Foundations of Physics Letters, Vol. 6, No. 3, 1993

C O N S I S T E N T HISTORIES, R O B U S T HISTORIES, A N D V E R I F I A B L E HISTORIES

C. H. Woo

Department of Phys{cs University of Maryland College Park, Maryland 20742

Reeeived April 3, 1992; revised December 9, 1992

A distinction is made among consistent histories in general, and those that are robustly consistent, and finally those that are classically verifiable. In the ease of an individual system, the Copenhagen view would regard only its verifiable history to have actually taken place. We analyze the conse- quences if instead one associates reality with a finer and yet robust history. i h e r e are distinct diiadvantages. In general one should probabilistieally sum over the fme-grained consistent histories, eren when the events have already happened.

Key words: consistent histories, Copenhagen interpretation

The status of the Copenhagen interpretation in quantum mechanics is in flux: On the one hand, it still represents the orthodoxy and is taught in many textbooks; on the other hand, its validity is increasingly being questioned by some researchers. For the sake of clarity, it may be useful to distinguish between two propositions usually associated with the Copen- hagen interpretation: (i) the separate validity of classical physics and (ii) the distinction between potentiality and actuality, as exemplified by the central role of the completion of a measurement-like event in actualizing just one out of the many potential alternatives. The first proposition is indeed questionable, but the insight contained in the Copenhagen caution against attaching too rauch reality to what is only potential still deserves our close attention. By the "separate validity of classical physics" we mean the denial that quantum physics is capable of describing all elassical fea- tures under suitable conditions. It is not clear that such a denial is what the founding fathers had in min(t, but in any case the proposition when stated in this form is now suspect, since more and more of what may be

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called classical features have already been shown to emerge from q u a n t ~ l physicsf1-3] Because, however, an important ingredient for such demon- strations is the notion of decoherence, and because a split into "decoherent histories "[4] or into "consistent histories "[s'6} bears a certain similarity to wavefunction reductions, there seems to be some tendency to regard the specification of a family of consistent histories as rendering superfluous the Copenhagen distinction between the potentiM and the actual. Such an inference would be unjustified. To be sure, the introduction of consis- tent histories [5] represents a significant advance: it focuses out attention on those alternatives which do not give rise to unnecessary conceptuM dilelmnas; but until one of these alternatives gets actualized the consistent histories that can be theoretically defmed are still potential ones.

A history is specified by a chain of Heisenberg-picture projections (E~l( t l )E~2(t2 ). . .E~"(tn)) together with a knowledge of the inltial density matrix p(to), where each set of E~'s for fixed i provides a resolution of the identity. A family of histories is said to be consistent if the probabilities for histories obey additivity rules, and consistency holds if

R « T r ( E n (tù). . .E 1 ( t l )p( to)E 1 ( t l) . . .E~ ( t h ) ) - - 0

whenever the set {a~} differs from {a~} and the sum of the two histories is another history. We shall think of the quantum theory to be self-contained, and so the projectors operate on a Hilbert space that includes any appa- ratus or whatever else needed for a closed description. 0ne indication of the "potential" nature of consistent histories is the lack of uniqueness: there can be different and complementary divisions into alternative histo- .ries that are equally refined and that are each consistent among the family members. Another indication is the fragility of the consistency of fine- grained histories: a fanfily of consistent histories can lose that consistency as a result of further projections for t > th. The meaningfulness of a sequence of events that have already actually occurred ought not be tlds ephemeral. Part of the problem is that in theory any fanfily of continua- tions of projections (E~ ' ( t l «" ~"+' )ù.E,~ (t,~)Eù+ 1 (t,~+l)...) may be considered.

