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arXiv:quant-ph/9412012

Quantization Failure in Unified Field Theories

Daniel C. Galehouse

Published Jan 1, 1995• 2 citations
arXivHTMLPDF

Abstract

Studies of geometrical theories suggest that fundmental problems of quantization arise from the disparate usage of displacement operators. These may be the source of a concealed inconsistency in the accepted formalism of quantum physics. General relativity and related theories cannot be quantized by the classical procedure. It is necessary to avoid the construction of differential equations by operators applied algebraically. For such theories, Von Neumann's theorem concerning hidden variables is avoided. A specified alternative class of gravitational-quantum-electrodynamic theories is possible.

DOI
10.48550/arxiv.quant-ph/9412012

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Studies of geometrical theories suggest that fundmental problems of quantization arise from the disparate usage of displacement operators. These may be the source of a concealed inconsistency in the accepted formalism of quantum physics. General relativity and related theories cannot be quantized by the classical procedure. It is necessary to avoid the construction of differential equations by operators applied algebraically. For such theories, Von Neumann's theorem concerning hidden variables is avoided. A specified alternative class of gravitational-quantum-electrodynamic theories is possible.
The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.
It has recently been argued that quantization can be established within classical theory as a consequence of lost information. In this view, classical mechanics is regarded as a union of quantum mechanics and what are called 'hidden variables'. Hidden variable theories were first considered some years ago and abandoned because they did not produce new physics. They have now been revived however, because they appear to provide a logically sound means of relating classical and quantum theories. It is argued here that the Heisenberg uncertainty relations constitute a necessary condition for gauge invariance in the 5-dimensional Kaluza-Klein theory, where the fifth dimension of 5-spacetime is hidden as a 5-dimensional theory of general relativity is projected onto 4-spacetime.
This short document presents no new results. It is a collection of related ideas that may lead to a way of quantizing gravity. To achieve this it may be relevant to use my recent new approach to quantum theory, which is based on quantum events. Given that most standard quantum texts do not define a mathematical object which corresponds to a physical event, the definition of a quantum event is motivated and discussed in detail. This in turn leads to the idea that the mathematical object that contains enough quantum events for describing a quantum system is a von Neumann algebra. The next idea is that the kinematics of special relativity (as embodied in flat Minkowski space) should be quantized as a first step in the program of the quantization of general relativity, that is, quantizing Einstein's equation. It must be emphasized that the events of special relativity are viewed as part of the kinematics, rather than of the dynamics of that theory. The same holds for general relativity. Similarly, I am viewing quantum events as part of the kinematics of quantum theory, even though some find this to be counterintuitive. An Appendix shows how ``collapse'' is easily understood in this new approach to quantum theory.
Forty years after the advent of quantum mechanics the problem of hidden variables, that is, the possibility of imbedding quantum theory into a classical theory, remains a controversial and obscure subject. Whereas to most physicists the possibility of a classical reinterpretation of quantum mechanics remains remote and perhaps irrelevant to current problems, a minority have kept the issue alive throughout this period. (See Freistadt [5] for a review of the problem and a comprehensive bibliography up to 1957.) As far as results are concerned there are on the one hand purported proofs of the non-existence of hidden variables, most notably von Neumann’s proof, and on the other, various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2]. One of the difficulties in evaluating these contradictory results is that no exact mathematical criterion is given to enable one to judge the degree of success of these proposals. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this review, we would like to highlight the three known no-go theorems in quantum physics in relation to the processof quantization that maps classical observables to quantum ones. The quantization approach considered is a mixture ofIsham’s group-theoretic quantization and geometric quantization with special emphasis on underlying compact phasespace geometry of spheres. The first is Groenewold-van Hove theorem that states the obstruction of quantizing the fullalgebra of observables and in the sphere case, only limited to the spin observables plus the constant functions. Theother two are theorems of Bell and Kochen-Specker stating that the only hidden variable theories allowed by quantumphysics are nonlocal and contextual ones. We give simple examples of these no-go theorems and indicate someinteresting problems arising from them for the field of quantization.
With all its remarkable achievements, Quantum Mechanics as established in the fundamental papers written by Heisenberg, de Broglie, Schro¨dinger, Dirac, etc. contains some inconsistencies and internal contradictions. Quantum mechanics deals with quantum processes in which the number and the type of particles do not vary during their interaction, but only their energy and momentum states. The formalism of Quantum Mechanics is not capable of describing neither the variation of the number of particles, nor the transformation of one type of particle into another type. Even if in some cases the number of particles is allowed to change, this is somewhat forced and inconsistent. For example, Quantum Mechanics is not able to give a satisfactory explanation of the spontaneous emission, in which an electron situated on an excited energy level emits a quanta γ and decays to a lower energy level.