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Decoherent Histories


The decoherent histories (DH) approach was initiated in 1984 by Robert Griffiths [10] (who spoke, however, of consistent histories) and independently proposed by Roland Omnès [11] a little later; it was subsequently rediscovered by Murray Gell-Mann and James Hartle [12], who made crucial contributions to its development. (DH should not be confused with the environment-induced superselection approach of Wojciech Zurek [13], in which decoherence also plays a crucial role, but one that differs significantly from its role in DH. In this approach the environment is fundamental, acting in effect as an observer, so that it is difficult to regard this proposal as genuinely providing a quantum theory without observers.)

DH may be regarded as a minimalist approach to the conversion of the quantum measurement formalism to a theory governing sequences of objective events, including, but not limited to, those that we regard as directly associated with measurements. Where the Copenhagen interpretation talks about finding (and thereby typically disturbing) such and such observables with such and such values at such and such times, the decoherent histories approach speaks of such and such observables having such and such values at such and such times. To each such history h, DH assigns the same probability P(h) of happening that the quantum measurement formalism--the wave function reduction postulate for ideal measurements together with the Schrödinger evolution--would assign to the probability of observing that history in a sequence of ideal (coarse-grained) measurements of the respective observables at the respective times: If the (initial) wave function of the system is tex2html_wrap_inline1050 ,


where tex2html_wrap_inline1058 with tex2html_wrap_inline1060 the Heisenberg projection operators corresponding to the time-ordered sequence of events defining the history h. For example, for a spin 1/2 particle initially (at t=0) in the state tex2html_wrap_inline623 with tex2html_wrap_inline625 , we might consider the history h for which tex2html_wrap_inline629 at t=1 and tex2html_wrap_inline633 at t=2. For Hamiltonian H=0, (1) then yields P(h)=1/4.

DH can be regarded as describing a stochastic process, a process with intrinsic randomness. Think, for example, of a random walk, with histories corresponding to a sequence of jumps and probabilities of histories generated by the probabilities for the individual jumps. The histories with which DH is concerned are histories of observables, for example of positions of particles. While Schrödinger's spherical wave impinges continuously on a screen over its full expanse, the screen lights up at one instant at one spot because it is precisely with such events that DH is concerned and to such events that DH assigns nonvanishing probability.

To understand DH one must appreciate that the histories with which it is concerned are not histories of wave functions. For DH the wave function is by no means the complete description of a quantum system; it is not even the most important part of that description. DH is primarily concerned with histories of observables, not of wave functions, which play only a secondary role, as a theoretical ingredient in the formulation of laws governing the evolution of quantum observables via the probabilities assigned to histories. Thus DH avoids the measurement problem in exactly the manner suggested by Einstein.

It should come as no surprise that the consistent development of the DH idea, of assigning probabilities to objective histories, is not so easy to achieve. After all, Bohr and Heisenberg were no fools; they surely would not have insisted that all is observation were such a radical conclusion easily avoidable. It is only as a first approximation that DH can be regarded as merely describing a stochastic process. There are, in fact, some very significant differences. Perhaps the most crucial of these concerns the role of coarse graining. Because of quantum interference effects, coarse graining plays an essential role for DH, not just for the description of events of interest to us, but in the very formulation of the theory itself. A fine-grained history--given for a system of particles by, for example, the precise specification of the positions of all particles at all times in some time-interval--will normally not be assigned any probability. In fact, most coarse-grained histories won't either.

For example, for the two-slit experiment DH assigns no probability to the history in which the particle passes unobserved through, say, the upper slit and lands in a small neighborhood of a specific point on the scintillation-screen. Nor, indeed, does it assign any probability to the spin history that differs from the one described after formula (1) only by the replacement of tex2html_wrap_inline633 by tex2html_wrap_inline643 at t=2 . This is because (1) yields the value 1/4 also for this history, which is inconsistent with the value 0 for the corresponding coarse-grained history with t=1 ignored! (These values involve no inconsistency for the usual quantum theory, in which they concern the results of measurements, since the measurement of tex2html_wrap_inline649 at t=1 would be expected to disturb tex2html_wrap_inline653 .)

DH assigns probabilities only to histories belonging to special families tex2html_wrap_inline1064 , closed under coarse graining, which satisfy a certain decoherence condition (DC)


guaranteeing that P(h) is additive on tex2html_wrap_inline1064 and hence provides a consistent assignment of probabilities to elements of tex2html_wrap_inline1064 . (The decoherence condition in fact has several versions, the differences between which I shall here ignore. There is also a perhaps simpler version of (1), with a linear dependence on E(h), that involves a much more robust decoherence condition than DC [14].) Whether or not a family tex2html_wrap_inline1064 satisfies the DC depends not only on a sequence of times and coarse-grained observables at those times, but also upon the (initial) wave function tex2html_wrap_inline1050 (or density matrix tex2html_wrap_inline1078 ) as well as the Hamiltonian H of the relevant system, so it is convenient to regard also these as part of the specification of tex2html_wrap_inline1064 . DH thus assigns probabilities P(h) to those histories h that belong to at least one decoherent family tex2html_wrap_inline1064 (as I will call those families satisfying the DC).