As emphasized by Onmès [7] , however, not all events that are mathemati- cally definable are operationally ascertainable: e.g., a determination of the microscopic stare of the universe at a given time is something that, though in theory describable as a projection, carmot be operationally ascertained within a world of finite resources. Since it has been shown that under appropriate conditions certain quasi-projections do exhibit classical deter- ministic behavior, in the sense spelled out in ref. 7, one is justified in in- voking such classical notions as the reliability of records that are correlated with a 1arge number of other deterministicaUy behaving variables, and in speaking of some historical events as belag anchored in lasting "classical records". Of course, the classical behavior that emerges from a quantum description is only approximate, and a purist may find the promotion of a certain type of probability-near-one behavior into "classical determinism" unsatisfactory [sl. But it would be difficult to speak of objective reality

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at all if one refuses to accept the "for all practical purposes" kind of de- terminism; and, as Onmès says:"Without it(deternfinism)...it would eren be impossible to check the probabilities that are predicted by quantmn mechanics. ''[7] It is in the spirit of considering a very large quantitative difference ai tantamount to a qualitative one that we speak of classical certainty here.

A family of histories for which each projection corresponds to a non- trivial event supported by classical records may be called "verifiable his- tories." Verifiable histories are automatically consistent and remain con- sistent upon extension into longer verifiable histories, to the extent that records last. A translation of the Copenhagen emphasis on actuality into the language of histories would be that for an individual system only its verifiable history is actually true. Since, however, in terms of the degree of refinement the gap between verifiable histories and some ~ne-grained histories is buge, an inevitable question is whether some consistent history finer than a fully verifiabte one may not provide a better description of the objective reality. This is a very complicated question, but a question that one must ultimately confront. Here we restrict ourselves to a specific form of this query: we shaU consider what happens if one associates reality with an intermediate type of consistent history that is said to be "robust."

We will first define what we mean by a nontrivial projection, a no- tion already used earlier. In any chain of projections that defines a history, one can insert an additional projection without changing the probability of occurrence for that history(e.g., the insertion may correspond to an extrap- olation by dynamical evolution from either the previous or the subsequent projection). We call such projections trivial only in this limited sense; the dynanfical evolution may be nontrivial in other respects. Then a non- trivial projection is one the insertion of which changes the probability of occurrence. We shall say that a family of histories is robustly consistent if they are consistent histories specified exclusively by nontrivial projections, some of which are tied to classical records and others tied to nonretrievable orthogonal correlates, so that the consistency will persist upon extension of the histories by verifiable projections (the nonretrievable variables being traced over). A simple example will clarify the distinction among general consistent histories, robust histories, and verifiable histories. Suppose a particle during its passage through a two-slit arrangement emitted a pho- ton, which went oft in different directions depending on which slit the particle emerged from. At time t~ the particle hit the observation screen, while the photon was undetected. Then the two histories describing the particle alone which, starting from some given state, passed through either the right slit or the left one before arriving at the screen, would be consis- tent alternatives up to time th. If, after t,~, the photon is scattered in such a way that, though the initial possible directions were divergent, the scat- tered photon can end up in a conmmn detector, the extended histories may be such that the two earlier paths no longer constitute consistent alterna- tives. If, on the contrary, the photon has a clear path out to infinity and can never be detected, with the photon degrees traced out the two alterna-

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tives for the particle path are not only consistent but robustly consistent. Finally, if the photon is detected and its state is recorded before it can scatter, one can deduee thereby which slit the particle did pass through~ and one of the two alternatives becomes the actual verifiable history.

A possible challenge to the Copenhagen viewpoint may consist of ask- ing whether a robust history might not be a bettet representation for reality - "better" in the sense of being more detailed - in the situation where the photon escapes irretrievably to infmity. Although the definition of robust histories is not free of ambiguity when the probabilities of crossing the two slits are equal, one way to resolve the ambignity is by extrapolation from the cases when the probabilities are unequal. In the following diseussion we do refer to equal probability situations for the sake of simplicity, with the supposition that the choice of robust histories has been determined from considerations of nondegenerate situations. With that understaùud- ing, we now compare the Copenhagen position with the view that reality corresponds to only one of these two robust histories which describe the particle as having passed through either the right slit or the left one. Since there is no verifiable record of which slit was traversed, the Copenhagen view would hold that, even though one is dealing with a single particle and with what has already happened, the best description of the situation is still a probabilistic one: "The particle has one half probability of having arrived at the detection screen via the right slit, and one half probability via the left slit." On the other hand, the "robust history as reality" view would be:"Although the half-half description is suitable for an ensemble, the single particle under the robustness condition must have gone through only one of the slits. The lack of recoverable records should not be taken to mean that the objective reality itself is fuzzy, particularly since there is no interference effects which reqnire such fuzziness in the intermediate stage."

