I first came across the work of Roger Penrose 25 years ago, when I was marooned in a sea of mud and snow that was to become the campus of the new university of Louvain-la-Neuve. After years of rivalry, the ancient university of Leuven in the Flemish-speaking part of Belgium had split off what was to be its French-speaking Walloon equivalent to be constructed on a greenfield site south of Brussels. Apparently the bad feeling was such that the library had to distribute the even volumes of the Physical Review to one site and the odd ones to the other. I was a visiting professor in theoretical physics and spent many evenings in the one functioning bar talking with a middle-aged "mature" research student who was studying one of Penrose's many papers on general relativity. Almost like a refrain to a song, he repeatedly shook his head and marvelled, saying, "Oh, that Penrose!" Later that same year I attended a seminar Penrose gave at Cern at which he introduced his work on "twistors". His motivation was to rid quantum field theory of the "infinities" that plague the conventional approach via Feynman diagrams. To obtain numbers that make sense from these diagrams then requires all the elaborate rigmarole of the "renormalisation" programme - which Paul Dirac, and Richard Feynman himself, felt was merely sweeping the real problem "under the carpet". Penrose believes that the troublesome infinities arise at the start from the use of individual points in space-time in the formulation of the theory. His twistor approach "smears out" points in a way which he thinks is more likely to be consistent with quantum theory and the uncertainty principle. If this approach is successful, twistor diagrams - without any divergence problems - would supersede Feynman diagrams as a calculational technique for quantum field theory. Twenty-five years on, as Penrose admits in his afterword to this collection, the twistor programme has not yet really matured into a serious physical theory. But, as Toby Bailey remembers in his contribution, Penrose used to draw "a picture of twistor theory as a tree, which he was trying to train mainly towards fundamental physics, but which was always sending off vigorous shoots in the direction of pure mathematics."

This volume brings together contributions by many distinguished scientists presented at a symposium organised in honour of Professor Sir Roger Penrose in his 65th year. The goal of the conference was to examine the interaction between geometry and physics and highlight the contribution made by Penrose to many areas of mathematics and physics. The articles cover a vast range of topics from differential geometry and integrable systems to gravity and quantum mechanics. The collection is prefaced by a lovely "Laudatio" by John Wheeler, remembering the young Penrose in Princeton, and a personal appreciation by Michael Atiyah, his fellow mathematics professor at Oxford, but, apart from these articles, this volume is for experts only. The language of mathematics is wonderfully arcane, even for a lapsed theoretical physicist such as myself. For a full appreciation of all the papers included here, one would need more than a nodding acquaintance with a whole slew of equations - Nahm, Halpern, Seiberg-Witten, Bogomol'nyi, Painleve's sixth, Schmid - and invariants - Chern-Simons, Casson, Godbillon-Vey, Reshetikin-Turaev, Jones Polynomial, Donaldson. Not to mention Radon transforms, de Rahm cohomology classes, the Codazzi-Raychaudhuri identity, hypoelliptic calculus, Betti numbers, Reidermeister moves, Hochschild cycles, the Cheeger-Gromoll splitting theorem and Toda chains, to list only a few of the terms sprinkled throughout the book.

Although the main themes of this book - and of Penrose's research - are the problems of general relativity and quantum mechanics, and their union into a true theory of quantum gravity, there is an interesting article on "Penrose tilings", curious non-periodic, space-filling patterns of tiles that can now be bought as jigsaw puzzles. This apparently "academic" observation by Penrose has led to the realisation that there may be new forms of solids - quasi-crystals - with symmetries long thought to be forbidden by Bravais's laws of crystal symmetry developed more than 150 years ago. Paul Steinhardt's article concludes with the statement that "this demanding problem is Roger Penrose's inadvertent legacy to the world of solid state physics".

