1 ON THE ORIGIN OF QUANTUM MECHANICS C. Darwin Hepburn 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. 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 Light................................................................................................ 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 UNCERTAINTY PRINCIPLE.................................................................................22 VIWAVE-PARTICLE DUALITY.......................................................................................................24 VI.AThe Wavelength Of A Particle.................................................................................................24 VI.BIntegrity Of Particle At Source And Detection Points............................................................25 VI.CDouble Slit Example................................................................................................................26 VI.DPhoton Emission From Electron 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 phase 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 speed of light (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 operator, 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