Physics Today
Consistent treatments of quantum mechanics
Yakir Aharonov, Sandu Popescu, Jeff Tollaksen
Published Oct 1, 2011• 6 citations
Abstract
Aharonov, Popescu, and Tollaksen reply: We are grateful to David Mermin for stating loud and clear, in his unmistakable style, that “the authors are correct.” As for the rest of his letter, a lesson on clear writing from a great master of the subject is always welcome. We did strive for clarity in our article, but there is always space for improvement, and Freudian slips of Newspeak often creep in—that’s their very nature.However, as Mermin certainly knows, analyzing sentences in isolation without the many clues found in context doesn’t do them justice. The initial setup is indeed elementary, and we put in all the details, so it is hard to see how it could be misinterpreted. The situation becomes far more interesting and surprising, though, when weak measurements are taken into account; to ignore them, as Michael Nauenberg and Art Hobson did, is to completely miss the point. We believe that no true understanding of quantum mechanics can be obtained without taking weak measurements seriously.The letter by Robert Griffiths is mainly concerned with the merits of the consistent histories approach to quantum mechanics. It is an interesting subject but one that has very little to do with our article. We presented a number of quantum effects that arise in the context of pre- and postselection, first and foremost the effects that appear when weak measurements are involved. And no, there is no error in reasoning when we claim that in a situation like that described in our article, spin measurements may, in a consistent way, indicate that spin-1⁄2 particles have value 100; that is a statement of fact about results of measurements, which incidentally have been confirmed experimentally. One is free to disagree with the meaning we attach to those facts but cannot deny the facts themselves. Of course, since our effects are purely quantum mechanical, they can be analyzed using any interpretation of quantum mechanics, including consistent histories. However, before having such a discussion, it helps if the effects are first discovered! Their existence is our main message.© 2011 American Institute of Physics.
- DOI
- 10.1063/pt.3.1273
Related papers
Time-symmetric quantum mechanics questioned and defended
Yakir Aharonov, Sandu Popescu, Jeff Tollaksen
Aharonov, Popescu, and Tollaksen reply: We thank the letter writers for their interest and for the opportunity to better clarify our ideas.Michael Nauenberg and Art Hobson make essentially the same point—namely, that our ideas are completely wrong. To put their criticism in the right context, we point out that the outcome of our research program is twofold. First, we have discovered an entirely new class of quantum effects; second, we present a new way of thinking about quantum mechanics.The fact that quantum mechanics predicts the effects we discovered is just that, a fact. The effects are computed using standard quantum mechanics, without additions or modifications. As such, their prediction by quantum mechanics is beyond doubt (unless one suspects algebraic mistakes). Furthermore, many of our effects have been verified experimentally; in particular, different versions of our amplification method have been used as novel technological tools. Both Nauenberg and Hobson completely ignore our effects. But one should not ignore them. They are novel and they are strange. Even more, they don’t appear in isolation, but they form a well-structured pattern. Surely there is a lesson here that quantum mechanics wants to teach us; one ignores it at one’s peril.On the other hand, our way of looking at quantum mechanics is certainly unconventional; it introduces new concepts, and it approaches old concepts in a new way. That is essentially what the two letter writers point out, Hobson most emphatically when he writes that our article “is riddled with errors.” We are criticized for thinking in a different way and for asking new questions. But our way of thinking leads to the same predictions as the conventional way, so as far as experiments are concerned they are completely equivalent. As Richard Feynman says in his book The Character of Physical Law (Modern Library, 1994), suppose we have “two theories” that “have all the consequences . . . exactly the same. . . . How are we going to decide which one is right? There is no way by science, because they both agree with the experiment