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The second measurement problem

Particle physicists are always designing and building their experiments under the assumption that a measurement carried out in the real world can accurately reflect the state of affairs that existed just before the measurement. From a string of sparks or bubbles, for example, they infer the prior passage of an ionizing particle through the chamber. Extrapolating the tracks of several ionizing particles backward, they locate the point where the collision that produced the particles took place. But according to many textbook accounts of the quantum measuring process, retrodictions that use experimental results to infer what the particle was doing before this kind of measurement was made are not possible. Should we conclude, then, that experimenters don't take enough courses in quantum theory?

The consistent-histories analysis shows that the experimenters do, in fact, know what they are doing, and that such retrodictions are perfectly compatible with quantum theory. It also provides general rules for carrying out retrodictions safely, without producing contradictions or paradoxes. The consistent-histories approach even offers some insight into why the textbooks have often regarded retrodiction as dangerous.

The basic idea can be illustrated once again by reference to figure 1. Suppose the photon has been detected by detector C. In which channel was it just prior to detection: channel c or d? The very nature of the question tells us that (3) is of no help; we must resort to the histories in (4). But even they are inadequate, because they tell us nothing about what the photon is doing at intermediate times. To address that question, we must consider the following refinements of the histories in (4):

(5) |a>|C>|D> arrow right |c>|C>|D> arrow right |C[sup*]>|D>, |a>|C>|D> arrow right |d|C>|D> arrow right >|C>|D[sup *]>,

in which intermediate events have been added to describe the photon after it passes through the beam splitter, but before it is detected. The consistent-histories rules assign a probability of 1/2 to each of these histories. That means it is impossible, given the initial state, to predict whether the photon will leave the beam splitter through channel c or d. But if the final detector state is |C[sup *]>|D>, meaning that C has detected the photon, then the first history in (5), not the second, is the one that actually occurred. So, at the intermediate time, the photon was in state |c> rather than |d>. That is to say, it was in the c channel.

Why has this rather obvious way of solving the second measurement problem been overlooked for so long? Probably because a quantum physicist who grew up with the standard textbooks will describe the situation in figure 1 by means of a pair of histories

(6) |a>|C>|D> arrow right |s>|C>|D> arrow right |C[sup *]>|D>, |a>|C>|D> arrow right |s>|C>|D> arrow right |C>|D[sup *]>,

in which, at the intermediate time, the photon is in the superposition state |s> defined in (1). He will wait until the measurement takes place and then "collapse" the wavefunction for reasons that he may not understand very well. But at least they make more sense to him than does the macroscopic quantum superposition state |S> of (3).

From the standpoint of consistent histories, such a physicist is, in effect, employing the histories in (6), which are perfectly good quantum beables, as part of a stochastic quantum description. However, if the photon is in the superposition state |s> at the intermediate time, quantum incompatibility implies that it makes no sense to ask whether it is in the c channel or the d channel. That question can be asked only in the context of the histories in (5).

The existence of a quantum description employing the set of histories in (6), in which the question of the relationship between the measurement result and the location of the photon before the measurement is meaningless, does not invalidate the conclusion reached by means of the histories in (5), which provide a definite answer to that question. It is a quite general feature of quantum reasoning that various questions of physical interest can be addressed only by constructing an appropriate quantum description. That is quite unlike classical physics, where a single description, such as specifying a precise point in the phase space of a mechanical system, suffices to answer all meaningful questions.

Consistency conditions

The beables in consistent-histories quantum theory are a collection of mutually exclusive histories to which probabilities are assigned by the dynamical laws of quantum mechanics (Schrodinger's equation). If the histories involve just two times, as in (4), these probabilities are given by the usual Born rule--namely, the absolute square of the inner product of the time-evolved initial state and the final state in question. Histories involving three or more times, as in (5), require a generalization of the Born rule and additional consistency conditions to assure that the probabilities make physical sense.

Not all collections of mutually exclusive histories satisfy the mathematical conditions of consistency. The consistent-histories approach ascribes physical meaning only to histories that satisfy the consistency conditions. Other cases are regarded as meaningless; that is to say, they are rather like trying to simultaneously ascribe values for S[sub x] and S[sub z] to a spin-1/2 particle. (See the box above for additional remarks on consistency conditions.)

Consistency conditions are needed for a consistent discussion of the quantum double-slit experiment,(n8) in which a wavepacket approaches the slits at time t[sub 1], it passes through one or the other slit just before t[sub 2], and it arrives at t[sub 3] at some point in the interference zone, where waves from the two slits interfere with each other. It turns out that histories in which the particle passes through a particular slit and then arrives at a particular point in the interference zone do not satisfy the consistency conditions, and thus do not constitute acceptable quantum beables. That will come as no surprise to generations of students who have been taught that asking which slit the particle passes through is not a sensible question. In this respect, the consistency conditions support the physicist's usual intuition at the same time as they provide a precise mathematical formulation applicable in other situations where intuitive arguments are not sufficient for precise analysis.

