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arXiv:0910.0344

Further on Pilot-Wave Theories

Elemér Elad Rosinger

Published Oct 2, 2009• 0 citations
arXivHTMLPDF

Abstract

In [2], a detailed argument is presented on a version of pilot-waves, given by at Theory of Exclusively Local Beables. What the author of [2] considers to be his crucial proposal is described in the title of section 3 as 'complicated, ugly, and highly contrived'. The reason for such a proposal is claimed to be the widespread perception among quantum physicists, according to which 'those sympathetic to the pilot-wave ontology no doubt expected that, for a system of N particles moving in three spatial dimensions, the theoretical description would be of N wave-particle pairs - each pair consisting of a point particle guided in some way by an associated wave propagating in 3-space. But Schroedinger's wave function for such an N-particle system was emphatically not a set of N (interacting) waves, each propagating in 3-space. It was, rather, a single wave propagating in the 3N-dimensional configuration space for the system', [2, p. 5]. In the present paper it is argued that the mentioned widespread expectation need not be seen as being betrayed by what actually happens. In particular, the construction in section 3 of [2] may be avoidable.

DOI
10.48550/arxiv.0910.0344

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In [2], a detailed argument is presented on a version of pilot-waves, given by at Theory of Exclusively Local Beables. What the author of [2] considers to be his crucial proposal is described in the title of section 3 as 'complicated, ugly, and highly contrived'. The reason for such a proposal is claimed to be the widespread perception among quantum physicists, according to which 'those sympathetic to the pilot-wave ontology no doubt expected that, for a system of N particles moving in three spatial dimensions, the theoretical description would be of N wave-particle pairs - each pair consisting of a point particle guided in some way by an associated wave propagating in 3-space. But Schroedinger's wave function for such an N-particle system was emphatically not a set of N (interacting) waves, each propagating in 3-space. It was, rather, a single wave propagating in the 3N-dimensional configuration space for the system', [2, p. 5]. In the present paper it is argued that the mentioned widespread expectation need not be seen as being betrayed by what actually happens. In particular, the construction in section 3 of [2] may be avoidable.
Quantum theory and the de Broglie – Bohm pilot-wave theory are empirically equivalent. In addition to other objections to the pilot-wave theory, many physicists (and some philosophers) take this to be enough to dismiss the pilot-wave theory, as they say it adds nothing to the standard theory. In this short paper I review some objections and replies to the pilot-wave theory. In particular I respond to the empirical equivalence challenge arguing that, given their mutual relationship, there is no reason to expect the two theories to make different predictions, even if they actually might.
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).
One of the major concerns of Schrodinger, Lorentz, Einstein, and many others about the wavefunction is that it is defined on the 3N3\mathbf{N}3N-dimensional configuration space, rather than on the 333-dimensional physical space. This gives the impression that quantum mechanics cannot have a 333D-space or spacetime ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wavefunction as a physical entity, in particular the many worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory. Here, a representation of the many-particle states is given, as multi-layered fields defined on the 333-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wavefunctions as living in the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the non-explicit and informal intuition of many working quantum physicists, who, by considering the wavefunction sometimes to be defined on the configuration space, and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schrodinger cats just like the usual one. But it may help various interpretations to solve these problems, through inclusion of the wavefunction as (part of) their primitive ontology. In an appendix, it is shown how the multi-layered field representation can be extended to quantum field theory.
Some two centuries before the quantum revolution, Newton (1) suggested that corpuscles of light generate waves in an aethereal medium like skipping stones generate waves in water, with their motion then being affected by these aether waves. Times have changed. Light corpuscles are now known as photons, and the majority of physicists have dispensed with the notion of aether. Nevertheless, certain features of Newton's metaphor live on in one particular version of quantum mechanics. According to pilot wave theory, first proposed by de Broglie (2) and later developed by Bohm (3) with Einstein's encouragement, microscopic elements such as photons and electrons consist of both particle and wave, the former being guided by the latter. Although this physical picture has not been widely accepted, it has had some notable proponents, including Bell (4). Its principal appeal is that it restores realism and determinism to quantum mechanics, its weakness that the physical nature of the guiding wave field remains unclear. At the time that pilot wave theory was developed and then overtaken by the Copenhagen interpretation as the standard view of quantum mechanics, there was no macroscopic pilot wave analog to draw upon. Now there is.
This paper provides a detailed and unified historical account of early debates over wave-function realism, the modern term for the view that the wave function of quantum theory is physically real. As this paper will show, the idea of physical waves associated with particles had its roots in work by Einstein and de Broglie, who both originally thought of these waves as propagating in three-dimensional physical space. De Broglie quickly turned this wave--particle duality into an early pilot-wave theory, on which a particle's associated ``phase wave'' piloted or guided the particle along its trajectory. Schrödinger built on de Broglie's phase-wave hypothesis to provide a comprehensive account of the nascent quantum theory. However, Schrödinger's new ``undulatory mechanics'' came at the cost of replacing de Broglie's phase waves propagating in physical space with a wave function propagating in a system's abstract configuration space. The present work will argue that this move from three-dimensional physical space to a many-dimensional configuration space was a decisive reason why the founders of quantum theory uniformly abandoned the physical reality of the wave function. This paper will further clarify that de Broglie introduced two distinct pilot-wave theories, and will then argue that it was Bohm's rediscovery of the second of these two pilot-wave theories over two decades later, as well as Bohm's vociferous defense of wave-function realism, that were responsible for resurrecting the idea of an ontological wave function. Finally, this paper will describe how the influence of Schrödinger and Bohmended up playing a central role in Everett's development of the ``many worlds'' interpretation.