It is an inevitable consequence of the standard (linear and unitary) formalism of quantum mechanics that the treatment of any measurement of attributes of quantum entities by a macroapparatus hAs to culminate in a pure state (superposition of macroscopically distinct outcomes) corresponding to a homogeneous ensemble comprising indistinguishable members. This is clearly at variance with a ubiquitous fact of experience - the definiteness of an individual measurement outcome. In a nutshell, the standard formalism of quantum mechanics seems actually to forbid a measurement to take place. Not surprisingly, Steven Weinberg has called this "the most important puzzle in the interpretation of quantum mechanics".

An orthodox response is that since interference between different macrostates of an apparatus is difficult (if not impossible) to observe in practice, the "pure" state at the end of a measurement process behaves as if it were a mixed state corresponding to a heterogeneous ensemble. However, critics point out that this does not "solve" the problem in principle because the absence of any observable interference between macroscopically distinct alternatives does not by itself imply that an individual alternative has actually occurred- in other words, the way a "pure" state is interpreted cannot be abruptly changed merely because the relevant evidence is difficult to obtain in practice.

Since there is no fundamental reason why the physics involved in "measurements" should be different from any other kind of physics, it is the very legitimacy of such a conceptual discontinuity or "cut", not so much whether its precise position can be specified or not, that is the crux of the issue.

The dissatisfaction with the measurement paradox has spurred a number of nonorthodox approaches (Bohm model, wave function collapse models, environment-induced decoherence models, consistent histories approach and their variants) whose salient features have been admirably analysed, succinctly and incisively, by Bernard d'Espagnat. In particular, his analysis of the environment-induced decoherence models is illuminating in highlighting the "conceptual inadequacies" of such models which are often obscured by mathematical details.

Apart from stressing that there is "no general rule of physics" which justifies assuming that the environmental properties are not observable, d'Espagnat drives home a key technical point that the reduced density operator pertaining to an ensemble of macrosystems (after the environmental states have been traced out) corresponds to not one but an infinity of mixed states, most of which do not correspond to position localised states of macro-objects. Hence the fact that individual members of one of these ensembles do exhibit the "localisation" features is not derived from such models. Related to this point, d'Espagnat clarifies in detail the often misunderstood reasons why various versions of the (so-called) ensemble interpretation "do not contribute to a better understanding of the conceptual foundations of quantum theory".

A well-crafted critique of the different formulations of "consistent histories" (a la R. B. Griffiths, R. Omnes, M. Gell-Mann and J.B. Hartle) is another noteworthy feature of d'Espagnat's book. His central reservation about these interpretations is that they are based on a restrictive criterion for truth, viz that truth (even of macroscopic past events) is relative to what "we prefer to discuss". This conflicts with the usual concept of realism (in the sense that d'Espagnat calls "strong objectivity"): that propositions are true or false independent of whether we are able to discern which they are or of the way we try to do so.

D'Espagnat interprets the difficulties plaguing all the proposed "realist" models of quantum mechanics (including the problems of ensuring Lorentz invariance at the level of individual events described by the ontological approaches such as the Bohm scheme and the models of wave function collapse) as implying that "independent reality" is necessarily veiled, the true nature of which cannot be completely comprehended. What we can perceive and comprehend is what d'Espagnat calls "empirical reality", of which knowable features are structured.

To what extent the distinction between "empirical reality" and "veiled independent reality" signifies a substantial conceptual advance beyond the currently available "realist" models of quantum mechanics is, of course, highly debatable, though the author articulates the related philosophical nuances with marked competence. Even if one does not agree with the author's philosophical position, the book is undoubtedly an important contribution to the literature on the foundations of quantum mechanics and should provide valuable insights to anyone seriously interested in this subject.

Apart from the quantum measurement paradox, the question as to how to reconcile the enigmatic features of quantum nonlocal connections with the requirements of special relativity theory is another controversial issue. The book by Tim Maudlin examines the nature of this conflict with an impressive clarity and provides an in-depth analysis. After explaining what is so peculiar about quantum nonlocality and highlighting the importance of the new variants of Bell's theorem, the author sets about correcting a number of popular misconceptions about what form of faster-than-light connection is really ruled out by relativity theory.

First, it is pointed out that no particle which travels below the speed of light can ever be accelerated to, or beyond that speed because that would require infinite energy. However, tachyons (particles with speed always greater than light) are permitted to exist, but with very unusual properties (in some reference frames they can travel backwards in time). Can tachyons be used to understand how Bell's inequality is violated?