It turns out, naturally enough, that a family of histories describing the results of a sequence of measurements will normally be decoherent, regardless of whether or not we actually observe the measurement devices involved. Moreover, interaction with a measurement device is incidental; satisfaction of the DC may be induced by any suitable interaction--or by none at all.

In fact, families defined by conditions on (commuting) observables at a single time are always decoherent. After all, it is for precisely such families that textbook quantum mechanics supplies perfectly straightforward probability formulas--via spectral measures. It is important to bear in mind, however, that even for such standard families, the textbook probability formulas have an entirely different meaning for DH than for orthodox quantum theory, describing the probability distribution of the actual value of the relevant observable at the time under consideration, and not merely the distribution of the value that would be found were the appropriate measurement performed. This difference is the source of a very serious difficulty for DH.

The difficulty arises already for the standard families, involving observables at a single time. The problem is that the way that the probabilities P(h) are intended in the DH approach, as probabilities of what objectively happens and not merely of what would be observed upon measurement, is precisely what is precluded by the no-hidden-variables theorems of, e.g., Gleason [15, 25, 33], Kochen and Specker [16, 33], or Bell [17, 25, 33]! It is a consequence of these theorems that the totality of joint quantum mechanical probabilities for the various sets of commuting observables is genuinely inconsistent: the ascription of these probabilities to actual joint values, as relative frequencies over an ensemble of systems--a single ensemble, defined by the wave function under consideration, for the totality--involves a contradiction, albeit a hidden one. For example, the correlations between spin components for a pair of spin 1/2 particles in the singlet state, if consistent, would have to satisfy Bell's inequality. They don't.

A simple and dramatic example of the sort of inconsistency I have in mind was recently found by Lucien Hardy [18]. For almost all spin states of a pair of spin 1/2 particlesgif there are spin components A, B, C, and D such that the quantum probabilities for appropriate pairs would imply that in a large ensemble of such systems (1) it sometimes happens that A=1 and also B=1; (2) whenever A=1, also C=1; (3) whenever B=1, also D=1; and (4) it never happens that C=1=D. The quantum probabilities are thus inconsistent: there clearly is no such ensemble. (The probability that A=1=B is about 9% with optimal choices of state and observables.) The inconsistency implied by violation of Bell's inequality is of a similar nature.

Thus, as so far formulated, DH is not well defined. For a given system, with specified Hamiltonian and fixed initial wave function, the collection of numbers P(h), with h belonging to at least one decoherent family, cannot consistently be regarded as intended--as the probability for the occurrence of h. Too many histories have been assigned probabilities: to be well defined, DH must restrict, by some further condition or other, the class of decoherent families whose elements are to be assigned probabilities. It is not absolutely essential that there be only one such family. But if there be more than one, it is essential that the probabilities defined on them be mutually consistent.

Without directly addressing this problem of mutual inconsistency, Gell-Mann and Hartle [12, 19] have emphasized that for various reasons the DC alone allows far too many families. They have therefore introduced additional conditions on families, such as ``fullness'' and ``maximality,'' and have proposed distinguishing families according to certain tentative measures of ``classicity.'' With such ingredients they hope to define an optimization procedure--and hence what might be called an optimality condition--that will yield a possibly unique ``quasiclassical domain of familiar experience,'' a family that should be thought of as describing familiar (coarse-grained) macroscopic variables, for example hydrodynamical variables. When the probability formula P(h) is applied to this family, it is hoped that the usual macroscopic laws, including those of phenomenological hydrodynamics, will emerge, together with quantum corrections permitting occasional random fluctuations on top of the deterministic macrolaws (and classical fluctuations).

GMH do not seem to regard their additional conditions as fundamental, but rather merely as ingredients crucial to their analysis of a theory they believe already defined by the DC alone. While I've argued that there is no such theory, a physical theory could well be defined by the decoherence condition together with suitable additional conditions (DC+) like those proposed by GMH, also regarded as fundamental. In this way, DH becomes a serious program for the construction of a quantum theory without observers.

It is true that much work remains to be done in the detailed construction of a theory along these lines, even insofar as nonrelativistic quantum mechanics is concerned. It is also true that many questions remain concerning exactly what is going on in a universe governed by DH, particularly with regard to the irreducible coarse graining. Nonetheless it seems likely that the program of DH can be brought successfully to completion. It is, however, not at all clear that the theory thus achieved will possess the simplicity and clarity expected of a fundamental physical theory. The approach to which I shall now turn has already led to the construction of several precise and reasonably simple versions of quantum theory without observers.

next up previous
Next: Spontaneous Localization Up: Quantum Theory Without Observers Previous: Introduction

Shelly Goldstein
Wed Aug 13 17:22:41 EDT 1997