to the same extent.” So the criticism is baseless.At the same time, if our approach is completely equivalent to the standard one, why bother? Again Feynman gives the best answer: “For psychological reasons . . . , these two things may be far from equivalent, because one gives a man different ideas from the other. . . . There will be something, for instance, in theory A that talks about something, . . . but to find out what the corresponding thing is . . . in [theory] B may be very complicated—it may not be a very simple idea at all.” As a consequence, a new way of thinking allows one to ask new questions that, although they could be asked in the old theory as well, would have been very difficult to even envisage. That is precisely what we did. First, we raised the issue of the physics in pre- and postselected ensembles. And to reply to Hobson, no, there is nothing “erroneous” in the process of postselection. Postselection is a question about results of experiments, and every question about the results of actual measurements is legitimate. Subsequently, we discovered the concept of weak measurements, which in turn led us to discover the various effects we presented. Since the power of any new approach is given by its ability to predict new effects, one should conclude that ours is strong indeed.Furthermore, as many physicists agree, an intuitive understanding of quantum mechanics is still missing. That is why quantum physicists are surprised over and over again by the discovery of strange and unexpected fundamental effects. We hope that our new way of thinking is a step toward the long-sought intuition. Even more important, the new way of thinking may give us new ideas about what to change, if experiments ever turn out to contradict quantum mechanics and therefore require its modification. In particular, since we tinker with the idea of time— one of the most important concepts in physics—starting the change from there may be a very potent method.Shaul Mukamel refers to our experiment in which the component along some given axis of a spin-1⁄2 particle is found, by a weak measurement, to have the value √2‾/2 which is √2‾ times larger than the largest eigenvalue. He suggests an alternative explanation based on a classical vector model of spin. According to his explanation, values up to √3‾/2 should be possible. However, we presented the experiment showing √2‾/2 only because it was mathematically simple; by choosing a different postselection, we could have obtained, as results of weak measurements, values as large as we wanted. Hence the above simple classical-vector view doesn’t work.Robert Griffiths points out that there are two other time-symmetric formulations of quantum mechanics besides ours—“consistent histories” and “decoherent histories.” In particular, Griffiths is right when he emphasizes a major difference in spirit between our theory and consistent histories. The main goal of consistent histories is to find an explanation for the (apparent?) collapse of the wavefunction during a quantum measurement; although that solution is hotly disputed, as are all other solutions to the collapse problem, it is certainly a very ingenious one. However, an answer to the collapse problem is not our primary interest (though we are starting to see glimmers of an alternative answer to it using our approach).The letter by Griffiths, however, runs the risk of being misread as implying that solving the collapse problem will by itself clarify most or all of the counterintuitive aspects of quantum mechanics. That conclusion would be wrong. Quantum mechanics is strange and unusual and defies intuition in many ways; solving the collapse problem is by no means its only interesting fundamental issue. Nor can the solution of that one problem lead to a complete understanding of quantum phenomena. That particles tunnel in the first place is surprising by itself; even more so is the fact that, as we showed, perfectly good measuring devices, working with arbitrarily high precision, indicate consistently that the tunneling particles have negative kinetic energy. Equally surprising is that we can arrange a situation in which perfectly good measurements, made with as high a precision as we want, indicate that spin-1⁄2 particles have arbitrarily large spin. And these are only two examples. As far as we are aware, none of the many proposed solutions to the collapse problem make these effects seem less surprising, let alone predict them.© 2011 American Institute of Physics.
