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Physics Today

Interpretations of quantum mechanics

Robert B. Griffiths

Published Oct 1, 2019• 5 citations
Publisher

Abstract

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.

DOI
10.1063/pt.3.4306

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Interpretations of quantum mechanics

Jean Bricmont

Bricmont replies: I find it odd, as the reviewer of a book, to be criticized because I did not discuss a theory that was not mentioned in the book. But I’ll answer Robert Griffiths point by point.In Bohmian mechanics (BM), it is simply a mathematical fact, noted by John Bell in chapter 14 of reference 1 of the Griffiths letter, that in the delayed double-slit experiment Griffith describes, particles cannot cross a symmetry plane but instead bounce back from it.11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Offering a different theory in which particles follow a different trajectory does not refute that fact.In standard quantum mechanics (QM), particles do not have trajectories; that fact was emphasized by, among others, Richard Feynman22. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. and Lev Landau and Evgeny Lifshitz.33. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2.The theory by Griffiths, based on the idea of “consistent histories,” is therefore not standard QM but is instead, like BM, an attempt to complete QM, by adding histories that consist of real events that occur independently of any measurements made on the quantum system (a measurement means an interaction with that system that may affect what would happen to it in the absence of measurements). Unfortunately, Griffiths’s attempt runs into contradictions, as shown in particular by Sheldon Goldstein in his two-part feature for Physics Today (March 1998, page 42, and April 1998, page 38).Griffiths misses the first step in Bell’s proof of nonlocality: the Einstein-Podolsky-Rosen (EPR) dilemma that perfect correlations between distant events cannot be explained unless one supposes either that some form of action at a distance occurs or that the events are predetermined by antecedent causes. The latter assumption is not a “classical” one; it is one part of the EPR dilemma. But Bell showed with his inequalities that the assumption leads to a contradiction. Hence, nonlocality follows. For more details, see, for example, references 11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. and 44. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). and Travis Norsen’s book.ReferencesSection:ChooseTop of pageReferences <<1. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Google ScholarCrossref2. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. Google Scholar3. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2. Google Scholar4. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). Google Scholar© 2019 American Institute of Physics.

Physics Today•Oct 1, 2019•2 citations

Bell on Bohm

Sheldon Goldstein

Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.

Cambridge University Press eBooks•Aug 31, 2016•0 citations

Measured responses to quantum Bayesianism

Michael Nauenberg

Some time ago, John Bell and I wrote a tongue-in-cheek article on the interpretation of quantum mechanics;11. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270;reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. in it we remarked that only a minority of physicists had any interest in the topic and that the typical physicist thought he would understand it “if ever he can spare twenty minutes to think about it.” According to David Mermin, however, that situation has changed dramatically, and “new interpretations [of quantum mechanics] appear every year.”Actually, most textbooks on the subject continue to be based on the well-established Born interpretation of quantum mechanics, which postulates that the absolute square of the Schrödinger wavefunction, projected onto a particular quantum state, gives the probability for the occurrence of that state after an appropriate measurement has been made. Mermin indicated that the problem with the Born interpretation involves a confusion, “prevalent among physicists,” regarding the nature of probability. He presented his preference for a “personalist Bayesian view,” which leads separate “agents,” like Eugene Wigner and his friend, to make different probability assignments based on their beliefs. But as Richard Feynman remarked, “Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not.”22. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7.Indeed, as applied to experiments, probability simply corresponds to the relative frequency for the occurrence of a given final state after identical states are prepared initially and measurements are made by appropriate detectors—devices that, through an irreversible macroscopic process, provide a permanent record of the outcome. Hence, the results of an experiment are independent of any observers, whose only role is to set up appropriate measurement devices, calculate the probabilities from quantum theory, and then write papers based on the recorded results.As Max Born also emphasized, in classical mechanics the situation is analogous, when one takes into account that in practice initial conditions in an experiment are known only imprecisely.33. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Hence the observed evolution of the system is also determined by a probability function that satisfies the Liouville equation. But in that case, after a measurement is performed, no one talks about the “collapse” of this classical function, whereas in quantum mechanics it is common to use that controversial expression for the Schrödinger wavefunction.Mermin discussed a “shifty split” between the classical and the quantum domain, a split that supposedly “haunts quantum mechanics.” He claimed that “regardless of what is split from what, all versions of the shifty split are vague and ambiguous.” However, that is not always the case, as a simple example shows. Consider an initial quantum wavepacket for a particle under an attractive central force that varies inversely with the square of the radial distance—the classic problem of planetary motion from the viewpoint of quantum mechanics, for example, or an excited electron in a Rydberg atom. For a finite time, the wavepacket will follow a classical Keplerian elliptic orbit, but like the Liouville distribution for the analogous classical problem, from the outset it will disperse due to the spread in initial position and momentum.For the classical distribution to be in close correspondence with the quantum wavepacket, its initial spread is assumed to satisfy the uncertainty relation. Eventually, however, the head of the wavepacket, like in the classical distribution, will catch up with its tail. Afterwards, further spreading of the wavepacket leads to interference effects between the two components, and the classical–quantum correspondence ceases to be valid. The time when such interference effects first appear is of order nτ/3, where n is the mean principal quantum number of the wavepacket and τ is the Kepler period.44. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133 Experimentally, that transition has been observed to agree with theory in Rydberg atoms55. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44 for n = 100, τ = 152 picoseconds, and nothing shifty about this classical–quantum correspondence is found. For planetary orbits, however, n is of order 1070, and therefore the answer to Albert Einstein’s often repeated remark “Is the moon there when nobody looks?” (see David Mermin, Physics Today, April 1985, page 38) is yes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270; Google Scholarreprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. Google Scholar2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7. Google Scholar3. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Google Scholar4. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133, Google ScholarCrossref, CAS5. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44, Google ScholarCrossref© 2012 American Institute of Physics.

