It is customary to dismiss the foundational issues of quantum mechanics on the grounds that they stem primarily from subjective predilections, aesthetic considerations or from mere "classical" prejudices. But what is not appreciated by many practising physicists is that several key foundational problems have become more precisely formulated, tinged with hard-core physical relevance. A number of them either are already amenable to experimental studies or promise to be so in the near future, with ingenious new ideas being explored.

The conceptual inadequacies of the Bohr-Heisenberg "standard" interpretation have sprouted various alternative approaches that have been developed with much vigour in recent times. In particular, such "nonstandard" schemes have been conceived not only qualitatively, but have been subjected to precise quantitative formulations. These in turn give rise to the exciting possibility of testable predictions beyond the standard framework. But, of course, more refinements of all these schemes are necessary to reach a stage where decisive empirical judgements could be possible.

Here it may be useful to recall the background of the genesis of the general theory of relativity. The trigger was Einstein's realisation that the equivalence principle implied that one could not construct a theory of gravitation compatible with the principle of relativity by restricting the space-time transformations to those that belong to the Lorentz group. This was then a mere conceptual lacuna in the fabric of the Newtonian theory that was otherwise mathematically coherent and compatible with all the relevant empirical data known at the time. But the need to remove this flaw at the conceptual level led (at least in hindsight) to the birth of general relativity that in turn predicted new testable results not envisaged by the Newtonian theory. The lesson to be drawn from this story is that a "conceptual dilemma" can become "creative" if it is formulated in a sufficiently precise manner, both mathematically and conceptually. The present phase of investigations into the foundations of quantum mechanics is at this sort of exploratory stage. Viewed from this broader perspective, the books under review are definitely significant contributions to the ongoing research pursuits in this field.

It was long believed that it was not possible to have any alternative interpretation of quantum mechanics that was internally consistent and empirically equivalent to the "standard" scheme. This complacency was severely jolted by the seminal papers of David Bohm in 1952 that succeeded in showing decisively that a "realist" interpretation of quantum mechanics embodying the action of a quantum-wave function on a localised entity ("particle") having well-defined dynamical variables at all instants could indeed be developed, fully consistent with the standard formalism of quantum mechanics and with all the experimentally verified quantum predictions. Some call this approach the "De Broglie-Bohm" model. But Bohm's formulation in the form now used has some major technical differences with de Broglie's original ideas (1926-), though the basic spirit is the same. Bohm himself in an appendix to one of his papers with Basil Hiley (Foundations of Physics, Volume 12, 1982) mentioned that he had "independently" developed his model "without being aware of de Broglie's work of 1926-"; it was only after sending the preprints of his 1952 papers to de Broglie that Bohm came to know about the latter's work, which he then acknowledged in the introduction to his published paper.

It is thus justified that the title of the book refers to Bohm's scheme as Bohmian mechanics, though, interestingly, in one of the well-researched articles in this book, A. Valentini tries to examine in depth the extent to which de Broglie's 1926- pilot-wave model contained the seeds of Bohm's 1952 scheme.

The volume is conceived in a form that is indeed a timely contribution in the context of a refreshing resurgence of interest in Bohm's model. It is not a mere collection of papers on this model. Instead, it is judiciously grouped under different categories: the concise and lucid presentations of the basic elements of the Bohmian approach, its applications, careful analyses of the relevant historical and conceptual aspects and instructive comparisons with other "nonstandard" schemes such as the dynamical theory of wave function collapse, consistent histories, and modal interpretations. As mentioned by the editors in their preface, not all contributors to this volume are "enthusiastic supporters" of Bohm's scheme. In fact, the points of dissent and diverse viewpoints easily discernible in the various articles should stimulate new thoughts and promise to provide useful directions for finer refinements of this model. All the contributors are competent and well-known research workers in this field. The editors deserve praise for this and for being able to gather their articles in suitable form for this volume.

Now that several "nonstandard" models of quantum mechanics have been developed that can compete with the "standard" scheme, the question of utmost importance is whether they are at all feasible and, if so, how one can empirically discriminate between them; or, will it be possible to discover new empirical signatures not envisaged by the "standard" scheme? In this context, the article by C. R. Leavens on the "tunnelling-time problem" should suggest intriguing possibilities. Among many other provocative articles in this volume (paucity of space prevents me from discussing even a few of them), let me just mention the one by P. R. Holland who presents a persuasive argument that a truly "universal theory" of the mechanical world should contain both the quantum and classical theories as special cases of a more general structure. Holland gives specific examples to illustrate his point that it is an intrinsic feature of the quantum formalism (not an artefact of any particular interpretation) that quantum and classical mechanics remain essentially different (conceptually as well as numerically) under any limiting condition.