Physics Today•May 1, 2011•7 citations
Consistent treatments of quantum mechanics
Robert B. Griffiths
The Letters reply by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen gives a misleading impression of the consistent histories interpretation of quantum mechanics. As I pointed out in the same issue, consistent histories is an alternative to their time-symmetric formulation. It resolves the “collapse problem” of quantum foundations, which Aharonov and coauthors admit they have failed to do. However, that is not its only or even its principal virtue. It also solves what I call the second quantum measurement problem: how to infer from the measurement outcome the state of the measured system at an earlier time before the measurement took place. Thus the consistent histories approach, without any fuzzy references to “ensembles,” provides a proper probabilistic framework for inferring that Sx for a spin-1⁄2 particle actually was −1⁄2 before the measurement took place. This approach evades the criticisms expressed by Michael Nauenberg and Art Hobson in their letters in the same issue.Furthermore, consistent histories, unlike certain aspects of the approach of Aharonov and coauthors, is fully consistent with the Hilbert-space structure of quantum mechanics introduced by John von Neumann and now taught to all students of the subject. Indeed, with a consistent histories analysis one can locate fairly precisely11. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 the error in reasoning that leads to claims such as that the measured spin of a spin-1⁄2 particle can equal 100.22. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 Claims about what weak measurements actually measure, when made by those who have not yet solved either the collapse problem or the second measurement problem, should be treated with great caution.The claim that the consistent histories approach is controversial was true back in the 1990s, and the controversy led to a clearer formulation of the interpretation. However, no significant problems have come to light in the decade since the publication of my book.33. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. The fact that something has been ignored does not imply it is controversial. Admirers of Richard Feynman, some of whose writings were cited by Aharonov and coauthors in their reply, may be interested to know that shortly before his death he was quite sympathetic to an early form of the consistent histories or decoherent histories approach to understanding quantum mechanics (see PHYSICS TODAY, February 1999, page 11)—an open-mindedness worth imitating.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 , Google ScholarCrossref, ISI2. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 , Google ScholarCrossref, ISI3. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. Google Scholar© 2011 American Institute of Physics.
Physics Today•Oct 1, 2011•1 citation
Time-symmetric quantum mechanics questioned and defended
Robert B. Griffiths
Readers of the article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen might be interested in an alternative time-symmetric formulation of quantum mechanics, known as “consistent histories,” that was developed over roughly the same time period as Aharonov’s work (see the article by Robert Griffiths and Roland Omnès, PHYSICS TODAY, August 1999, page 26). Closely related is the “decoherent histories” approach of Murray Gell-Mann and James Hartle,11. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. but as that is not usually formulated in a way that is transparently time-symmetric, the following remarks refer to the consistent histories approach; see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. for an up-to-date formulation.Both the consistent histories approach and that of Aharonov and coworkers pay attention to events at several different times, are formulated in a way that is time-symmetrical, and address a number of quantum paradoxes. Both are consistent with the calculational procedures taught to students in a typical quantum mechanics course, so they are “standard quantum mechanics,” without the additional variables of de Broglie–Bohm or the additional collapses of Ghirardi-Rimini-Weber. And both approaches do not accept the “shut up and calculate” mentality that alas continues to dominate much classroom instruction. So far as I can tell, all the results mentioned by Aharonov and coauthors and in the earlier work they cite are fairly readily translated into the language of consistent histories, though the reverse is not true (see below); therefore, the consistent histories view is more general.In the treatment by Aharonov and coauthors, measurement, as in textbook quantum theory, remains a black box: It collapses the wavefunction, but nothing more can be said. And for good reason: The textbook approach of introducing probabilities by reference to measurement yields what appear to be insoluble difficulties if one attempts to apply quantum theory to the measurement process itself—that is, to actual apparatus constructed out of entities that are quantum mechanical. In the consistent histories approach, that difficulty does not arise, because it treats quantum dynamics as fundamentally probabilistic, not deterministic, and the same rules apply to measurements as to all other physical processes. Speaking metaphorically, the probabilistic approach used in consistent histories allows one to open the black measurement box and watch the quantum gears turn.The other major difference between the two approaches is their treatment of quantum paradoxes. We owe many of the most striking and delightful paradoxes of quantum theory to Aharonov and his coworkers, and he and Daniel Rohrlich have written a book on the topic.33. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . But he leaves the paradoxes largely unresolved; the reader is encouraged to study but not unravel them. The consistent histories approach is exactly opposite: Paradoxes should be—and a large number of them have been—resolved by the correct application of well-formulated and fully consistent quantum principles (see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. , chapters 19–25).Students new to quantum theory are often confused and deserve reasoned responses to their queries. Although paradoxes are valuable illustrations of how the quantum world differs from our everyday experience, I prefer to provide students with the conceptual tools needed to resolve and make sense of them. In particular, students benefit from learning a fully consistent approach to probabilities in the quantum domain, one not based on measurements but on general quantum principles. A colleague and I have just finished using that approach in teaching the first term of our introductory graduate quantum mechanics course. Although it requires extra time and effort to learn how to think about quantum processes rather than just do calculations, the reward comes in a deeper understanding of how the real (quantum) world works.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. Google Scholar2. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. Google Scholar3. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . Google ScholarCrossref© 2011 American Institute of Physics.
