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.
