For example, Kent below writes:

"There is, though, no parallel in standard quantum mechanics for the prediction and retrodiction of contradictory propositions, and many might feel that no acceptable interpretation of quantum theory should allow this sort of self-contradiction. … A scientific theory which maintains that all crows are black and that all crows are pink is surely in deep trouble, even if for some reason it is unable to show that pink implies not-black. … If we reject these defences we seem to be left with the conclusion that the self-contradictions of the consistent histories formalism make it impossible to take it seriously as a fundamental theory in its present form."

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gr-qc/9604012 4 Apr 96

DAMTP/96-18 gr-qc/9604012

http://xxx.lanl.gov/abs/gr-qc/9604012

Consistent Sets Contradict

written by

Adrian Kent

University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K.

Abstract In the consistent histories formulation of quantum theory , the probabilistic predictions and retrodictions made from observed data depend on the choice of a consistent set.

We show that this freedom allows the formalism to retrodict several contradictory propositions which correspond to orthogonal commuting projections and which all have probability one.

We also show that the formalism makes contradictory probability one predictions when applied to generalised time-symmetric quantum mechanics.

1. Introduction

The consistent histories approach to quantum theory , pioneered by Griffiths, Omnes, and Gell-Mann and Hartle, is arguably the best attempt to date at a precise formulation of quantum theory that involves no "hidden" auxiliary variables and can be applied to closed systems.

Since modern ideas in cosmology and quantum gravity require an interpretation of the quantum theory of the universe, the approach has naturally attracted a good deal of interest. It seems that without new axioms, whose precise form is presently unknown, no interpretation of the consistent histories formalism can unambiguously reproduce the standard predictions of classical mechanics and Copenhagen mechanics, although these predictions are not contradicted.

It should be said, though, that debate over the scientific status of the formalism continues and that its proponents tend to regard its lack of predictive power with more equanimity than do its critics.

This letter, however, looks at the logical properties of the consistent histories formalism rather than interpretational questions. The somewhat counterintuitive properties of consistent sets of histories have, of course, already been extensively investigated in the original literature and elsewhere.

Here we describe a feature which seems to have gone unnoticed, namely that different consistent sets extending a given history can imply , with probability one, propositions which are actually contradictory .

It is well known that the predictions and retrodictions made in different sets generally correspond to non- commuting projections and so are incompatible. The inferences we consider here, though, correspond to commuting but orthogonal projections.

The fact that they are nonetheless each assigned probability one is a result with no parallel in standard quantum theory.

It raises the question of whether the present version of the consistent histories formalism is a natural generalisation of Copenhagen quantum mechanics.

2. The Consistent Histories Formalism

W e begin with a brief description of the simplest version of the consistent histories formulation of non-relativistic quantum mechanics, in which sets of histories correspond to sets of projective decompositions.

While consistent histories can be defined abstractly on any Hilbert space H , it is generally assumed that operators corresponding to the Hamiltonian H and other physically interesting observables, such as position, momentum and spin, are given. The dynamics, however, are irrelevant to the examples we consider, so that we will take H = 0 and will not need to distinguish any particular operators as simple physical observables.

We are interested in a closed system whose initial density matrix is given. W e will also be interested in applying the formalism to a version of the time-symmetric generalisation of quantum mechanics first discussed by Aharonov, Bergmann and Lebowitz.

This is the non-relativistic version of the theory obtained by imposing initial and final conditions in quantum cosmology , and requires an initial density matrix, which we take to have the standard normalisation and write as initial and final boundary conditions defined by a positive semi-definite matrix normalised so that its Trace = 1.

The initial and final matrices then give boundary conditions for the system at times t i and t f , with t i less than t f . These times are irrelevant to our example, but for definiteness they may b e taken to be 0 and 1 . The physical propositions we are interested in correspond to members of sets of orthogonal Hermitian projections...

These projective decompositions of the identity should be thought of as being applied at definite times.

However, as our results depend only on the time ordering, we will omit explicit time labels and take sets of the form S.to be ordered with time increasing from left to right. The projections correspond to propositions about the system in the usual way.

For example, a projection on to the z = 1/2 eigenspace of a spin-1 /2 particle applied at time t corresponds to the statement that the particle was in the z = 1/2 eigenstate at the relevant time.

The consistent histories formulation differs from Copenhagen quantum mechanics, however, in that there is no dynamical projection postulate attached to statements of this type.