Since the "robust history as reatity" interpretation yields the same pre- dietions as the Copenhagen interpretation for repeated event,s, the question is whether it leads to any different expectations for nonrepeated events. The answer is "yes" in appropriate contexts, and we will now exhibit one such context. Although words like "real" and «potential" sound rather vague, what is not vague is the rule that the potential paths axe to be sununed. The situation in which a difference can arise is best explained with the help of a concrete (though highly artificial) example. Consider, instead of two slits, N slits arranged in a circle in a plane perpendicular to the path of the incident particle, which has equal probability for pass- ing through any of the N slits. Suppose the theory predicts that, given that the particle defmitely passed through the ith slit, then the probability for the particle to reach a certain region on the detection screen directly opposite the ith slit is 1 - e, and the probability that it lands in a re- gion on the screen directly opposite to the center of the circle is e, with e << 1. Assume furthermore that 1/N < e, that the condition for ro- bustness is met, and that there can be only one trial. (This «experiment" is eminently repeatable, and therefore the assumption that there can be

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only one trial is unrealistic. But the central issue here concerns quantum mechanics as the basis of a complete description of nature, and one can at least contemplate theoretieally a quantum description for the world as a whole, in which there are events that are not repeated either over t ime or through the existence of many sinfilar parts. We made the artificial assumption that there can be only a single trial in order to simulate a situation with an unrepeated sequence of events.) If now the outcome is that the particle is detected in the ith region, there would be no essential difference between the two views. Both interpretations would regard this as verification, within an error e, that the particle did cross the ith slit. If on the other hand the particle is detected in the central region, the two interpretations would view the outcome differently. Aceording to the "ro- bust history as objective reality" view, the single particle must have gone through only one of the slits, and no mat ter which slit it went through~ the probability for it to land at the screen center is miniscule compared to the probability for it ~o land opposite to the slit, which is near one. Therefore one concludes that the observed situation must be a fluke, an exceptional event. From the Copenhagen view, however, which does not insist that the particle went through one definite slit and which does al- low the probabilities for different paths to be summed even for what has already happened, the theory makes no strong prediction for the case at hand. For the stated situation, any individuM outcome must have only a small probability in any case, with "landing at the center" being more probable than any other individual outeome. Thus the observed outcome is regarded as an ordinary oecurrence, no more fluky than any other out- come. (One can of course take the safe position that for unrepeated events all theoretical predictions which are not near one in probability are to be ignored[9]; but by doing this the opportunities for checking whether one~s theoretical modeling is on the right frack become severely rednced, com- pared to the more common practice in which the goodness of a model is assessed from comparing observations and theoretical expectations even when the latter are not of the probability-near-one type.)

Some may object that "landing somewhere on the circle" is much more probable than "land ing at the center" eren according to the Copenhagen view, but this is a prediction about categories. The example of slits on a circle was used only fór simplicity; one could have constructed a less sym- metrical example in which the N + I outcomes do not divide naturally into ~categories. One notes that the probability for "landing at the center" is t he same with either interpretation; it is only in assessing its competit ion that the two interpretations come to different conclusions. Although both interpretations agree that the probability for "some outcome other than landing at the center" is near one, with the Copenhagen interpretation that probabili ty is dispersed among the N possible outcomes, and "landing at the center" is in competition against another individual outcome and not against the group. In contrast, by insisting that the particle taust have passed through only one slit, the "robust history as reality" interpreta- tion concentrates that probability, immediately after the particle's passage

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through a slit, to a single landing region. It is because of this concentra- tion of probability that "landing at center ' , by comparison, appears to be a fluke.

