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arXiv:quant-ph/0609109

Could quantum mechanics be an approximation to another theory?

Lee Smolin

Published Sep 14, 2006• 47 citations
arXivHTMLPDF

Abstract

We consider the hypothesis that quantum mechanics is an approximation to another, cosmological theory, accurate only for the description of subsystems of the universe. Quantum theory is then to be derived from the cosmological theory by averaging over variables which are not internal to the subsystem, which may be considered non-local hidden variables. We find conditions for arriving at quantum mechanics through such a procedure. The key lesson is that the effect of the coupling to the external degrees of freedom introduces noise into the evolution of the system degrees of freedom, while preserving a notion of averaged conserved energy and time reversal invariance. These conditions imply that the effective description of the subsystem is Nelson's stochastic formulation of quantum theory. We show that Nelson's formulation is not, by itself, a classical stochastic theory as the conserved averaged energy is not a linear function of the probability density. We also investigate an argument of Wallstrom posed against the equivalence of Nelson's stochastic mechanics and quantum mechanics and show that, at least for a simple case, it is in error.

DOI
10.48550/arxiv.quant-ph/0609109

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We consider the hypothesis that quantum mechanics is an approximation to another, cosmological theory, accurate only for the description of subsystems of the universe. Quantum theory is then to be derived from the cosmological theory by averaging over variables which are not internal to the subsystem, which may be considered non-local hidden variables. We find conditions for arriving at quantum mechanics through such a procedure. The key lesson is that the effect of the coupling to the external degrees of freedom introduces noise into the evolution of the system degrees of freedom, while preserving a notion of averaged conserved energy and time reversal invariance. These conditions imply that the effective description of the subsystem is Nelson's stochastic formulation of quantum theory. We show that Nelson's formulation is not, by itself, a classical stochastic theory as the conserved averaged energy is not a linear function of the probability density. We also investigate an argument of Wallstrom posed against the equivalence of Nelson's stochastic mechanics and quantum mechanics and show that, at least for a simple case, it is in error.
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Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.
We derive the classical Hamilton-Jacobi equation from first principles as the natural description for smooth stochastic processes when one neglects stochastic velocity fluctuations. The Schrödinger equation is shown to be the natural exact equation for describing smooth stochastic processes. In particular, processes with up to quadratic stochastic fluctuations are electromagnetically coupled quantum point particles. The stochastic derivation offers a clear geometric picture for Quantum Mechanics as a locally realistic hidden variable theory. While that sounds paradoxical, we show that Bell's formula for local realism is incomplete. If one includes smooth stochastic fluctuations for the hidden variables, local realism is preserved and quantum mechanics is obtained. Quantum mechanics should therefore be viewed as a "nondeterministic, non-Bell locally realistic hidden variable theory". Since the description is simply a stochastic process, it should be relatively straightforward to create mesoscopic analogue systems that show all the hallmarks of Quantum Mechanics, including super-Bell correlations. In fact, any system that can be described by the linear time evolution of a density matrix is both a stochastic and a Quantum system from our point of view, since we show that the existence of a transition probability density directly implies the existence of a density matrix. Systems for which the stochastic degrees of freedom vary smoothly over time have quantum Hamiltonians with standard kinetic terms.
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system's initial condition and dynamics. Such sets of histories are said to decohere. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained. We describe how a quasiclassical domain described by averages of densities of approximately conserved quantities could be an emergent feature of an initial condition of the universe that implies the approximate classical behavior of spacetime on accessible scales.
Quantum mechanics is traditionally formulated in terms of postulates inferred from experimentalobservations. Here we propose that the statistical regularities of quantum mechanics arise as macro-scopic manifestations of stochastic fluctuations of dynamical parameters in microscopic systems gen-erated by a non-inertial physical background. Although the specific nature of this background mayvary, the universality of quantum statistics is ensured by invariant properties of stochastic dynam-ics - analogous to the role of the equivalence principle in general relativity - and by the existence ofa fundamental action scale, Planck constant h. By extending classical mechanics to include higher-order derivatives induced by non-inertial fuctuations and imposing a stability condition on admissibletrajectories, the minimal action scale h naturally emerges. In this framework quantum mechanicsappears as an effective statistical description of systems embedded in a fluctuating non-inertial envi-ronment while preserving all standard predictions of the theory.