Math (Несколько текстов для зачёта), страница 8

The two measurement problems, and the consistent-histories approach to solving them, can be understood by referring to the simple gedanken experiment shown in figure 1. A photon (or neutron, or some other particle; it makes no difference) enters a beam splitter in the a channel and emerges in the c and d channels in the coherent superposition:

(1) |a> arrow right |s> = (|c> + |d>)/square root of 2.

Here |a>, |c>, and |d> are wavepackets in the input and output channels, and |s> is what results from |a> by unitary time evolution (that is, by solving the appropriate Schrodinger equation) as the photon passes through the beam splitter.

The photon will later be detected by one of two detectors, C and D. To describe this process in quantum terms, we assume that |C> is the initial quantum state of C, and that the process of its detecting a photon in a wavepacket |c> is described by

(2) |c>|C> arrow right |C[sup *]>,

where |C[sup *]> is the triggered state of the detector after it has detected the photon. Once again, the arrow indicates the unitary time evolution produced by solving Schrodinger's equation. It is helpful to think of |C> and |C[sup *]> as physically quite distinct: Imagine that a macroscopically large pointer, initially horizontal in |C>, is moved to a vertical position in the state |C[sup *]> when the photon has been detected.

By putting together the processes (1), (2), and the counterpart of (2) that describes the detection of a photon in the d channel by detector D, one finds that the unitary time development of the entire system shown in figure 1 is of the form

(3) |a>|C>|D> arrow right |s> = (|C[sup *]>|D> + |C>|D[sup *]>)/ square root of 2.

Ascribing some physical significance to the peculiar macroscopic-quantum-superposition state |S> in (3) poses the first measurement problem in our gedanken experiment. The difficulty is that |S> consists of a linear superposition of two wavefunctions representing situations that are visibly, macroscopically, quite distinct: The pointer on C is vertical and that on D is horizontal for |C[sup *]>|D>, whereas for |C>|D[sup *]> the D pointer is vertical and the C pointer is horizontal. In Schrodinger's famously paradoxical example, the two distinct situations were a live and a dead cat. A great deal of effort has gone into trying to interpret |S> as meaning that either one detector or the other has been triggered, but the results have not been very satisfactory.(n5)

The first measurement problem is an almost inevitable consequence of supposing that, in quantum theory, a solution of Schrodinger's equation represents a deterministic time evolution of a physical system, in the same way as does a solution of Hamilton's equations in classical mechanics. That was undoubtedly Schrodinger's point of view when he introduced his equation. The probabilistic interpretation now universally accepted among quantum physicists was introduced shortly thereafter by Max Born. Since then, chance and determinism have maintained a somewhat uncomfortable coexistence within quantum theory, with many scientists continuing to share Einstein's view that resorting to probabilities is a sign that something is incomplete.

A stochastic theory

By contrast, the consistent-histories viewpoint is that quantum mechanics is fundamentally a stochastic or probabilistic theory, as far as time development is concerned, and that it is not necessary to introduce some deterministic underpinning of this randomness by means of hidden variables. The basic task of quantum theory is to use the time-dependent Schrodinger equation, not to generate deterministic orbits, but instead to assign probabilities to quantum histories--sequences of quantum events at a succession of times--in much the same way that classical stochastic theories assign probabilities to sequences of coin tosses or to Brownian motion. This perspective does not exclude deterministic histories, but those are thought of as arising in special cases in which the probability of a particular sequence of events is equal to 1.

For the gedanken experiment in figure 1, the consistent-histories solution to the first measurement problem consists of noting that a perfectly good description of what is happening is provided by assuming that the initial state is followed at a later time by one of two mutually exclusive possibilities: |C[sup *]>|D> or |C>|B[sup *]>. They are related to each other in much the same way as heads and tails in a coin toss. That is to say, the system is described by one (and, in a particular experimental run, only one) of the two quantum histories:

(4) |a>|C>|D> arrow right |C[sup *]>|D> or |a>|C>|D> arrow right |C>|D[sup *]>,

where the arrow no longer denotes unitary time development. Quantum theory assigns to each history a probability of 1/2. (Of course, to check this prediction, one would have to repeat the experiment using several photons in succession, each time resetting the detectors.)

