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arXiv:quant-ph/9705024

Origin of quantum randomness in the pilot wave quantum mechanics

Yuri Shtanov

Published May 14, 1997• 3 citations
arXivHTMLPDF

Abstract

We account for the origin of the laws of quantum probabilities in the de Broglie-Bohm (pilot wave) formulation of quantum theory by considering the property of ergodicity likely to characterise the dynamics of microscopic quantum systems.
DOI
10.48550/arxiv.quant-ph/9705024

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We study the origin of the Born probability rule ρ = | ψ | 2 in the de Broglie–Bohm pilot–wave formulation of quantum theory. It is argued that quantum probabilities arise dynamically, and have a status similar to thermal probabilities in ordinary statistical mechanics. This is illustrated by numerical simulations for a two–dimensional system. We show that a simple initial ensemble, with a non–standard distribution ρ ≠ | ψ | 2 of particle positions, evolves towards the quantum distribution to high accuracy. The relaxation process ρ →| ψ | 2 is quantified in terms of a coarse–grained H –function (equal to minus the relative entropy of ρ with respect to | ψ | 2 ), which is found to decrease approximately exponentially over time, with a time constant that accords with a simple theoretical estimate.
The de Broglie-Bohm (pilot wave) formulation of quantum theory appears to be free from the conceptual problems specific to quantum mechanics (problem of measurement) and to quantum cosmology (problem of time). We further study the extension of this formulation to the case of gravity and demonstrate that the foliation of spacetime by spacelike hypersurfaces obtained during the solution of the quantum problem turns out to be distinguished in general. This means that quantum pilot wave dynamics is not invariant with respect to an arbitrary change of foliation, or, in other terms, that quantum nonlocality takes place. We also discuss the general structure of a realistic pilot wave theory that could describe the universe.
As we discussed in Section 2.4, in his Solvay lecture of 1927 de Broglie presented the pilot-wave dynamics of a non-relativistic many-body system, and outlined some simple applications of his ‘new dynamics of quanta’ (to interference, diffraction and atomic transitions). Further, as we saw in Section 10.2, contrary to a widespread misunderstanding, in the general discussion de Broglie's reply to Pauli's objection contained the essential points needed to treat inelastic scattering (even if Fermi's misleading optical analogy confused matters): in particular, de Broglie correctly indicated how definite quantum outcomes in scattering processes arise from a separation of wave packets in configuration space. We also saw in Section 10.4 that de Broglie was unable to reply to a query from Kramers concerning the recoil of a single photon on a mirror: to do so, he would have had to introduce a joint wave function for the photon and the mirror. De Broglie's theory was revived by Bohm 25 years later (Bohm 1952a,b) (though with the dynamics written in terms of a law for acceleration instead of a law for velocity). Bohm's truly new and very important contribution was a pilot-wave account of the general quantum theory of measurement, with macroscopic equipment (pointers, etc.) treated as part of the quantum system. In effect, in 1952, Bohm provided a detailed derivation of quantum phenomenology from de Broglie's dynamics of 1927 (albeit with the dynamical equations written differently).
The de Broglie-Bohm pilot-wave theory promises not only a realistic description of the microscopic world (in particular, a description in which observers and observation play no fundamental role) but also the ability to derive and explain aspects of the quantum formalism that are, instead, (awkwardly and problematically) postulated in orthodox versions of quantum theory. Chief among these are the various "measurement axioms" and in particular the Born rule expressing the probability distribution of measurement outcomes. Compared to other candidate non-orthodox quantum theories, the pilot-wave theory suffers from something of an embarrassment of riches in regard to explaining the Born rule statistics, in the sense that there exist, in the literature, not just one but two rather compelling proposed explanations. This paper is an attempt to critically review and clarify these two competing arguments. We summarize both arguments and also survey some objections that have been given against them. In the end, we suggest that there is somewhat less conflict between the two approaches than existing polemics might suggest, and that indeed elements from both arguments may be combined to provide a unified and fully-compelling explanation, from the postulated dynamical first principles, of the Born rule.
The original version of the de Broglie-Bohm pilot-wave theory, also called Bohmian mechanics, attempted to treat the wave function or pilot wave as a part of the physical ontology of nature. More recent versions of the de Broglie-Bohm theory appearing in the last few decades have tried to regard the pilot wave instead as an aspect of the theory's nomology, or dynamical laws. This paper argues that neither of these views is correct, and that the de Broglie-Bohm pilot wave is best understood as a collection of latent variables in the sense of a hidden Markov model, a construct that was not available when de Broglie and Bohm originally formulated what became their pilot-wave theory. This paper also discusses several other challenges for the ontological view of the pilot wave. One such challenge is due to Foldy-Wouthuysen gauge transformations, which connect up with the Deotto-Ghirardi ambiguity in the de Broglie-Bohm theory. Another challenge arises from the freedom to carry out canonical transformations in the wave function's own notion of phase space, as defined by Strocchi and Heslot.