Three more books on quantum mechanics. What more can be or needs to be said about a theory that is now a century old? But these books address three different audiences.

* Speakable and Unspeakable in Quantum Mechanics * is a collection of papers on "quantum philosophy" by the master, John Bell. I was fortunate enough to work with Bell at Cern in the 1970s and to hear him give three of these papers "live" in seminars in 1974 and 1975. This second edition of his papers has an introduction by Alain Aspect, the French experimental physicist who did so much to realise an experimental test of Bell's inequalities, together with two "new" papers that appeared after the first edition was published.

Aspect's essay, "John Bell and the second quantum revolution", presents a delightful and authoritative overview of the development of quantum theory subsequent to Bell's seminal 1964 paper on the Einstein-Podolsky-Rosen (EPR) paradox. This paper highlighted the extraordinary features of quantum entanglement, namely, that "quantum mechanics describes a pair of entangled objects as a single global quantum system, impossible to be thought of as two individual objects, even if the two components are far apart".

Aspect claims that this paper showed the new generation of physicists that the conceptual understanding of quantum mechanics achieved by Bohr and Heisenberg by the 1940s was not necessarily the end of the story. The resulting studies of entanglement and the application of quantum mechanics to single objects, rather than being restricted to making statistical predictions applicable for ensembles of objects, form the basis for what Aspect calls the "second quantum revolution". Physicists now routinely exploit the bizarre properties of entanglement in applications such as quantum cryptography and quantum computation.

Ten of the 24 papers included in this volume are concerned with entanglement and with the consequences of events at one place capable of propagating to other places at faster-than-light speed. Four papers discuss the possibility that the present formulations of quantum mechanics are incomplete and may need to be supplemented by so-called hidden variables - albeit hidden variables that allow the faster-than-light effects of entanglement. Apart from an idiosyncratic paper advocating the reintroduction of the ether with length contractions and time dilations in the teaching of special relativity, the remainder of the papers deal with the problem of quantum jumps and measurements in quantum mechanics. The speakable and unspeakable of the title represent the "shifty split" that Bell so detested in Bohr's Copenhagen interpretation: the arbitrary division of the world into a speakable classical measuring apparatus, whose properties we are allowed to talk about, and an unspeakable quantum system, whose properties we cannot talk about.

The two new papers show Bell at his trenchant best. The first, "Against measurement", analyses the ambiguities and hidden assumptions of the treatment of quantum measurement in four well-known "good books" by Dirac, Landau and Lifshitz, Gottfried and Van Kampen. Bell focuses on the problem of the observer in his usual forthright way: "What exactly qualifies some physical systems to play the role of 'measurer'? Was the wavefunction of the world waiting to jump for thousands of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system... with a PhD?" The paper ends with a discussion of attempts to define a "precise" quantum mechanics by modifying the Schrodinger equation so that large "classical" objects - such as Schrodinger's cat - are never left long in a quantum limbo but are "very quickly killed or spared"! It concludes with a memorable quotation from Richard Feynman: "We do not know where we are stupid until we stick our necks out."

The second paper is entitled "Nouvelle cuisine" and is "respectfully dedicated to the great chef". In fact the paper is concerned with the faster-than-light aspects of quantum entanglement. Bell begins with an anecdote about a seminar he once gave at the University of Hamburg under the title "What cannot go faster than light": some graffiti artist had added "John Bell, for example!" Bell picks this up and ponders whether this statement applies to the electrons in his body? With the renunciation of the idea of a classical trajectory in quantum theory, the whole concept of "the velocity of an electron" is somewhat problematic. He also highlights problems for causality - the "effect" coming before the "cause" - which hypothetical objects such as tachyons that can travel faster than light would pose for causality, by analysing the "perfect tachyon crime". Bell concludes with Einstein's judgement on the new cookery of quantum mechanics: "in my opinion it contains all the same a certain unpalatability".

* Beyond Measure * by Jim Baggott is intended as an introduction for undergraduate and graduate physical science students "to the fundamental conceptual and philosophical problems of quantum theory at a reasonably rigorous level". It sets out to bridge the gap that Baggott perceives between popular presentations of quantum mechanics that emphasise the "inherent weirdness of the quantum world" and a "serious one with its abstract mathematical formalism that all students of physical science must learn how to apply without worrying too much about what it all means..."

