, with respect for his minimalist approach and appreciation of experimental process, from the ending of Einstein’s paper On the Influence of Gravitation on the Propagation of Light, Annalen der Physik, 35, 1911. 105 Ref. 49, pp. 713-715; and I. Aitchison & A. Hey, Gauge Theories In Particle Physics, Adam Hilger (1989) pp. 463-466. 106 Feynman’s comments on antimatter in Elementary Particles And The Laws Of Physics, The 1986 Dirac Memorial Lectures, Cambridge Univ. Press (1987), appear to be commensurate with the antimatter implications of the γ -dance. 107Paraphrase of H. Minkowski, in Space and Time, a translation of an address delivered at the 80th Assembly of German Natural Scientists and Physicians, at Cologne, 21 September, 1908, reprinted in The Principle of Relativity, by Lorentz, Einstein, Minkowski, and Weyl, Dover Publications, Inc., 1952. 108 As Ernest Hemingway spoke through his autobiographical character in his posthumously published novel The Garden Of Eden: “It is all well and good to write simply, but don’t think so damned simply. Think the complex thought, and then express it simply.”