The breadth of Penrose's interests is well illustrated by the inclusion of two articles by leading exponents of the new fields of quantum cryptography and quantum computing. Artur Ekert begins his article with a fascinating discussion of the encryption techniques of the ancient Spartans and of Julius Caesar, before reviewing Vernam ciphers, one-time pads and public key cryptographic systems. Remarkably, Albert Einstein and John Bell both have something to contribute to new quantum key distribution systems with their focus on the mysterious, intrinsically quantum-mechanical property of "entanglement". The theme of entanglement is enlarged upon by Richard Josza in his discussion of quantum computers. These are computers that can manipulate quantum information to solve problems much more efficiently than on classical computers. So efficiently, in fact, that if we could build a sizeable quantum computer, the basis of many modern cryptographic systems would be under threat - which explains the interest in this subject from certain government agencies around the world. More exciting perhaps is the prospect of exploring the limitations of quantum mechanics. Because quantum computers derive their extraordinary power from the specifically non-local entanglement property of quantum mechanics, they "may provide particularly acute opportunities to witness a possible breakdown of the current conventional quantum formalism".

Quantum computers running new quantum "programs" could generate many surprises. For example, if we want to determine whether or not a coin is genuine, with a "head" on one side and a "tail" on the other, or counterfeit, with two heads or two tails, we need to make two "measurements' and look at both sides of the coin. As first pointed out by David Deutsch and Richard Josza, a quantum computer could solve the analogous problem with only one measurement. The fundamentals of quantum mechanics are further explored by Lev Vaidman in his paper on interaction-free measurements. This is illustrated by what Penrose has called the Elitzur-Vaidman bomb-testing problem. Using the mysteries of quantum mechanics it is possible to determine whether an ultra-sensitive bomb is dud or not without causing it explode: classically, just shining light at a mirror attached to a "good" bomb will cause it to explode. One can even design a scheme for a computer that will allow knowing the result of a computation without computing.

Which brings us back to one of the main themes of Roger Penrose's research over the years - the possible role of quantum gravity in the "collapse of the wave function" in quantum mechanics. Anyone who has read Penrose's tour de force survey of mathematics and physics in his popular book The Emperor's New Mind knows he has a wonderful way with words. Abhay Ashtekar begins his paper with a quotation from a 1976 paper by Penrose: "If we remove life from Einstein's beautiful theory by steam-rollering it first to flatness and linearity, then we shall learn nothing from attempting to wave the magic wand of quantum theory over the resulting corpse." This illustrates very well Penrose's belief that new ideas are needed to resolve the problems of quantum gravity and indeed of quantum mechanics itself. Needless to say, his suggestion - with biologist Stuart Hameroff, who also contributes to this volume - that the source of macroscopic quantum effects may lie in neuronal "microtubules" and may be related to the problem of consciousness is still controversial.

I have just read the book On Giants' Shoulders, a pleasing collection of biographical sketches of 12 famous scientists put together by Melvyn Bragg and inspired by the involvement of scientists in his late-lamented Start the Week radio programme. The collection begins with Archimedes and ends with Crick and Watson, and, although I deplore the omission of Kepler, one of my heroes, Bragg's selection is eminently defensible. That is, apart from one statement at the end of the chapter on Marie Curie, where he places Penrose on a par with Copernicus, Kepler and Maxwell. Much though I admire Penrose's achievements and thought-provoking research, I do not think he is yet ready for such canonisation. Nevertheless, as Atiyah says in his introduction, I do agree that we should be grateful that Penrose is "one of those scientists who are helping to diversify our 'gene pool' of ideas in science". This volume is a fitting tribute to his endeavours over the past 40 years.

Tony Hey is professor of computation, University of Southampton.

## The Geometric Universe: Science, Geometry and the Work of Roger Penrose

Editor - S. A. Huggett, L. J. Mason, K. P. Tod, S. T. Chou and N. M. J. Woodhouse

ISBN - 0 19 850059

Publisher - Oxford University Press

Price - £29.50

Pages - 431

## Have your say