Physics Today•Nov 30, 2012•0 citations

More on the quantum measurement problem

Gregory N. Derry

Although I have neither an objection to nor an endorsement of David Mermin’s major premise in his Quick Study, “There is no quantum measurement problem” (Physics Today, June 2022, page 62), I would like to make several contextual points.Mermin’s argument has an extensive history, along with a comparable history of rebuttal. Five decades ago Max Jammer wrote that “measurements in quantum mechanics are no more or less problematic than in classical physics, for the Hilbert space vector is only a purely formal device for relating the statistics associated with these arrangements to the physics of observations in classical physics.” That seems like a condensed version of the argument Mermin presents. On the following page, however, Jammer notes that a “problematic aspect whose serious implications were only gradually understood was the fact that as long as a quantum mechanical one-body or many-body system does not interact with macroscopic objects, as long as its motion is described by the deterministic Schrödinger time-dependent equation, no events could be considered to take place in the system.”11. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474.Niels Bohr giving a lecture in Copenhagen in April 1929. (Photograph by Samuel Goudsmit, courtesy of the AIP Emilio Segrè Visual Archives, Goudsmit Collection.)PPT|High resolutionRegarding the quotations from Niels Bohr, Mermin’s point is entirely valid and represents Bohr’s position accurately, but Bohr’s thought is always subtle and difficult to capture completely. Another quotation from Bohr, which is consistent with everything else he wrote, is, “The decisive point is that in neither case does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience.”22. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. That quotation seems inconsistent with Mermin’s paragraph that includes the passage “Why should I insist that my interpretation of science, which I use to make sense of the world that I experience, should never make any mention of me?” Bohr always insisted that his framework was a rational “generalization” of classical objectivity.The last and most conceptually important point concerns the ontological status of probability. I don’t necessarily disagree with Mermin and Bruno de Finetti, but it is by no means obvious that they are correct either. David Hawkins, for example, begins 50 pages of rigorous analysis with a brief statement of a major premise.The obvious and natural interpretation of probability in this context is that it introduces, not something that has to do with the nature of the physical system, but something that has to do with the incompleteness of our knowledge….But this opinion turns out to be false. Ignorance of dynamical regularity may be a motive for resorting to probabilities, but it does not explain the probabilities to which we resort.33. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135.Another example can be found in Jammer’s well-known historical account of quantum theory: “For Einstein the notion of probability … was the traditional conception of classical physics, a mathematical objectivization of the human deficiency of complete or exact knowledge but ultimately a creation of the human mind…. For Born probability, as far as it was related to the wave function, was not merely a mathematical fiction but something endowed with physical reality.”44. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286.Of course, such examples don’t mean that Mermin’s position is incorrect, but I do think that they demand a better argument than a comparison to “Fairies and Witches.” I am confident that Mermin has such an argument, and I look forward to reading it as I continue working toward a coherent position on the unresolved interpretational issues of quantum theory.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474. Google Scholar2. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. Google Scholar3. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135. Google Scholar4. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286. Google Scholar© 2022 American Institute of Physics.

