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Journal of Mathematical Physics

Symplectic hidden-variables theories—The missing link in algebraic contextual approaches

Mićo Đurđević, Milan Vujičić, Fedor Herbut

Published Nov 1, 1991• 4 citations
Publisher

Abstract

The three known demonstrations that there exist contextual hidden variables (HV) that reproduce quantum mechanics (the HV theories of Bohm–Bub, Wiener–Siegel, and Gudder) are shown to be special cases of a generalized Gudder’s theory defined in this work. Within this scheme, a symplectic HV theory is introduced for which it is argued that it is physically more useful than Gudder’s theory because it makes possible the definition of a Hamiltonian subquantum dynamical law that reproduces in a natural way the Schrödinger equation. On the other hand, the symplectic HV theory contains as concrete realizations a slightly modified theory of Bohm–Bub and the theory of Wiener and Siegel.

DOI
10.1063/1.529055

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The three known demonstrations that there exist contextual hidden variables (HV) that reproduce quantum mechanics (the HV theories of Bohm–Bub, Wiener–Siegel, and Gudder) are shown to be special cases of a generalized Gudder’s theory defined in this work. Within this scheme, a symplectic HV theory is introduced for which it is argued that it is physically more useful than Gudder’s theory because it makes possible the definition of a Hamiltonian subquantum dynamical law that reproduces in a natural way the Schrödinger equation. On the other hand, the symplectic HV theory contains as concrete realizations a slightly modified theory of Bohm–Bub and the theory of Wiener and Siegel.
An abstract definition of a general hidden-variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm-Bub hidden-variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden-variable impossibility proofs. The general definition is used in the consideration of quantum-mechanical ordering and the measurement process.
We propose two quantum experiments - modified Bell tests - that could detect contextual hidden variables underlying quantum mechanics. The experiments are inspired by hydrodynamic pilot-wave systems that mimic a wide range of quantum effects and exhibit a classical analog of contextuality. To justify the experiments, we show that contextual hidden variables are 'physics as usual' if a unification between quantum mechanics and general relativity is possible. Accordingly, contextual theories can bypass Bell's theorem in a way that is both local and non-conspiratorial. We end with a note on the relevance of exploratory experiments in the foundations of quantum physics.
Abstract We propose two quantum experiments – modified Bell tests – that could detect contextual hidden variables underlying quantum mechanics. The experiments are inspired by hydrodynamic pilot-wave systems that mimic a wide range of quantum effects and exhibit a classical analog of contextuality. To justify the experiments, we show that contextual hidden variables are inevitable and ‘physics as usual’ if a unification between quantum mechanics and general relativity is possible. Accordingly, contextual theories can bypass Bell’s theorem in a way that is both local and non-conspiratorial. We end with a note on the relevance of exploratory experiments in the foundations of quantum physics.
We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite‐dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite‐dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows.
An hidden variable (hv) theory is a theory that allows globally dispersion free ensembles. We demonstrate that the Phase Space formulation of Quantum Mechanics (QM) is an hv theory with the position q, and momentum p as the hv. Comparing the Phase space and Hilbert space formulations of QM we identify the assumption that led von Neumann to the Hilbert space formulation of QM which, in turn, precludes global dispersion free ensembles within the theory. The assumption, dubbed I, is: "If a physical quantity A\mathbf{A}A has an operator A^\hat{A}A^ then f(A)f(\mathbf{A})f(A) has the operator f(A^)f(\hat{A})f(A^)". This assumption does not hold within the Phase Space formulation of QM. The hv interpretation of the Phase space formulation provides novel insight into the interrelation between dispersion and non commutativity of position and momentum (operators) within the Hilbert space formulation of QM and mitigates the criticism against von Neumann's no hidden variable theorem by, virtually, the consensus.