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arXiv:quant-ph/0406134

A Newtonian Hidden Variable Theory

Bruno Galvan

Published Jun 18, 2004• 0 citations
arXivHTMLPDF

Abstract

A new hidden variable theory is proposed, according to which particles follows definite trajectories, as in Bohmian Mechanics or Nelson's stochastic mechanics; in the new theory, however, the trajectories are classical, i.e. Newtonian. This result is obtained by developing the following concepts: (i) the essential elements of a hidden variable theory are a set of trajectories and a measure defined on it; the Newtonian HCT will be defined by giving these two elements. (ii) The universal wave function has a tree structure, whose branches are generated by the measurement processes and are spatially disjoined. (iii) The branches have a classical structure, i.e. classical paths go along them; this property derives from the fact that the paths close to the classical ones give the main contribution to the Feynman propagator. (iv) Classical trajectories can give rise to quantum phenomena, like for instance the interference phenomena of the two-slit experiment, by violating the so called Independence Assumption, which is always implicitely made in the conceptual analysis of these phenomena.

DOI
10.48550/arxiv.quant-ph/0406134

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A new hidden variable theory is proposed, according to which particles follows definite trajectories, as in Bohmian Mechanics or Nelson's stochastic mechanics; in the new theory, however, the trajectories are classical, i.e. Newtonian. This result is obtained by developing the following concepts: (i) the essential elements of a hidden variable theory are a set of trajectories and a measure defined on it; the Newtonian HCT will be defined by giving these two elements. (ii) The universal wave function has a tree structure, whose branches are generated by the measurement processes and are spatially disjoined. (iii) The branches have a classical structure, i.e. classical paths go along them; this property derives from the fact that the paths close to the classical ones give the main contribution to the Feynman propagator. (iv) Classical trajectories can give rise to quantum phenomena, like for instance the interference phenomena of the two-slit experiment, by violating the so called Independence Assumption, which is always implicitely made in the conceptual analysis of these phenomena.
The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.
Quantum mechanics is often praised as a theory which unifies classical mechanics and classical wave theory. Quanta are said to behave either as particles or waves, depending on the type of experiment they are subjected to. But where in the standard formalism can a pointlike particle structure actually be found? Perhaps only to some degree in the reduction postulate applied to position measurements. In reaction to this unsatisfactory state of affairs, Bohmian mechanics introduces a mathematically precise particle concept into quantum mechanical theory. The fuzzy wave functions are supplemented by sharp particle world-lines. Through this additional structure some quantum phenomena like the double slit experiment have lost their mystery.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.
In this paper, I describe a general strategy for deriving Newtonian mechanics from Bohm’s theory in the classical macroscopic regime. The strategy is based on two main steps: (i) the use of open systems, leading to the formation of independent subsystems; (ii) the analysis of the quantum potential for the emergent subsystems. The first step leads to and explains (within the Bohmian framework) the well-known phenomenon of decoherence. The second one provides a precise criterion for the emergence of Newtonian trajectories. In particular, I will show that, under reasonable classicality conditions, the quantum potential of the emergent subsystems becomes negligible. As a consequence, these systems will follow an approximately Newtonian dynamics.
Quantum uncertainty emerged from two strikingly different routes to quantum mechanics. Heisenberg's matrix mechanics replaced the classical description of atomic orbits by relations among observable transition quantities and led, in 1927, to the uncertainty relation and its influential measurement-based interpretation. Schrödinger's wave mechanics grew instead from the Bose--Einstein and de Broglie developments and restored a continuous space-time description through matter waves. The two formalisms were soon shown to be closely related, but neither historical route by itself settles the ontological question of whether particles may possess underlying trajectories. After reviewing these developments, including Einstein's criticism of a simple observability doctrine and Mott's wave-mechanical account of straight cloud-chamber tracks, we distinguish precise quantum uncertainty relations from the stronger claim that quantum mechanics excludes definite microscopic histories. We then compare Feynman paths, Bohmian trajectories, Nelson's stochastic mechanics, and finite-speed persistent Goldstein--Kac dynamics. The Kac process has continuous, piecewise differentiable finite-speed paths; its diffusion limit approaches Wiener kinematics, while analytic continuation of its two-sector dynamics is related to Dirac evolution. These examples show that quantum uncertainty relations constrain observable statistics without, by themselves, uniquely determining an underlying path geometry.
Maintaining the position that the wave function ψψψ provides a complete description of state, the traditional formalism of quantum mechanics is augmented by introducing continuous trajectories for particles which are sample paths of a stochastic process determined (including the underlying probability space) by ψψψ. In the resulting formalism, problems relating to measurements and objective reality are solved as in Bohmian mechanics (without sharing its weak points). The pitfalls of Nelson's stochastic mechanics are also avoided.