But, as a concession to the less mathematically minded reader, Baggott confines all the equations to optional appendices so that the book succeeds in having a broader appeal than might be imagined from the above description.

The book is divided into five parts and begins with the origins of quantum theory before moving on to a discussion of the formalism. Here, I was somewhat surprised to see Dirac's classic text on quantum mechanics dismissed as insufficiently rigorous and pride of place given instead to Von Neumann's text. The nature of this debate can be illustrated as follows. Dirac introduced what is now known as the Dirac delta function - a function that is zero everywhere except at one point where it is infinite in such a way that its integral is unity. On hearing Dirac talk about his delta function, the great French mathematician Laurent Schwartz is reputed to have said: "With such people, collaboration is impossible."

Nevertheless, this did not prevent Schwartz from providing a mathematically rigorous basis for delta functions with his invention of distribution theory and generalised functions. In practice, I would think that most practising physicists are happy with Dirac's level of rigour. In "Against measurement", Bell says: "It is not mathematical precision, but physical, with which I will be concerned here. I am not squeamish about delta functions. From the present point of view, the approach of Von Neumann's book is not preferable to that of Dirac's."

Baggott's section on "Meaning" starts with a discussion of philosophy with the positivist or antirealist position of Bohr, together with other antirealist philosophies such as constructive empiricism, pragmatism and social constructivism, contrasted with the realist philosophy of Einstein.

The debate is given a more concrete basis with a careful account of the famous Einstein-Bohr debate, leading up to Einstein's "bolt from the blue" with the EPR paradox and with Bell's contribution. A nice feature of the book is the section on recent experiments including Aspect's famous series of tests of the Bell inequalities. The concluding part surveys the alternatives to the Copenhagen interpretation. These include the De Broglie-Bohm pilot waves with their curious quantum potential, as well as sections on decoherence and on modifications to the Schrodinger equation.

Chapters on consciousness and on the ever-popular many-worlds interpretation of quantum theory form the conclusion.

By contrast, Kenneth Ford's * The Quantum World * has the subtitle "Quantum Physics for Everyone", which makes clear its target audience. Unlike John Polkinghorne's excellent book of the same title, this book is at least as much about elementary particle physics as about quantum mechanics. Although the explanations and the diagrams are at an elementary level, I suspect that the battery of terminology Ford introduces early on will be hard going for some readers. Thus, in short order, we have a whole menagerie of leptons, baryons, mesons and hadrons, fermions and bosons, the muon and the tau with their neutrinos, quarks and gluons, pions, kaons, lambda, sigma and omega particles. There is also a discussion of conservation laws and invariance principles including isospin and the discrete symmetries T, C and P. In between, there are chapters on "Quantum lumps", "Quantum jumps" and "Waves and particles" before a concluding chapter discusses such things as dark matter and dark energy.

The tone is determinedly light and is enlivened by anecdotes and historical asides, often relegated to footnotes. A nice example is the quotation from astrophysicist John Bahcall, who reportedly said that if the experimenters saw any fewer solar neutrinos, they would have proved that the sun is not shining! Notwithstanding these positive aspects, the most extraordinary omission of Ford's book is its total neglect of any of the developments and problems discussed in the books above. Bell's contribution to our understanding of the EPR paradox and entanglement receives no mention, astonishingly. There is no discussion of the measurement problem or of alternative interpretations challenging the Copenhagen orthodoxy. Even in a book aimed at a mass audience, these are serious omissions.

In summary, the new edition of Bell's papers is a must-buy for anyone seriously interested in the fundamentals of quantum mechanics and in understanding Bell's contribution to our grasp of the subject. The book by Baggott provides a helpful complement to Bell's original papers and contains a comprehensible and detailed overview of both the origins of quantum theory and of the new experiments that are now subjecting it to ever more detailed scrutiny. Finally, Ford's book is pitched at a much more popular level but for me has a rather old-fashioned feel and omits any mention of the genuinely puzzling conceptual problems faced by quantum mechanics.

Tony Hey is professor of computation, Southampton University, and director of e-Science, EPSRC.

## Speakable and Unspeakable in Quantum Mechanics

Author - J. S. Bell

Publisher - Cambridge University Press

Pages - 248

Price - £55.00 and £22.95

ISBN - 0 521 81862 1 and 52338 9

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