Physics Today•May 1, 2011•1 citation
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
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
Time-symmetric quantum mechanics questioned and defended
Shaul Mukamel
The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics' fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors' figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.
Physics Today•May 1, 2011•3 citations
Aharonov, Popescu, and Tollaksen reply: We are grateful to David Mermin for stating loud and clear, in his unmistakable style, that “the authors are correct.” As for the rest of his letter, a lesson on clear writing from a great master of the subject is always welcome. We did strive for clarity in our article, but there is always space for improvement, and Freudian slips of Newspeak often creep in—that’s their very nature.However, as Mermin certainly knows, analyzing sentences in isolation without the many clues found in context doesn’t do them justice. The initial setup is indeed elementary, and we put in all the details, so it is hard to see how it could be misinterpreted. The situation becomes far more interesting and surprising, though, when weak measurements are taken into account; to ignore them, as Michael Nauenberg and Art Hobson did, is to completely miss the point. We believe that no true understanding of quantum mechanics can be obtained without taking weak measurements seriously.The letter by Robert Griffiths is mainly concerned with the merits of the consistent histories approach to quantum mechanics. It is an interesting subject but one that has very little to do with our article. We presented a number of quantum effects that arise in the context of pre- and postselection, first and foremost the effects that appear when weak measurements are involved. And no, there is no error in reasoning when we claim that in a situation like that described in our article, spin measurements may, in a consistent way, indicate that spin-1⁄2 particles have value 100; that is a statement of fact about results of measurements, which incidentally have been confirmed experimentally. One is free to disagree with the meaning we attach to those facts but cannot deny the facts themselves. Of course, since our effects are purely quantum mechanical, they can be analyzed using any interpretation of quantum mechanics, including consistent histories. However, before having such a discussion, it helps if the effects are first discovered! Their existence is our main message.© 2011 American Institute of Physics.
Aharonov, Popescu, and Tollaksen reply: We thank the letter writers for their interest and for the opportunity to better clarify our ideas.Michael Nauenberg and Art Hobson make essentially the same point—namely, that our ideas are completely wrong. To put their criticism in the right context, we point out that the outcome of our research program is twofold. First, we have discovered an entirely new class of quantum effects; second, we present a new way of thinking about quantum mechanics.The fact that quantum mechanics predicts the effects we discovered is just that, a fact. The effects are computed using standard quantum mechanics, without additions or modifications. As such, their prediction by quantum mechanics is beyond doubt (unless one suspects algebraic mistakes). Furthermore, many of our effects have been verified experimentally; in particular, different versions of our amplification method have been used as novel technological tools. Both Nauenberg and Hobson completely ignore our effects. But one should not ignore them. They are novel and they are strange. Even more, they don’t appear in isolation, but they form a well-structured pattern. Surely there is a lesson here that quantum mechanics wants to teach us; one ignores it at one’s peril.On the other hand, our way of looking at quantum mechanics is certainly unconventional; it introduces new concepts, and it approaches old concepts in a new way. That is essentially what the two letter writers point out, Hobson most emphatically when he writes that our article “is riddled with errors.” We are criticized for thinking in a different way and for asking new questions. But our way of thinking leads to the same predictions as the conventional way, so as far as experiments are concerned they are completely equivalent. As Richard Feynman says in his book The Character of Physical Law (Modern Library, 1994), suppose we have “two theories” that “have all the consequences . . . exactly the same. . . . How are we going to decide which one is right? There is no way by science, because they both agree with the experiment to the same extent.” So the criticism is baseless.At the same time, if our approach is completely equivalent to the standard one, why bother? Again Feynman gives the best answer: “For psychological reasons . . . , these two things may be far from equivalent, because one gives a man different ideas from the other. . . . There will be something, for instance, in theory A that talks about something, . . . but to find out what the corresponding thing is . . . in [theory] B may be very complicated—it may not be a very simple idea at all.” As a consequence, a new way of thinking allows one to ask new questions that, although they could be asked in the old theory as well, would have been very difficult to even envisage. That is precisely what we did. First, we raised the issue of the physics in pre- and postselected ensembles. And to reply to Hobson, no, there is nothing “erroneous” in the process of postselection. Postselection is a question about results of experiments, and every question about the results of actual measurements is legitimate. Subsequently, we discovered the concept of weak measurements, which in turn led us to discover the various effects we presented. Since the power of any new approach is given by its ability to predict new effects, one should conclude that ours is strong indeed.Furthermore, as many