Physics Today•Nov 1, 2022•2 citations

Deterministic Interpretations of Quantum Mechanics

Peter E. Hodgson

The dominance of the Copenhagen interpretation is the result of a historical accident (Cushing, 1994). At the Solvay conference in 1927, Louis de Broglie put forward the deterministic pilot wave theory, described later. Never slow to criticize, Pauli jumped up and demolished de Broglie’s proposal, and de Broglie was so discouraged by this that he did not develop his interpretation any further. The Copenhagen interpretation was then developed by Bohr, Heisenberg, Pauli and many others, as described in the previous chapter. It was strongly supported by von Neumann’s proof of the impossibility of hidden variables. Years later, in 1952, Bohm published a deterministic hidden variable theory, showing that there must be something wrong with von Neumann’s proof. The error was found by John Bell in 1966, as described in the last chapter, and he also showed how Pauli’s objections to de Broglie’s original idea can be answered. If this had been done at the conference in 1927, quantum mechanics could have been interpreted deterministically from the beginning. The essential mistake in the Copenhagen interpretation is to treat it as a complete account of the behaviour of each individual system. It is frequently claimed that it enables us to calculate, at least in principle, everything that can be measured. Thus Hooft (1997, p. 11) said that ‘The laws of quantum mechanics have been formulated very accurately. We know exactly how to compute anything we would like to know.’ Also Peierls (1997, p. 25) has said that quantum mechanics ‘had become a complete and consistent scheme capable of giving a unique answer to any questions relating to actual or possible observations’. Even if this were true, it would not imply that it is the final complete theory, for it is always possible that some new phenomenon might be found that cannot be calculated quantum-mechanically. However, it is not true: there are many phenomena such as the time of decay of a radioactive nucleus, or the direction a particle is scattered by a nucleus, that cannot be calculated, even in principle. Quantum mechanics is therefore an incomplete theory. We can only calculate the statistical properties of these systems, such as the half-life of a radioactive decay or the differential scattering cross-section that gives the probabilities that a particle is scattered through various angles. All measurements of quantum systems are of this statistical character. It might be thought that the possibility of making measurements on a single electron provides an exception to this. However, even in such cases the electron is continually bathed in a fluctuating background radiation from nearby atoms. Since this is variable, that electron is a member of an ensemble of electrons subject to different fluctuations. All these examples clearly demonstrate the statistical character of quantum mechanics. ‘I am rather firmly convinced’, Einstein (in Schilpp, 1949, pp. 666, 671) remarked, ‘that the essentially statistical character of contemporary quantum theory is solely to be ascribed to fact that this theory operates with an incomplete description of physical systems’. As a result, ‘the c-function is to be understood as the description not of a single system but of an ensemble of systems’. So, ‘if the statistical quantum theory does not pretend to describe the individual system completely, it appears unavoidable to look elsewhere for a complete description of the individual systems’. Thus ‘the difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as a description of an ensemble of systems’. If this is achieved, ‘the statistical theory would, within the framework of future physics, take an approximately analogous position to statistical mechanics within the framework of classical mechanics’. ‘Thus, essentially, nothing has changed since Galileo or Newton or Faraday concerning the status of the ‘‘observer’’ or our ‘‘consciousness’’ or of our ‘‘information’’ in physics’ (Popper, 1982, p. 46). Once this is accepted, the quantum paradoxes that plagued the Copenhagen interpretation are easily resolved, as described below. In any textbook of quantum mechanics problems such as the calculation of the energy levels of the hydrogen atom are solved as if that hydrogen atom is alone in the universe. As Feynman (1972) remarked, ‘when we solve a quantum-mechanical problem, what we really do is to divide the universe into two parts – the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the whole universe.’ In fact, however, every hydrogen atom on which measurements are made is surrounded by other atoms and exposed to their radiations. Quantum mechanics somehow takes this into account and gives the average behaviour of an ensemble of hydrogen atoms. A more detailed theory takes these influences into account. One attempt to do this is stochastic electrodynamics, described in a later section.