Suppose now we have a list of sets of this form. Then the histories given by choosing one projection from each ... in all possible ways are an exhaustive and exclusive set of alternatives.

Axiom of choice used here? Is there an alternative "non-standard" axiom that GMH needs to use to get out of the inconsistency? (J. Sarfatti) We use Gell-Mann and Hartle's decoherence condition, and say that S is a consistent set of histories if T or,

in the time-symmetric case, .... When S is consistent, p ( i 1 : : : i n ) is the probability of the history ..... W e say the set So ..... is a consistent extension of a consistent set of histories ......

Suppose now that we have a collection of data defined by the history ..... which has non-zero probability and belongs to the consistent set S . T his history might, for example, describe the results of a series of experiments or the observations made by an observer. To make scientific use of the formalism we then want to make further inferences from the data.

In the standard formalism, this can only be done relative to a choice of consistent extension So of S.

Once So is fixed w e can make probabilistic inferences conditioned on the history H .

For example, if So has the above form, the histories extending H in So are.....and the history H i has conditional probability ......

We use the convention that the calculation is made at the time of the last event n from the history , so that any projection occurring before this last event is a retrodiction.

Thus if k = n then ...is the probability with which the proposition corresponding to the projection Q ...is predicted; if k less than n it is the probability with which the proposition is retrodicted.

The different So are to be thought of as different, equally valid, possible pictures of the past and future physics of the system, or more formally as different and generally incompatible logical structures allowing different classes of inferences from the given data.

W e consider only the standard formalism here. It is possible to amend the formalism by appending axioms which identify natural retrodictions. .....

If only these retrodictions are allowed, contradiction is a voided. However, this would also exclude almost all scientifically desirable retrodictions.

3. Contradictory retrodictions and predictions W e now give two simple examples of contradictory inference in the consistent histories formalism.

The Hilbert space H is taken to be infinite-dimensional and separable. Example 1

It is not hard to verify that So is consistent and that the conditional probability of .... given H is 1. It is also easy to see that there are at least two mutually orthogonal vectors b satisfying . ... Thus this construction produces consistent sets which give contradictory probability one retrodictions.

Above was a contradiction of GNH for orthodox quantum mechanics. Next comes a similar contradiction for Aharonov’s et-al post-orthodox theory.

Example 2 Now consider the formalism applied to generalised quantum mechanics.

.... Example 1. As above, the conditional probability of ... given H is 1, so that we obtain consistent sets which give mutually contradictory probability one predictions.

Note that it is impossible to produce an example in which the formalism makes contradictory predictions when applied to ordinary quantum mechanics.

In this context, if P is predicted with probability 1 from the history H ....

Thus if a projection Q orthogonal to P belongs to any consistent set then its probability in that set, conditional on the history H , is zero.

Note also that we can construct examples in which any number of consistent sets make mutually contradictory retrodictions or, in the case of generalised quantum mechanics, predictions by choosing ..... sufficiently small.

4. Conclusions

The incompatibility of the logics corresponding to different consistent sets is generally described as a natural generalisation of the principle of complementarity in Copenhagen quantum mechanics: a discussion making precisely this point can be found, for example, in Chapter 5.4 of Omnes' recent book.

There is, though, no parallel in standard quantum mechanics for the prediction and retrodiction of contradictory propositions, and many might feel that no acceptable interpretation of quantum theory should allow this sort of self-contradiction.

Indeed, Omnes comments that

"The worst event would be if two different ways of reasoning could lead to different conclusions when one is using two different consistent logics. In view of this danger, which would mean that the present approach is completely wrong, we shall initially discuss how two different logics can be related to each other."

Now Omnes has in mind here a slightly different possibility , namely that if two propositions both belong to two distinct consistent sets, and one implies the other in one set, the implication might fail in the other set.

This cannot happen in the consistent histories formalism. It is not possible, for example, to use the same set of data to predict the proposition P in one set and its negation in another, both with probability one.

At first sight it ma y seem as though the above examples do precisely this. The reason why they fail to do so is that, in the consistent histories formalism, if we ha v e t w o propositions corresponding to projection operators P included in Q (i.e. the range of P is a subspace of that of Q ) and if P is predicted with probability one, it does not follow that Q is predicted with probability one (or with any other probability).

It might possibly be argued that this last feature is less of a flaw, and that the examples above are less worrying, than the type of contradiction Omnes considers, but it is hard to see why.