Thus we have displayed one context in which a difference does exist: Given that the particle has landed at the center, the "robust history as reality" view would conclude that the observed situation is a fluke, whereas the Copenhagen view would not. (There can be other difference-revealing circumstances involving a system near its ground stute, but we will not discuss those situations hefe.)

If one were to place just one bet for a specific landiug region in a single trial, the bet should be on "landing at the center" according to either view. Although it may seem strange that the "robust history as reality" viewpoint would regard the same outcome as both probable and fluky, this by itself is not a logical contradiction that would invalidates that view. The reasoning used there was just the classical additivity of probabilities, and there can be circumstances in which it is probable that a fluke will occur, in the sense illustrated above. What does mutter, however, is the fact that for a probabilistic theory to be useful in practice, the implicit assumption is always there that the system in question is not highly exceptionM. Therefore, interpreting unnecessarily terrain types of events as fiukes is something very much to be avoided. From this perspective, the Copenhagen viewpoint remains preferable. (We should add that we have considered here only the challenge to the Copenhagen interpretation arising out of the existence of fme-grained consistent histories. There is also a different and more fundamental challenge, based on the idea that the initial condition of the world may be such that the quantmn to classical connection is deterministic. [1°] In that case one may say that, contrary to Bohr's assertion, there is a quantum world; hut we cannot enter into that subject in this brief note.)

We conclude that although there are circumstances where one has a choice to regard either only the verifiable history or a rauch fmer robustly consistent history as describing the objective reality» the latter choice car- ries the price that a greater nmnber of outcomes may appear in some tight as flukes. Moreover, as it is not easy to ascertain whether unobserved correlates are retrievable, in practice one will often be unsure whether fu- tu.re evidence will not destroy the consistency of presently consistent fme- grained histories. Therefore in general one should probabilistically sum over the fine-grained consistent histories, despite the fact that the proba- bilities of different consistent histories can be added in the same way as in classical probability theory, and classically the customary th~nking is that what has already happended in the past of an individuM system is objectively definite.

Although the example we considered is obviously m'lrealistic, the issue itself is not purely academic. For example, in considering the effects of inflation-era quantum fluctuations on present day observation~~ one may find certain observed data to require such an implausible occurrence in the intermediate stage as to almost rule out a given theoretical model; but in

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fact, within the given model, the assessment of the degree of plausibility can depend on whether one identifies the early universe with a single fme- grained consistent history, or with a probabilistic sum of such lüstories.

I thank Jim Hartle for a series of correspondences and a lunch dis- cussion; this does not mean, of course, that he necessarily agrees with the content of this note.

R E F E R E N C E S

1. E. Joos and H. D. Zeh,Z. Phys. B59, 223 (1985). 2. A.D. Caldeira and A.J. Leggett,Physics A121, 587 (1983). 3. R. Oranès, J. Star. Phys.57, 357 (1989). 4. M. Gell-Mann and J.B. Hartle, "Quantum mechanics in the light of quantum cosmology," in Complexity, Entropy and the Physics of Informa- tion, W. Zurek, ed. (Addison Wesley, Redwood City, 1991). 5, R. Griffiths, J. Star. Phys.36, 219 (1984). 6. R. Omnès, J. Star. Phys.53, 893 (1988). 7. R. Omnès, Rev. Mod. Phys.64, 339 (1992). 8. J.S. Bell, Physics Wortd, August 1990, p.33. 9. J.B. Hartle, "Prediction in quantum cosmology," in Gravitation in As- trophysics ( NATO Advanced Sununer Institute, Cargese), B. Carter and J.B. Hartle, eds. (Plenum, New York, 1986). 10. L.S. S¢hulman, Phys. Lett.Al02, 396 (1984).