The troublesome macroscopic quantum superposition state |S> of (3) appears nowhere in (4). Indeed, as we discuss below, the rules of consistent-histories quantum theory mean that |S> cannot occur in the same quantum description as the final detector states employed in (4). Therefore, the first measurement problem has been solved (or, at least it has disappeared) if one uses the stochastic histories in (4) in place of the deterministic history in (3).

The fundamental beables of consistent histories quantum theory--that is, the items to which the theory can ascribe physical reality, or at least a reliable logical meaning--are consistent quantum histories: sequences of successive quantum events that satisfy a consistency condition about which more is said below. A quantum event can be any wavefunction--that is to say, any nonzero element of the quantum Hilbert space. The two histories in (4), as well as the single history in (3), are examples of consistent quantum histories. They are thus acceptable quantum descriptions of what goes on in the system shown in figure 1.

At this point, the reader may be skeptical of the claim that the first measurement problem has been solved. We have simply replaced (3), with its troublesome macroscopic quantum superposition state, by the more benign pair of histories in (4). But as long as (3) is an acceptable history--as is certainly the case from the consistent-histories perspective--how can we claim that (4) is the correct quantum description rather than (3)? Or is it possible that both (3) and (4) apply simultaneously to the same system? Before attempting an answer, let us take a slight detour to introduce the concept of quantum incompatibility, which plays a central role in the consistent-histories approach to quantum theory.

Quantum incompatibility

The simplest quantum system is the spin degree of freedom of a spin-1/2 particle, described by a two-dimensional Hilbert space. Every nonzero (spinor) wavefunction in this space corresponds to a component of spin angular momentum in a particular direction taking the value 1/2 in units of h. Thus the quantum beables of this system, in the consistent-histories approach as well as in standard quantum mechanics, are of the form S[sub w] = 1/2, where w is a unit vector pointing in some direction in three-dimensional space, and S[sub w] is the component of spin angular momentum in that direction. (Actually, S[sub w] = 1/2 corresponds to a whole collection of wavefunctions obtained from each other through multiplication by a complex number, and thus to a one-dimensional subspace of the Hilbert space.)

The nonclassical nature of quantum theory begins to appear when one asks about the relationship of these beables, or quantum states, for two different directions w. If the directions are opposite, for example +z and -z, the states S[sub z] = 1/2 and S[sub -z] = 1/2 are two mutually exclusive possibilities, one of which is the negation of the other. Thus they are related in the same way as the results of tossing a coin: if heads (S[sub z] = 1/2) is false, tails (S[sub z] = -1/2) is true, and vice versa. This means, in particular, that the proposition "S[sub z] = 1/2 and S[sub z] = -1/2" can never be true. It is always false.

That this is a reasonable way of understanding the relationship between S[sub z] = 1/2 and S[sub z] = -1/2 is confirmed by the fact that if a spin-1/2 particle is sent through a Stern-Gerlach apparatus with its magnetic field gradient in the z direction, the result will be either S[sub z] = 1/2 or -1/2, as shown by the position at which the particle emerges. Precisely the same applies to any other component of spin angular momentum. Thus, for example, S[sub x] = 1/2 is the negation of S[sub x] = -1/2. (As an amusing aside, we note that when Otto Stern proposed in 1921 to demonstrate the quantization of angular-momentum orientation, Born assured him that he would see nothing, because such spatial quantization was only a mathematical fiction.(n6))

But what is the relationship of beables that correspond to components of spin angular momentum for directions in space that are not opposite to each other? How, for example, is S[sub x] = 1/2 related to S[sub z] = 1/27 In consistent-histories quantum theory, "S[sub x] = 1/2 and S[sub z] = 1/2" is considered a meaningless expression, because it cannot be associated with any genuine quantum beable, that is, with any element of the quantum Hilbert space. Note that every non-zero element in that space corresponds to S[sub w] = 1/2 for some direction w, so there is nothing left over that could describe a situation in which two components of the spin angular momentum both have the value 1/2.