A scientific theory which maintains that all crows are black and that all crows are pink is surely in deep trouble, even if for some reason it is unable to show that pink implies not-black.

Nor can the fact that the theory stipulates that the pictures corresponding to different sets are incompatible alternatives be used as a defence here without allowing the same defence in the case of Omnes' hypothetical disaster.

Another possible view is that, in the end, scientists need only worry about predictions, and contradictory predictions can be avoided by restricting the formalism to standard, rather than generalised, quantum mechanics.

One difficulty with this line of defence is that it is the retrodictive cosmological applications of the consistent histories formalism that are presently the most interesting.

Unlike other approaches to quantum theory , the formalism allows us to discuss series of past cosmological events and to assign probabilities to them, even when some or all of the events occurred before the formation of classical structures.

Prediction, on the other hand, is where the consistent histories formalism is at its weakest. No coherent interpretation of the formalism has been found which unambiguously implies the standard predictions of Copenhagen quantum mechanics, although those predictions (among many others) can be reproduced by calculations within the formalism.

Moreover, though the formalism allows many different predictive calculations, those which are new seem to be physically irrelevant except in highly implausible scenarios and, possibly, in the case of generalised time-symmetric quantum cosmology . This, though, is precisely the case in which contradictory predictions arise.

If we reject these defences we seem to be left with the conclusion that the self-contradictions of the consistent histories formalism make it impossible to take it seriously as a fundamental theory in its present form.

This means that further constraints beyond consistency are needed in order to construct a natural generalisation of the Copenhagen interpretation to closed systems.

Whether physically sensible and mathematically precise constraints can b e found in standard versions of the formalism, such as the one above, is an important and intriguing op n question. It might also be interesting to in vestigate the analogous problem in the more abstract schemes characterising the logical structure of consistent histories which have recently been developed.

For a clearly and carefully expressed alternative view the reader is encouraged to consult Ref. [10] in which it is argued that the consistent histories formalism requires the use of non-standard logic and necessitates a revision of our ideas as to the scope of science.

Acknowledgements

I would like to thank Fay Dowker for helpful discussions and Bob Griffiths for valuable comments and a critical reading of the manuscript. This work was supported b y a Royal Society University Research Fellowship.

References

[1] R.B. Griffiths, J. Stat. Phys. 36 (1984) 219 (1985)

[2] R.B. Griffiths, Found. Phys. 23 (1993) 1601.

[3] R. Omnes, J. Stat. Phys. 53 (1988) 893.

[4] R. Omnes, The Interpretation of Quantum Mechanics , Princeton University Press, Princeton (1994).

[5] M. Gell-Mann and J.B. Hartle in Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity, Vol. VI I I, W. Zurek (ed.), Addison Wesley , Reading (1990).

[6] M. Gell-Mann and J.B. Hartle in Proceedings of the NATO Workshop on the Physical Origins of Time Asymmetry, Mazagon, Spain, September 30-October 4, 1991, J. Halliwell, J. Perez-Mercader, and W. Zurek (eds.), Cambridge University Press, Cambridge (1994); gr-qc/9304023 http://xxx.lanl.gov/abs/gr-qc/9304023.

[7] M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.

[8] F. Dowker and A. Ken t, J. Stat. Phys. 82 (1996) 1575.

[9] A. Kent, "Quasiclassical Dynamics in a Closed Quantum System", gr-qc/9512023 http://xxx.lanl.gov/abs/gr-qc/9512023, submitted to Phys. Rev. A.

[10] R.B. Griffiths, "Consistent Quantum Reasoning", quant-ph/9505009 http://xxx.lanl.gov/abs/quant-ph/9505009.

[11] F. Dowker and A. Ken t, Phys. Rev. Lett. 75 (1995) 3038.

[12] Y. Aharonov, P . Bergmann and J. Lebowitz, Phys. Rev. B 134 (1964) 1410.

[13] R. Omnes, Found. Phys. 25 (1995) 605.

[14] F. Dowker and A. Kent, "Retrodiction in Quantum Mechanics", in preparation.

[15] A. Kent, "Consistent Records of Inconsistent Events", gr-qc/9604nnn, submitted to Proc. Roy . Soc. Lond.

[16] Page 161 of [4 ].

[17] C.J. Isham, J. Math. Ph ys. 23 (1994) 2157; C.J. Isham and N. Linden, J. Math. Phys. 35 (1994) 5452.

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