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arXiv:0903.3878

Bohmian particle trajectories contradict quantum mechanics

Michael Zirpel

Published Mar 23, 2009• 3 citations
arXivHTMLPDF

Abstract

The Bohmian interpretation of quantum mechanics adds particle trajectories to the wave function and ensures that the probability distribution of the particle positions agrees with quantum mechanics at any time. This is not sufficient to avoid contradictions with quantum mechanics. There are correlations between particle positions at different times which cannot be reproduced with real particle trajectories. A simple rearrangement of an experimental test of the Bell-CHSH inequality demonstrates this.

DOI
10.48550/arxiv.0903.3878

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Phys. 41, 705 (2011), and references therein. https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref4. J. von Neumann, Mathematical Foundations of Quantum Mechanics, R. T. Beyer, trans., Princeton U. Press (1955), chap. 3.5. Google Scholar© 2019 American Institute of Physics.

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The Bohmian interpretation of quantum mechanics adds particle trajectories to the wave function and ensures that the probability distribution of the particle positions agrees with quantum mechanics at any time. This is not sufficient to avoid contradictions with quantum mechanics. There are correlations between particle positions at different times which cannot be reproduced with real particle trajectories. A simple rearrangement of an experimental test of the Bell-CHSH inequality demonstrates this.
The violation of Bell type inequalities in quantum systems manifests that quantum states cannot be described by classical probability distributions. Yet, Bohmian mechanics is a realistic, non-local theory of classical particle trajectories that is claimed to be indistinguishable by observations from more traditional approaches to quantum mechanics. We set up a spatial version of the GHZ system with qubits realised as positional observables that demonstrates that the Bohmian theory fails to match predictions of textbook quantum mechanics (and most likely experients) unless enlarged by a microscopic theory of collapse of the wave function after observation. For this discrepancy to occur it is essential that positions at different times do not commute.
In Bohmian mechanics, particles follow continuous trajectories, so two-time position correlations have been well defined. However, Bohmian mechanics predicts the violation of Bell inequalities. Motivated by this fact, we investigate position measurements in Bohmian mechanics by coupling the particles to macroscopic pointers. This explains the violation of Bell inequalities despite two-time position correlations. We relate this fact to so-called surrealistic trajectories that, in our model, correspond to slowly moving pointers. Next, we emphasize that Bohmian mechanics, which does not distinguish between microscopic and macroscopic systems, implies that the quantum weirdness of quantum physics also shows up at the macro-scale. Finally, we discuss the fact that Bohmian mechanics is attractive to philosophers but not so much to physicists and argue that the Bohmian community is responsible for the latter.
Jean Bricmont’s review of Travis Norsen’s Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory (Physics Today, April 2018, page 58) stresses the virtues of Louis de Broglie and David Bohm’s pilot-wave interpretation of quantum mechanics, also referred to as Bohmian mechanics (BM), but the review omits mention of some of that interpretation’s rather serious defects. As Bricmont notes, in BM the standard Schrödinger wavefunction ψ(r, t) for a particle is augmented with a classical trajectory r(t) that is interpreted as the actual precise physical position of the particle as a function of time. Sometimes that Bohmian trajectory can provide a helpful intuitive picture, but in other cases it is quite misleading.John Bell presented an example of how Bohmian trajectories can mislead: a modified double-slit experiment with a lens placed right after each slit so that the emerging beams cross on their way to two distant detectors located beyond the crossing point.11. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge U. Press (1987). The beam from the lower slit arrives at the upper detector, and that from the upper slit at the lower detector. If a particle arrives at the upper detector, which slit did it pass through?Bell claims that the naive answer—that the particle came from the lower slit, based on the intuition that particles move in straight lines in a vacuum—is wrong. He notes that a Bohmian particle cannot cross the symmetry plane separating the upper slit and detector from their lower counterparts, and thus a particle arriving at the upper detector actually came from the upper slit: first it moved downward, but when it arrived in the region where the beams cross, it “bounced” back upward toward the upper detector.Years ago I published an analysis that contradicted Bell and showed that the answer he regarded as naive—the particle passes through the crossing region without bouncing—was perfectly good quantum mechanics.22. R. B. Griffiths, Phys. Lett. A 261, 227 (1999). https://doi.org/10.1016/S0375-9601(99)00542-3 I am still waiting for someone in the Bohmian community to publish a reply.Norsen is troubled because the nonlocal, often superluminal influences needed to make sense of BM are hard to reconcile with special relativity. Indeed, the widely accepted no-signaling principle of quantum mechanics asserts that those influences cannot transmit information, so a direct experimental detection of them is out of reach. Defenders of BM say they know that nonlocal influences exist because quantum mechanics violates Bell’s inequalities. But as I and others have pointed out, the hidden variables used in derivations of Bell inequalities are fundamentally classical,33. See R. B. Griffiths, Found. Phys. 41, 705 (2011), and references therein. https://doi.org/10.1007/s10701-010-9512-5 in contrast to the representation of quantum properties by Hilbert subspaces, as introduced by John von Neumann.44. J. von Neumann, Mathematical Foundations of Quantum Mechanics, R. T. Beyer, trans., Princeton U. Press (1955), chap. 3.5. Projectors on those subspaces do not generally commute, which clearly distinguishes quantum physics from classical. Thus that feature, rather than a failure of locality, is why Bell’s inequalities do not agree with quantum theory and experiment.The BM particle trajectories are instances of classical hidden variables, so the nonlocality of the pilot-wave interpretation of quantum mechanics is not surprising. Indeed, when proper account is taken of noncommutativity, one can show that quantum theory satisfies a principle of locality that Albert Einstein might well have agreed with: Objective properties of an isolated individual system do not change when something is done to another, noninteracting system.33. See R. B. Griffiths, Found. Phys. 41, 705 (2011), and references therein. https://doi.org/10.1007/s10701-010-9512-5 Thus a consistent Hilbert-space formulation of quantum theory without hidden variables removes any worry about a conflict with special relativity.Although Bohmian mechanics was worthy of consideration and has been useful in the development of quantum foundations, ignoring more recent developments is not the way to honor the memory of one of the great physicists of the last century. I hope that a second edition of Norsen’s well-written book will take account of more recent work.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge U. Press (1987). Google Scholar2. R. B. Griffiths, Phys. Lett. A 261, 227 (1999). https://doi.org/10.1016/S0375-9601(99)00542-3, Google ScholarCrossref3. See R. B. Griffiths, Found. Phys. 41, 705 (2011), and references therein. https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref4. J. von Neumann, Mathematical Foundations of Quantum Mechanics, R. T. Beyer, trans., Princeton U. Press (1955), chap. 3.5. Google Scholar© 2019 American Institute of Physics.
A number of assertions have recently been made that in two-particle interference devices, Bohmian trajectories may not reproduce exactly all statistical predictions of quantum mechanics. Specifically, let two identical bosons go through identical slits arranged symmetrically with respect to a plane, with wave functions transforming into each other under the symmetry operation. Then it is claimed that pairs of particles following Bohmian trajectories and registered on a screen opposite the slits may have observable spatial distribution properties not predicted by quantum mechanics. We point out that alleged discrepancies in fact originate from a specific sampling assumption on Bohmian particles that violates their general probability distribution.
Standard quantum mechanics famously disallows to simultaneously measure particle position and momentum with arbitrary precision. In contrast, Bohmian mechanics, a hidden-variable interpretation, at the same time ascribes reality to the positions and momenta of quantum particles at the cost of a nonlocal ontology [1]. It thus allows to conceive of definite particle trajectories, while being fully compatible with the standard theory in all empirical predictions. To record such trajectories [2] we consider two optical settings using weak measurements in order to investigate different features of particle propagation in the Bohmian picture.
Abstract In Bohmian mechanics it is assumed that quantum objects are particles that follow real physical paths. These Bohmian trajectories are obtained by integrating the velocity field that occurs in the continuity equation. However, in the course of deriving these trajectories, the independent position variable x, appearing in the velocity field, is arbtirarily replaced by the time-dependent trajectory q(t). We show under which restrictions such a replacement can be justified, leading to a totally different interpretation of the Bohmian trajectories. They are not real paths of physical particles but borders of regions of constant probability, so-called quantiles, and therefore must be seen in the context of descriptive statistics. Thus they provide a tool complementary to the conventional statistical formulation of quantum mechanics. A possible extension to include open dissipative systems is also discussed, showing that the inclusion of an environment does not change our interpretation of the Bohmian trajectories.