Putting it another way, there seems to be no sensible way to identify the assertion "S[sub x] = 1/2 and S[sub z] = 1/2," with S[sub w] = 1/2 for some particular direction w. (For a more detailed discussion, see section 4A of reference 7.) That agrees, by the way, with what all students learn in introductory quantum mechanics: There is no possible way to measure S[sub x] and S[sub z] simultaneously for a spin-1/2 particle. From the consistent-histories perspective, this impossibility is no surprise: What is meaningless does not exist, and what does not exist cannot be measured.

Meaningless or simply false?

It is very important to distinguish a meaningless statement from a statement that is always false. "S[sub z] = 1/2 and S[sub z] = 1/2" is always false, because S[sub z] = 1/2 and S[sub z] = -1/2 are mutually exclusive alternatives. The negation of a statement that is always false is a statement which is always true. By contrast, the negation of a meaningless statement is equally meaningless. The negation of the meaningless assertion "S[sub x] = 1/2 and S[sub z] = 1/2," following the ordinary rules of logic, is "S[sub x] =-1/2 or S[sub z] =-1/2." In consistent-histories quantum theory, this latter assertion is just as meaningless as the former. How, after all, would one go about testing it by means of an experiment?

This spin-1/2 example is the simplest illustration of quantum incompatibility: Two quantum beables A and B, each of which can be imagined to be part of some correct description of a quantum system, have the property that they cannot both be present simultaneously in a meaningful quantum description. That is, phrases like "A and B" or "A or B," or any other attempt to combine or compare A and B, cannot refer to a real physical state of affairs. Many instances of quantum incompatibility come about because of the mathematical structure of Hilbert space and the way in which quantum physicists understand the negation of propositions. Others are consequences of violations of consistency conditions for histories. In either case, the concept of quantum, incompatibility plays a central role in consistent histories. Failure to appreciate this has, unfortunately, led to some misunderstanding of consistent-histories ideas.

Now let us return to the discussion of the histories in (3) and (4). The two histories in (4) are mutually exclusive; if one occurs, the other cannot. Think of them as analogous to S[sub z] = 1/2 and S[sub z] = -1/2 for a spin-1/2 particle. On the other hand, each of the histories in (4) is incompatible, in the quantum sense, with the history in (3), which one can think of as analogous to SI = 1/2. Indeed, the relationship between the state |S> in (3) and the states |C>|D[sup *]> and |C[sup *]>|D> in (4) is formally the same as that between the state S[sub x] = 1/2 and the states S[sub z] = 1/2 and S[sub z] = -1/2. Consequently, the question of whether (3) occurs rather than, or at the same time as, the histories in (4) makes no sense.

It may be helpful to push the spin analogy one step further. Imagine a classical spinning object subjected to random torques of a sort that leave L[sub x], the x component of angular momentum, unchanged while randomly altering the other two components, L[sub y] and L[sub z]. In such a case, a classical history that describes only L[sub x] will be deterministic; it will have a probability of 1. L[sub z], on the other hand, can be described by a collection of several mutually exclusive histories, each having a nonzero probability.

Of course, classical histories of this kind can always be combined into a single history, whereas the deterministic quantum history in (3), corresponding to the L[sub x] history in this analogy, cannot be combined with the stochastic histories in (4), the analogs of the L[sub z] histories. Nevertheless, the analogy has some value in that it suggests that (3) and (4) might be regarded intuitively as describing alternative aspects of the same physical situation. Although all classical analogies for quantum systems break down eventually, this one is less misleading than trying to think of (3) and the set of histories in (4) as mutually exclusive possibilities. It helps prevent us from undertaking a vain search for some "law of nature" that would tell us that (4) rather than (3) is the correct quantum description.