Routledge eBooks•Sep 8, 2017•0 citations

Bohm’s approach to quantum mechanics: Alternative theory or practical picture?

Ángel S. Sanz

Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation ( entanglement ) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.

arXiv:1707.00609•Frontiers of Physics•Oct 2, 2018•45 citations
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.
Bricmont replies: I find it odd, as the reviewer of a book, to be criticized because I did not discuss a theory that was not mentioned in the book. But I’ll answer Robert Griffiths point by point.In Bohmian mechanics (BM), it is simply a mathematical fact, noted by John Bell in chapter 14 of reference 1 of the Griffiths letter, that in the delayed double-slit experiment Griffith describes, particles cannot cross a symmetry plane but instead bounce back from it.11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Offering a different theory in which particles follow a different trajectory does not refute that fact.In standard quantum mechanics (QM), particles do not have trajectories; that fact was emphasized by, among others, Richard Feynman22. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. and Lev Landau and Evgeny Lifshitz.33. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2.The theory by Griffiths, based on the idea of “consistent histories,” is therefore not standard QM but is instead, like BM, an attempt to complete QM, by adding histories that consist of real events that occur independently of any measurements made on the quantum system (a measurement means an interaction with that system that may affect what would happen to it in the absence of measurements). Unfortunately, Griffiths’s attempt runs into contradictions, as shown in particular by Sheldon Goldstein in his two-part feature for Physics Today (March 1998, page 42, and April 1998, page 38).Griffiths misses the first step in Bell’s proof of nonlocality: the Einstein-Podolsky-Rosen (EPR) dilemma that perfect correlations between distant events cannot be explained unless one supposes either that some form of action at a distance occurs or that the events are predetermined by antecedent causes. The latter assumption is not a “classical” one; it is one part of the EPR dilemma. But Bell showed with his inequalities that the assumption leads to a contradiction. Hence, nonlocality follows. For more details, see, for example, references 11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. and 44. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). and Travis Norsen’s book.ReferencesSection:ChooseTop of pageReferences <<1. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Google ScholarCrossref2. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. Google Scholar3. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2. Google Scholar4. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). Google Scholar© 2019 American Institute of Physics.
Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.
Some time ago, John Bell and I wrote a tongue-in-cheek article on the interpretation of quantum mechanics;11. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270;reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. in it we remarked that only a minority of physicists had any interest in the topic and that the typical physicist thought he would understand it “if ever he can spare twenty minutes to think about it.” According to David Mermin, however, that situation has changed dramatically, and “new interpretations [of quantum mechanics] appear every year.”Actually, most textbooks on the subject continue to be based on the well-established Born interpretation of quantum mechanics, which postulates that the absolute square of the Schrödinger wavefunction, projected onto a particular quantum state, gives the probability for the occurrence of that state after an appropriate measurement has been made. Mermin indicated that the problem with the Born interpretation involves a confusion, “prevalent among physicists,” regarding the nature of probability. He presented his preference for a “personalist Bayesian view,” which leads separate “agents,” like Eugene Wigner and his friend, to make different probability assignments based on their beliefs. But as Richard Feynman remarked, “Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not.”22. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7.Indeed, as applied to experiments, probability simply corresponds to the relative frequency for the occurrence of a given final state after identical states are prepared initially and measurements are made by appropriate detectors—devices that, through an irreversible macroscopic process, provide a permanent record of the outcome. Hence, the results of an experiment are independent of any observers, whose only role is to set up appropriate measurement devices, calculate the probabilities from quantum theory, and then write papers based on the recorded results.As Max Born also emphasized, in classical mechanics the situation is analogous, when one takes into account that in practice initial conditions in an experiment are known only imprecisely.33. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Hence the observed evolution of the system is also determined by a probability function that satisfies the Liouville equation. But in that case, after a measurement is performed, no one talks about the “collapse” of this classical function, whereas in quantum mechanics it is common to use that controversial expression for the Schrödinger wavefunction.Mermin discussed a “shifty split” between the classical and the quantum domain, a split that supposedly “haunts quantum mechanics.” He claimed that “regardless of what is split from what, all versions of the shifty split are vague and ambiguous.” However, that is not always the case, as a simple example shows. Consider an initial quantum wavepacket for a particle under an attractive central force that varies inversely with the square of the radial distance—the classic problem of planetary motion from the viewpoint of quantum mechanics, for example, or an excited electron in a Rydberg atom. For a finite time, the wavepacket will follow a classical Keplerian elliptic orbit, but like the Liouville distribution for the analogous classical problem, from the outset it will disperse due to the spread in initial position and momentum.For the classical distribution to be in close correspondence with the quantum wavepacket, its initial spread is assumed to satisfy the uncertainty relation. Eventually, however, the head of the wavepacket, like in the classical distribution, will catch up with its tail. Afterwards, further spreading of the wavepacket leads to interference effects between the two components, and the classical–quantum correspondence ceases to be valid. The time when such interference effects first appear is of order nτ/3, where n is the mean principal quantum number of the wavepacket and τ is the Kepler period.44. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133 Experimentally, that transition has been observed to agree with theory in Rydberg atoms55. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44 for n = 100, τ = 152 picoseconds, and nothing shifty about this classical–quantum correspondence is found. For planetary orbits, however, n is of order 1070, and therefore the answer to Albert Einstein’s often repeated remark “Is the moon there when nobody looks?” (see David Mermin, Physics Today, April 1985, page 38) is yes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270; Google Scholarreprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. Google Scholar2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7. Google Scholar3. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Google Scholar4. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133, Google ScholarCrossref, CAS5. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44, Google ScholarCrossref© 2012 American Institute of Physics.
Although I have neither an objection to nor an endorsement of David Mermin’s major premise in his Quick Study, “There is no quantum measurement problem” (Physics Today, June 2022, page 62), I would like to make several contextual points.Mermin’s argument has an extensive history, along with a comparable history of rebuttal. Five decades ago Max Jammer wrote that “measurements in quantum mechanics are no more or less problematic than in classical physics, for the Hilbert space vector is only a purely formal device for relating the statistics associated with these arrangements to the physics of observations in classical physics.” That seems like a condensed version of the argument Mermin presents. On the following page, however, Jammer notes that a “problematic aspect whose serious implications were only gradually understood was the fact that as long as a quantum mechanical one-body or many-body system does not interact with macroscopic objects, as long as its motion is described by the deterministic Schrödinger time-dependent equation, no events could be considered to take place in the system.”11. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474.Niels Bohr giving a lecture in Copenhagen in April 1929. (Photograph by Samuel Goudsmit, courtesy of the AIP Emilio Segrè Visual Archives, Goudsmit Collection.)PPT|High resolutionRegarding the quotations from Niels Bohr, Mermin’s point is entirely valid and represents Bohr’s position accurately, but Bohr’s thought is always subtle and difficult to capture completely. Another quotation from Bohr, which is consistent with everything else he wrote, is, “The decisive point is that in neither case does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience.”22. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. That quotation seems inconsistent with Mermin’s paragraph that includes the passage “Why should I insist that my interpretation of science, which I use to make sense of the world that I experience, should never make any mention of me?” Bohr always insisted that his framework was a rational “generalization” of classical objectivity.The last and most conceptually important point concerns the ontological status of probability. I don’t necessarily disagree with Mermin and Bruno de Finetti, but it is by no means obvious that they are correct either. David Hawkins, for example, begins 50 pages of rigorous analysis with a brief statement of a major premise.The obvious and natural interpretation of probability in this context is that it introduces, not something that has to do with the nature of the physical system, but something that has to do with the incompleteness of our knowledge….But this opinion turns out to be false. Ignorance of dynamical regularity may be a motive for resorting to probabilities, but it does not explain the probabilities to which we resort.33. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135.Another example can be found in Jammer’s well-known historical account of quantum theory: “For Einstein the notion of probability … was the traditional conception of classical physics, a mathematical objectivization of the human deficiency of complete or exact knowledge but ultimately a creation of the human mind…. For Born probability, as far as it was related to the wave function, was not merely a mathematical fiction but something endowed with physical reality.”44. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286.Of course, such examples don’t mean that Mermin’s position is incorrect, but I do think that they demand a better argument than a comparison to “Fairies and Witches.” I am confident that Mermin has such an argument, and I look forward to reading it as I continue working toward a coherent position on the unresolved interpretational issues of quantum theory.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474. Google Scholar2. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. Google Scholar3. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135. Google Scholar4. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286. Google Scholar© 2022 American Institute of Physics.