physicists agree, an intuitive understanding of quantum mechanics is still missing. That is why quantum physicists are surprised over and over again by the discovery of strange and unexpected fundamental effects. We hope that our new way of thinking is a step toward the long-sought intuition. Even more important, the new way of thinking may give us new ideas about what to change, if experiments ever turn out to contradict quantum mechanics and therefore require its modification. In particular, since we tinker with the idea of time— one of the most important concepts in physics—starting the change from there may be a very potent method.Shaul Mukamel refers to our experiment in which the component along some given axis of a spin-1⁄2 particle is found, by a weak measurement, to have the value √2‾/2 which is √2‾ times larger than the largest eigenvalue. He suggests an alternative explanation based on a classical vector model of spin. According to his explanation, values up to √3‾/2 should be possible. However, we presented the experiment showing √2‾/2 only because it was mathematically simple; by choosing a different postselection, we could have obtained, as results of weak measurements, values as large as we wanted. Hence the above simple classical-vector view doesn’t work.Robert Griffiths points out that there are two other time-symmetric formulations of quantum mechanics besides ours—“consistent histories” and “decoherent histories.” In particular, Griffiths is right when he emphasizes a major difference in spirit between our theory and consistent histories. The main goal of consistent histories is to find an explanation for the (apparent?) collapse of the wavefunction during a quantum measurement; although that solution is hotly disputed, as are all other solutions to the collapse problem, it is certainly a very ingenious one. However, an answer to the collapse problem is not our primary interest (though we are starting to see glimmers of an alternative answer to it using our approach).The letter by Griffiths, however, runs the risk of being misread as implying that solving the collapse problem will by itself clarify most or all of the counterintuitive aspects of quantum mechanics. That conclusion would be wrong. Quantum mechanics is strange and unusual and defies intuition in many ways; solving the collapse problem is by no means its only interesting fundamental issue. Nor can the solution of that one problem lead to a complete understanding of quantum phenomena. That particles tunnel in the first place is surprising by itself; even more so is the fact that, as we showed, perfectly good measuring devices, working with arbitrarily high precision, indicate consistently that the tunneling particles have negative kinetic energy. Equally surprising is that we can arrange a situation in which perfectly good measurements, made with as high a precision as we want, indicate that spin-1⁄2 particles have arbitrarily large spin. And these are only two examples. As far as we are aware, none of the many proposed solutions to the collapse problem make these effects seem less surprising, let alone predict them.© 2011 American Institute of Physics.
The Letters reply by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen gives a misleading impression of the consistent histories interpretation of quantum mechanics. As I pointed out in the same issue, consistent histories is an alternative to their time-symmetric formulation. It resolves the “collapse problem” of quantum foundations, which Aharonov and coauthors admit they have failed to do. However, that is not its only or even its principal virtue. It also solves what I call the second quantum measurement problem: how to infer from the measurement outcome the state of the measured system at an earlier time before the measurement took place. Thus the consistent histories approach, without any fuzzy references to “ensembles,” provides a proper probabilistic framework for inferring that Sx for a spin-1⁄2 particle actually was −1⁄2 before the measurement took place. This approach evades the criticisms expressed by Michael Nauenberg and Art Hobson in their letters in the same issue.Furthermore, consistent histories, unlike certain aspects of the approach of Aharonov and coauthors, is fully consistent with the Hilbert-space structure of quantum mechanics introduced by John von Neumann and now taught to all students of the subject. Indeed, with a consistent histories analysis one can locate fairly precisely11. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 the error in reasoning that leads to claims such as that the measured spin of a spin-1⁄2 particle can equal 100.22. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 Claims about what weak measurements actually measure, when made by those who have not yet solved either the collapse problem or the second measurement problem, should be treated with great caution.The claim that the consistent histories approach is controversial was true back in the 1990s, and the controversy led to a clearer formulation of the interpretation. However, no significant problems have come to light in the decade since the publication of my book.33. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. The fact that something has been ignored does not imply it is controversial. Admirers of Richard Feynman, some of whose writings were cited by Aharonov and coauthors in their reply, may be interested to know that shortly before his death he was quite sympathetic to an early form of the consistent histories or decoherent histories approach to understanding quantum mechanics (see PHYSICS TODAY, February 1999, page 11)—an open-mindedness worth imitating.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 , Google ScholarCrossref, ISI2. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 , Google ScholarCrossref, ISI3. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. Google Scholar© 2011 American Institute of Physics.