It is fitting to quote the concluding words of James Cushing's insightful introduction to this volume: "David Bohm's seminal views on and his considerable technical contributions to quantum theory are finally receiving the long overdue scrutiny they deserve to see whether or not they will prove to be consistent, empirically adequate and fruitful for further avenues of research." One only wishes that Bohm's remarkable papers of 1952 were also reprinted in this volume so that readers could readily compare his original thoughts with the current views on his scheme. It is interesting that Bohm himself strongly believed, as Aharonov and Vaidman mention at the beginning of their contribution to this volume, that "a man cannot find the final theory of world". Bohm's vision, as Aharonov and Vaidman rightly stress, was that his model "should suggest a way for generalisation to the next-level theory" which could be a "better approximation to the correct theory of the world".

The other book under review is written by a distinguished researcher, Jeffrey Bub, who was Bohm's graduate student during 1963-65 at Birkbeck College (curiously, in the preface, Bub says Bohm did not ever mention his 1952 model while Bub was his student). Bub's research interests over many years have centred on "quantum logic" and generalisation of "no collapse" interpretations. His earlier book, The Interpretation of Quantum Mechanics, was an important contribution in developing the subject of "quantum logic". Bub's present book begins with an interesting approach to the problem of quantum-classical transition. It is based on the idea that the non-Boolean structure for the properties of a quantum mechanical system is responsible for the "irreducible" probabilities arising in quantum mechanics. This is followed by a succinct statement of the essence of the measurement problem and (a rather too brief) critique of the "standard" Dirac-von Neumann approach to this problem.

In the next chapter, the Einstein-Podolsky-Rosen paper and Mermin's version of the Greenberger-Horne-Zeilinger argument illustrating quantum nonlocality have neatly been reviewed, followed by a concise discussion on Bell's theorem. A particularly instructive treatment has been given of the derivation of Bell's inequality for stochastic hidden variables. But it is surprising Bub does not discuss Hardy's ingenious argument nor any of its variants which show that a very wide class of entangled states of a pair of particles admits a demonstration of quantum nonlocality without requiring the use of Bell-type inequalities.

The third chapter provides a fairly elaborate and an up-to-date coverage of the Kochen-Specker theorem and its recent variants (including Bub's own proof) showing the "in-principle impossibility" of assigning fixed premeasurement values to certain finite sets of observables so that the value assignments would preserve the functional relations holding among the observables. This chapter should be quite useful for anyone wishing to be initiated into the study of this topic.

From the fourth chapter onwards Bub zooms in on his chosen window - viz how to resolve the measurement problem without using the models of wave function collapse. Much of the rest of the book is an extended discussion of Bub's collaborative work with Rob Clifton. They seek to prove that, subject to certain constraints, all "no-collapse" interpretations can be characterised in a uniform way (the Bub-Clifton "uniqueness theorem"). The crucial idea of this program is that a "preferred observable" and the quantum wave function at any instant t define a "non-Boolean sublattice" in the lattice of all possible subspaces of Hilbert space. This sublattice is "determinate" in the sense that it is a sublattice of all possible propositions that can be true or false at the instant t. Different choices of the "preferred observable" correspond to different "no-collapse" interpretations - in Bohm's scheme, for instance, this "preferred observable" is position.

In particular, Bub analyses in depth how the so-called "modal interpretation" (first proposed by Van Frassen around 1973-74) can be understood from this perspective. But since the entire scheme is couched in the language of Boolean algebra, one needs to be comfortably acquainted with the mathematical nuances of this algebra to assess this scheme properly and to judge to what extent it really signifies an advance compared with the other "nonstandard" approaches to the measurement problem. I am not sure the appendix sufficiently explains the required mathematical background. There are a number of sceptics (the present reviewer being one of them) who remain unenthusiastic about the modal interpretation and the Bub-Clifton "uniqueness theorem". For them, the discussions in this book will certainly provide a much better articulated presentation than that found in the relevant original papers.

The final chapter is an elegantly argued critique of the many-worlds and consistent-histories approaches that are based on the idea of "decoherence" arising from environmental interaction. Bub rebuts their claim to have "solved" the measurement problem. He calls these schemes "new orthodoxy" and rightly contends that they signify "no real advance" in tackling the measurement problem.

Both the books reviewed here testify the true state of quantum mechanics. There is ferment, there is disagreement, there is confusion; nevertheless, there is an ample measure of the intellectual vitality that brings a little of the magic of science to our lives.

Dipankar Home is a research scientist, Bose Institute, Calcutta.

## Bohmian Mechanics and Quantum Theory: An Appraisal

Editor - James T. Cushing, Arthur Fine and Sheldon Goldstein

ISBN - 0 7923 4028 0

Publisher - Kluwer Academic

Price - £108.00

Pages - 403

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