On the other hand, if there are detectors just behind the two slits, one's physical intuition says that it should be sensible to say which slit the particle passes through. Such intuition is used all the time in designing experiments in which collimators are placed in front of detectors. In that case, the relevant histories, which are the analogs of (5), turn out to be consistent. Furthermore, even if there are no detectors behind the slits, there are consistent histories in which the particle passes through a particular slit and then arrives in a spread-out wavepacket in the interference zone, rather than at a particular point. (See the box for more details in an analogous situation involving a Mach-Zehnder interferometer.)

The physical consequences of consistency conditions are still being explored, and there is not yet complete agreement even on their mathematical form. However, the different formulations one finds in references 9, 10, and 11 do not seem to make any significant difference in most physical applications.

Classical limit

Because classical mechanics provides an excellent description of the motion of macroscopic objects in the everyday world, one would expect that quantum theory, in an appropriate limit, would yield the laws of classical physics to very good approximation. This conclusion is supported by Paul Ehrenfest's argument, which one finds in elementary textbooks, to the effect that average values of certain quantum observables satisfy equations similar to those of classical mechanics. But that is not a satisfactory solution to the problem of the classical limit, for two reasons: One wants to know how individual systems behave, not just the ensemble to which such an average applies. Furthermore, such an average, in the usual textbook understanding of quantum theory, refers to the results of measurements, and is not valid when measurements are not made.

In the consistent-histories approach, the classical limit can be studied by using appropriate subspaces of the quantum Hilbert space as a "coarse graining," analogous to dividing up phase space into nonoverlapping cells in classical statistical mechanics. This coarse graining can then be used to construct quantum histories. It is necessary to show that the resulting family of histories is consistent, so that the probabilities assigned by quantum dynamics make good quantum mechanical sense. Finally, one needs to show that the resulting quantum dynamics is well approximated by appropriate classical equations.

Demonstrating all this in complete detail is a difficult problem. But so is the analogous problem of finding the behavior of a large number of particles governed by classical mechanics. Indeed, the problem of showing that a system of classical particles will exhibit thermodynamic irreversibility, a typical macroscopic phenomenon, has not yet been settled to everyone's satisfaction, despite a continuing effort that goes back to Ludwig Boltzmann's work a century ago. (See the articles by Joel Lebowitz in PHYSICS TODAY, September 1993, page 32, and by George Zaslavsky in this issue, page 39.)

Nonetheless, calculations carried out by one of us,(n11, n12) and by Gell-Mann and Hartle,(n10) indicate that, given a suitable consistent family, classical physics does indeed emerge from quantum theory. Of course the classical equations are only approximate. They must be supplemented by including a certain amount of random noise, as one would expect from the fact that quantum dynamics is a stochastic process. In many circumstances, this quantum noise will not have much influence, but it can be amplified in systems that exhibit (classical) chaotic behavior. Even so, because the classical dynamics of such systems is noisy for all practical purposes, even if it is deterministic in principle, they are not likely to exhibit distinctive quantum effects.

The consistency of a family of histories for a macroscopic system is often ensured by quantum decoherence, an effect closely related to thermodynamic irreversibility. (See the article by Wojciech Zurek in PHYSICS TODAY, October 1991, page 36.) Demonstrating that quantum systems actually exhibit irreversible behavior in the thermodynamic sense, on the other hand, is not trivial. There are conceptual and computational difficulties similar to those that arise when one considers a classical system of many particles. Nonetheless, there seems at present to be no difficulty, in principle, that prevents us from understanding macroscopic phenomena in quantum terms, including what happens in a real measurement apparatus. Thus, by interpreting quantum mechanics in a manner in which measurement plays no fundamental role, we can use quantum theory to understand how an actual measuring apparatus functions.

We are grateful to Todd Brun, Sheldon Goldstein, James Hartle, and Wojciech Zurek for comments on the manuscript. One of us (Griffiths) acknowledges financial support from the National Science Foundation through grant PHY 9602084.

Consistency Conditions: An Application

The consistency conditions as formulated in reference 9 are obtained by associating with each of the histories in a particular family a "weight" operator on the Hilbert space, and then requiring that the weight operators for mutually exclusive histories be orthogonal to each other--the operator inner product being generated by the trace. This somewhat abstract prescription is best understood by working through simple examples, such as the one in section 6C of reference 8. Here, we give an application of the consistency conditions to a situation of some physical interest.

Consider the Mach-Zehnder interferometer illustrated in figure 2. A wavepacket of light passing through the first beam splitter B[sub 1] is reflected by a pair of mirrors, C and D, onto a second beam splitter B2 preceding the output channels e and f. The effect of B[sub 1] on the wavepacket |a> of a photon in the initial a channel at time t[sub 1] is to produce, at a slightly later time t[sub 2], the same kind of superposition |s> of wavepackets |c> and |d> in the c and d arms of the interferometer as we had in equation (1). The effect of the second beam splitter is given by

(7) |c> arrow right (|e> + |f>)/ square root of 2 |d> arrow right (-|e>+ |f>)/square root of 2,

where |e> and |f> are wavepackets in the output channels at t[sub 3]. The optical paths have been so arranged that the two |e> components in (7) appear with opposite phases.

Therefore, when we combine (1) and (7), we see that the photon entering at a must emerge in channel f, corresponding to the three-time history