Readers of the article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen might be interested in an alternative time-symmetric formulation of quantum mechanics, known as “consistent histories,” that was developed over roughly the same time period as Aharonov’s work (see the article by Robert Griffiths and Roland Omnès, PHYSICS TODAY, August 1999, page 26). Closely related is the “decoherent histories” approach of Murray Gell-Mann and James Hartle,11. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. but as that is not usually formulated in a way that is transparently time-symmetric, the following remarks refer to the consistent histories approach; see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. for an up-to-date formulation.Both the consistent histories approach and that of Aharonov and coworkers pay attention to events at several different times, are formulated in a way that is time-symmetrical, and address a number of quantum paradoxes. Both are consistent with the calculational procedures taught to students in a typical quantum mechanics course, so they are “standard quantum mechanics,” without the additional variables of de Broglie–Bohm or the additional collapses of Ghirardi-Rimini-Weber. And both approaches do not accept the “shut up and calculate” mentality that alas continues to dominate much classroom instruction. So far as I can tell, all the results mentioned by Aharonov and coauthors and in the earlier work they cite are fairly readily translated into the language of consistent histories, though the reverse is not true (see below); therefore, the consistent histories view is more general.In the treatment by Aharonov and coauthors, measurement, as in textbook quantum theory, remains a black box: It collapses the wavefunction, but nothing more can be said. And for good reason: The textbook approach of introducing probabilities by reference to measurement yields what appear to be insoluble difficulties if one attempts to apply quantum theory to the measurement process itself—that is, to actual apparatus constructed out of entities that are quantum mechanical. In the consistent histories approach, that difficulty does not arise, because it treats quantum dynamics as fundamentally probabilistic, not deterministic, and the same rules apply to measurements as to all other physical processes. Speaking metaphorically, the probabilistic approach used in consistent histories allows one to open the black measurement box and watch the quantum gears turn.The other major difference between the two approaches is their treatment of quantum paradoxes. We owe many of the most striking and delightful paradoxes of quantum theory to Aharonov and his coworkers, and he and Daniel Rohrlich have written a book on the topic.33. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . But he leaves the paradoxes largely unresolved; the reader is encouraged to study but not unravel them. The consistent histories approach is exactly opposite: Paradoxes should be—and a large number of them have been—resolved by the correct application of well-formulated and fully consistent quantum principles (see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. , chapters 19–25).Students new to quantum theory are often confused and deserve reasoned responses to their queries. Although paradoxes are valuable illustrations of how the quantum world differs from our everyday experience, I prefer to provide students with the conceptual tools needed to resolve and make sense of them. In particular, students benefit from learning a fully consistent approach to probabilities in the quantum domain, one not based on measurements but on general quantum principles. A colleague and I have just finished using that approach in teaching the first term of our introductory graduate quantum mechanics course. Although it requires extra time and effort to learn how to think about quantum processes rather than just do calculations, the reward comes in a deeper understanding of how the real (quantum) world works.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. Google Scholar2. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. Google Scholar3. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . Google ScholarCrossref© 2011 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.
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.
The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics' fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors' figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.