The dominance of the Copenhagen interpretation is the result of a historical accident (Cushing, 1994). At the Solvay conference in 1927, Louis de Broglie put forward the deterministic pilot wave theory, described later. Never slow to criticize, Pauli jumped up and demolished de Broglie’s proposal, and de Broglie was so discouraged by this that he did not develop his interpretation any further. The Copenhagen interpretation was then developed by Bohr, Heisenberg, Pauli and many others, as described in the previous chapter. It was strongly supported by von Neumann’s proof of the impossibility of hidden variables. Years later, in 1952, Bohm published a deterministic hidden variable theory, showing that there must be something wrong with von Neumann’s proof. The error was found by John Bell in 1966, as described in the last chapter, and he also showed how Pauli’s objections to de Broglie’s original idea can be answered. If this had been done at the conference in 1927, quantum mechanics could have been interpreted deterministically from the beginning. The essential mistake in the Copenhagen interpretation is to treat it as a complete account of the behaviour of each individual system. It is frequently claimed that it enables us to calculate, at least in principle, everything that can be measured. Thus Hooft (1997, p. 11) said that ‘The laws of quantum mechanics have been formulated very accurately. We know exactly how to compute anything we would like to know.’ Also Peierls (1997, p. 25) has said that quantum mechanics ‘had become a complete and consistent scheme capable of giving a unique answer to any questions relating to actual or possible observations’. Even if this were true, it would not imply that it is the final complete theory, for it is always possible that some new phenomenon might be found that cannot be calculated quantum-mechanically. However, it is not true: there are many phenomena such as the time of decay of a radioactive nucleus, or the direction a particle is scattered by a nucleus, that cannot be calculated, even in principle. Quantum mechanics is therefore an incomplete theory. We can only calculate the statistical properties of these systems, such as the half-life of a radioactive decay or the differential scattering cross-section that gives the probabilities that a particle is scattered through various angles. All measurements of quantum systems are of this statistical character. It might be thought that the possibility of making measurements on a single electron provides an exception to this. However, even in such cases the electron is continually bathed in a fluctuating background radiation from nearby atoms. Since this is variable, that electron is a member of an ensemble of electrons subject to different fluctuations. All these examples clearly demonstrate the statistical character of quantum mechanics. ‘I am rather firmly convinced’, Einstein (in Schilpp, 1949, pp. 666, 671) remarked, ‘that the essentially statistical character of contemporary quantum theory is solely to be ascribed to fact that this theory operates with an incomplete description of physical systems’. As a result, ‘the c-function is to be understood as the description not of a single system but of an ensemble of systems’. So, ‘if the statistical quantum theory does not pretend to describe the individual system completely, it appears unavoidable to look elsewhere for a complete description of the individual systems’. Thus ‘the difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as a description of an ensemble of systems’. If this is achieved, ‘the statistical theory would, within the framework of future physics, take an approximately analogous position to statistical mechanics within the framework of classical mechanics’. ‘Thus, essentially, nothing has changed since Galileo or Newton or Faraday concerning the status of the ‘‘observer’’ or our ‘‘consciousness’’ or of our ‘‘information’’ in physics’ (Popper, 1982, p. 46). Once this is accepted, the quantum paradoxes that plagued the Copenhagen interpretation are easily resolved, as described below. In any textbook of quantum mechanics problems such as the calculation of the energy levels of the hydrogen atom are solved as if that hydrogen atom is alone in the universe. As Feynman (1972) remarked, ‘when we solve a quantum-mechanical problem, what we really do is to divide the universe into two parts – the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the whole universe.’ In fact, however, every hydrogen atom on which measurements are made is surrounded by other atoms and exposed to their radiations. Quantum mechanics somehow takes this into account and gives the average behaviour of an ensemble of hydrogen atoms. A more detailed theory takes these influences into account. One attempt to do this is stochastic electrodynamics, described in a later section.
Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation ( entanglement ) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.