This is a fascinating and original book, covering a vast domain of physics, mathematics and philosophy, which gives an excellent critical account of the problems being discussed by physicists at the end of this century. As the authors, T. Rothman and G. Sudarshan, write in their preface, the problems have been "compounded over the years by repeatedly exaggerated claims on the part of scientists for what science can deliver in the way of ultimate understanding or happiness. As a result, the public has become largely agnostic about science".

The authors imagine a modern version of Plato's academy in which they hold 11 debates with famous scientists of past and present, as well as with an assortment of students. These intriguing dialogues are sometimes friendly, sometimes violent, but are always conducted with good humour and a little irony about those who are convinced they have reached the ultimate truth.

Science, in my view, is now at the end of certainty. We are beginning to get to grips with a concept of nature far more complex than imagined by classical science. Classical physics hoped to give a geometrical description of nature (the foremost example being Einstein's theory of general relativity). But as a result of unexpected discoveries in the 20th century, such as the big bang, unstable particles and biological and molecular evolution, we are moving towards a more historical narrative of the universe.

The authors emphasise the close link between science and culture. The reader will find interesting remarks on Indian philosophy. This is not unexpected as Sudarshan, who has made important contributions to particle physics and quantum optics, is also an expert in Indian philosophy. Their remarks go far beyond the vague analogies with Indian philosophy found in many popular science books.

The second debate asks: "Is nature unreasonably mathematical?" Is mathematical reality outside us? An analogy with chemistry is useful. We produce millions of new molecules, which are the result of human activity; but some of them already exist in nature, while others do not. Similarly, mathematics is associated with brain activity; but the brain is part of nature. So it is not astonishing that a large part of the present mathematics seems inherent in nature. Yet, there may be mathematics never found in nature.

In the third debate, the authors discuss the question: "Is the world symmetrical?" It is a pity that they ignore most of non-equilibrium physics, except when they discuss chirality and the work of D. Kondepudi.

It is well known that the breaking of time symmetry as expressed by the second law of thermodynamics leads, far from equili-brium, to other symmetry breakings such as Turing structures, which correspond to space symmetry breaking. It is natural to speculate that time symmetry breaking appears also in the processes associated with the big bang. At this point it seems reasonable that the virtual particles of the quantum vacuum are transformed into real particles, with which we may associate an entropy.

The big bang would then correspond to an entropy explosion. Their coming into existence would therefore require no violation of energy conservation.

The price of the creation of the universe would be entropy.

In many places the authors discuss the role of chaos. They ask if chaos theory has lead to something new. However, they miss an essential point. It has been proved rigorously for deterministic chaotic maps that there exists, in addition to the Newtonian trajectory description, a second formulation in which probability plays the central role and time symmetry is broken. Deterministic chaos is the simplest example of the fact that classical mechanics, when applied to well-defined classes of dynamical systems, leads to different formulations. Probably the most important examples are "thermodynamical" systems in which, when the number of particles N and the volume V tend to infinity, the ratio N/V neverthless remains finite.

The distinction between extensive quantities proportional to N or V, such as energy and entropy, and intensive variables, such as pressure and chemical potential, is maintained when N becomes infinite. It can again be shown rigorously that these conditions cannot be satisfied by a trajectory description and require a probabilistic description including time symmetry breaking.

The authors' treatment of the Zeno effect needs comment. One of the unpleasant features of the orthodox formulation of quantum theory is its duality. On one side are the reversible processes as described by the Schrodinger equation, on the other measurement processes that are irreversible and increase the entropy. In a highly interesting paper, B. Misra and Sudarshan (1977) showed that by repeated observations at short time intervals the decay of an unstable particle, say a radioactive atom, might be slowed down. If this were to be observed, the Zeno effect would provide a justification for the duality of quantum mechanics.

In 1990, W. M. Itano and co-workers claimed to have observed the Zeno effect. But Rothman and Sudarshan omit to mention a subsequent demonstration that Itano's results can be accounted for by the usual dynamics if one includes the measurement device necessary for observation of the effect. In my view this is fortunate: for what would be the meaning of physics if mere "looking" at an atom could modify its lifetime?

I hope I can participate in future debates of the celebrated academy the authors have imagined. This book is a unique introduction to understanding aspects of nature at the limits of our scientific knowledge.

Ilya Prigogine, Nobel laureate, is director, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, University of Texas at Austin, United States.

## Doubt and Certainty

Author - T. Rothman and G. Sudarshan

ISBN - 0 738 20169 3

Publisher - Perseus

Price - £10.50

Pages - 320

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