Maudlin explains why the answer to this question is no - violation of Bell's inequality does not imply any form of faster-than-light transportation of matter or energy. However, superluminal causal connections at the level of individual events are involved in bringing about violations of Bell's inequality - the outome of a measurement of any one member of a correlated pair does depend on the outcome of a distant measurement on the other member, even though they may be space-like separated (ie they cannot be connected by light signals). This dependence is of a form that cannot be understood as a result of some common cause. Some experts, however, believe that such connections per se need not be called causal. Maudlin probes this point of view but chooses to disagree.

Within the framework of standard quantum mechanics, the causal connections underpinning quantum nonlocal correlations do not involve any identifiable distinction between cause and effect-hence such a superluminal connection does not imply that in some frames of reference backward (in time) causation occurs (for such symmetrical connections each frame can use its own judgement of time order to decide which event is cause and which is effect). Causal paradoxes therefore do not arise. However, if one tries to understand how this form of causation operates at the individual level (in terms of, say, the notion of wave function collapse or the Bohm model), it seems necessary that some reference frame be intrinsically preferred over others (the preferred frame is assumed to be undetectable by any empirical means). That is, the problem of Lorentz invariance crops up at a fundamental level.

All these points are well argued in Maudlin's book. My only qualm is about Chapter Six where the presentation could have been clearer. Here the author tries to justify the claim that the nonlocal dependence of a measurement outcome on the distant setting necessarily requires an "infinite" information transfer, though of a very subtle kind which is unusable for the purpose of signalling (transfer of usable information in a controlled way is what Maudlin calls signalling).

On the whole, the book is a stimulating piece of work and clearly the result of much deep thought. Maudlin concludes in a rather provocative way, "Quantum theory and relativity seem not to directly contradict one another, but neither can they be easily reconciled. Something has to give: either relativity or some foundational element of our world-picture must be modified." It is hard to guess what turn this intriguing story will take; and it is only proper that the author has used "intimations" rather than "implications" in his subtitle.

In the preface to his book, Asher Peres describes it as complementing standard texts on quantum mechanics which concentrate more on applications of the theory than on explaining the nuances of its content. Peres also make clear that his book "considers only standard quantum theory . . . Readers who are interested in deviant mutations will not be able to find them here." Within this constraint, Peres has come up with a superb treatise for anyone who is prepared to delve into quantum mechanics beyond the glib, matter-of-fact presentations in most textbooks.

The great strength of this book is its penetrating discussion of many fine technical points which are often side-stepped in quantum mechanics texts. For example, Peres shows that the generalised uncertainty relation ceases to be valid for states outside the domain of definition of the relevant commutator.

The subtleties involved in the use of unbounded operators and delta functions are carefully discussed. The way Peres ex£ the delicate mathematical aspects of the use of probabilities in quantum theory, motivating the discussion from relevant experiments, is indeed a fascinating model for teaching of such topics. Mention must also be made of his explanation of metric properties, truncated Hilbert space, and representations of space-time symmetries in quantum mechanics which encapsulate the elements useful for a physicist without being enmeshed in unnecessary mathematical intricacies.

Another high point of this book is a comprehensive treatment of formal aspects of the quantum mechanics of composite entangled systems, appropriately blended with meticulous attention to conceptual clarity. Apart from covering the Einstein-Podolsky-Rosen (EPR) argument and the well-known theorems of Bell, Gleason and Kochen-Specker, Peres discusses their modern variants including his own significant contributions to this subject.

Particularly instructive is the proof that quantum nonlocality is generic in the sense that any entangled state violates Bell's inequality provided one chooses appropriate pairs of correlated observables. Surprisingly, however, Peres eschews any reference to Lucien Hardy's charming variation of the EPR-Bell argument.

The final part of the book deals with a number of topics which have hitherto appeared only in specialised journals, such as the relationship of quantum theory to thermodynamics and to information theory, quantum aspects of classical chaos, the way "quantum chaos" differs from the more familiar classical deterministic chaos, emergence of irreversibility, quantum cryptography and teleportation, measurements of finite duration and time-of-flight measurements. In fine, here is a book which deserves not only to be an integral component of any graduate-level course on quantum mechanics but should also be useful to experts as a provocative defence of the pragmatic and instrumentalist position on the conceptual issues of quantum mechanics.

Dipankar Home is a Homi Bhabha fellow, Bose Institute, Calcutta.

## Quantum Theory: Concepts and Methods

Author - Asher Peres

ISBN - 0 7923 2549 4

Publisher - Kluwer Academic

Price - £88.00

Pages - 429

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