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Canadian Journal of Mathematics

Multiple Quantization

Adrian E. Scheidegger

Published Jan 1, 1953• 0 citations
Publisher

Abstract

The efforts of most theoretical physicists of this century have been directed towards that branch of the physical science which is commonly called “Quantum Theory.” Physically, Quantum Theory was postulated because of a vast amount of physical evidence which led to the postulates of states, observables, superposition, and commutation relations. From these four postulates, all quantum mechanics follows.

DOI
10.4153/cjm-1953-003-6

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The efforts of most theoretical physicists of this century have been directed towards that branch of the physical science which is commonly called “Quantum Theory.” Physically, Quantum Theory was postulated because of a vast amount of physical evidence which led to the postulates of states, observables, superposition, and commutation relations. From these four postulates, all quantum mechanics follows.
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent field-theoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck’s quantum of action and soon incarnated in quantization condition (commutation or anticommutation relations) and uncertainty relations.
Quantum mechanics is essentially a 20th century development that is based on a number of observations that defied classical explanations. While some of these experiments have semiclassical explanations, the triumph of quantum mechanics is that it gives precise verification for an overwhelming number of experimental observations. Its extensions into relativistic quantum mechanics through quantum electrodynamics (QED), electro-weak theory, and quantum chromodynamics (QCD) have led us to the Standard Model of today. While the number of unanswered questions remains approximately constant at each stage of development, the number of answered questions that relate theory with experiment continues to grow rapidly. In this chapter, some of the historical landmarks in the development of the theory are noted. The resolution of the dilemmas presented by classical theory will be dealt with in later chapters, but these are listed to motivate the break from classical mechanics. Because of the apparently unphysical nature of the postulates upon which quantum mechanics is founded, we supply motivation and justification for this break. In the first part of this chapter, we will review some of the experiments that confounded classical theory, and then introduce a formalism that provides some rationale for the postulates upon which quantum mechanics is based. In the end, these postulates will stand on their own.
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It is well recognized that two of the foundational theories of modern physics, relativity and quantum mechanics, are inconsistent. Efforts to overcome this through intensive pursuit of unifying theoretical frameworks or improvements in experimental precision as pursued by Robertson-Mansouri-Sexl and Standard-Model-Extension test theory protocols, have so far proven unsuccessful. Nonetheless, empirical evidence deriving from investigations inspired by the twentieth-century 'EPR' polemic, can now claim to have identified 'locality' and 'separability' as notions integral to special relativity theory inconsistent with quantum mechanics and experimental evidence supporting it. Following intimations of J.S. Bell, L. Smolin, Maudlin and others, the history and philosophy of science are engaged as research tools to uncover the origins of the assumptions underlying such incompatibility. In particular, known but prevalent fallacies and the deprecation of processes as described by Whitehead, together with recent insights into the functional distinctions between representative models and explanatory theories, are engaged. Analysis of the evolution of optical and electromagnetic theorizing is then suggested to demonstrate that the choice of a mechanically compatible model of electromagnetic radiation, as has conventionally underwritten development of the light-speed postulate of special relativity, is inappropriate. Pursuing options made available within a multiple models idealization of optical and electromagnetic phenomena, replacement of the sponsoring model by a nonlocal and non-separable wavefront process of radiation of the form advanced by Fresnel and Kirchhoff on the basis of Huygens' principle, suggests the possibility of reformulation of relativity theory without incurring such conflicts. However, choice of a wavefront electrodynamic model including the extinction theorem of Ewald and Oseen, equally suggests possible theory reformulation without necessitating a four-dimensional construct of space and time or postulating an absolute constancy of light-speed in vacuum. Such a constancy can in turn be considered as demonstrating an innate limitation on empirical observation resulting from the loss of relative velocity information during electromagnetic transmission due to the memorylessness of the Markov propagation process of refraction together with information flow discontinuity resulting from dielectric boundary extinction events. Such characteristics are shown to be applicable to electromagnetic radiation of all frequencies. Further, from a wavefront reformulated hypothesis of inertial relative motion, it appears possible to demonstrate that the Galilean principle of relativity can be preserved. However, Lorentz invariance would then require reinterpretation wherein the Lorentz transform group would be deemed to quantify the degree of information distortion inherently experienced by any empirical enquiry into electrodynamics involving relative motion.