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.1 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.2 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,3 in contrast to the representation of quantum properties by Hilbert subspaces, as introduced by John von Neumann.4 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.3 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.
