Relativity Needs a Manifold with a Metric to Introduce Spinors

NATURE VOL. 313 14 FEBRUARV 1985 607 ------BOOKREVIEWS,------::..: Advertisement Penrose's way with paid. While one needs only a differential manifold to speak of tensor fields, one Keep in touch with the foremost relativity needs a manifold with a metric to introduce spinors. However, in general relativity, laser research through Abhay Ashtekar space-time manifolds do, naturally, come with metrics which dictate physical Spinors and Space-Time. Vol. 1 Two­ processes such as propagation of light and Spinor Calculus and Relativistic Fields. ticking of clocks. In this setting, By R. Penrose and W. Rindler. conceptually, spinors are indeed a more Cambridge University Press: 1984. natural starting point than world-vectors Pp. 458. £45, $89.50. and tensors. The use of spinors also has technical advantages. Since the spin-space is a two-dimensional (complex) vector AN INTERNATIONAL JOURNAL ALTHOUGH spinors were introduced by space, it comes equipped with a unique (up Elie Cart an about 70 years ago, their use in to an overall multiplicative factor) non­ Editors:Asia Koichi Shimoda, Keio general relativity became common only degenerate 2-form. Together with its University, Japan. East Europ"e {';( USSR rather recently. It is in fact easy to trace the inverse, the 2-form can be used to identify V. S. Letokhov, USSRAcademyof history of this development. contravariant and covariant spinors, that Sciences. Western Europ"!;'J. Joussot· The first major event was the lecture is, to raise and lower indices. Further, a Dubien, Laboratoire de Chimie by Roger Penrose, entitled "General spinor which is anti-symmetric in any two Physique, Talence, France & R. J. Relativity in Spinor Form", given at the of its indices is proportional, in those Vetter, CNRS II, Orsay, France. ~ Canada PauiJ. Dagdigian, Johns third international conference on general indices, to the 2-form and the proportion­ Hopkins University & Ahmed H. Zewail, relativity and gravitation, held at ality factor is easily obtained by contracting California Institute of Technology, USA Royaumont, near Paris, in June 1959. The the two indices in question; consequently reason behind the success of spinor only symmetric spinors matter. These simpli­ '~ .. with the ferment ofactivity in laser methods is perhaps best explained in the fications are not shared by world-tensors research and the quality ofits opening sentence of Penrose's report in the because the tangent space at a space-time international board, this newjournal proceedings of the conference. It reads: point is four (real) dimensional. shoutd rapidly gain momentum and respectability" I propose to show that if a spinor calculus is used The monograph first introduces the - The Times Higher in general relativity, instead of the usual tensor reader to the geometry of spinors in Education Supplement calculus, many of the important expressions of Minkowski space, then treats in detail the theory take on a surprisingly simple form. LASER CHEMISTRY is the first journal spinor algebra and calculus in the abstract to unite international research activity The elegance and the power of the new index notation and concludes by applying in this rapidly developing field. It is techniques made an immediate impact. I the framework to the study of various intended to bridge the gap between am told that a group of younger par­ physical fields in curved space-time. It is physics and chemistry laser-related ticipants, several of whom were destined to written with a wonderful sense of balance. research. Thejournal publishes become leaders in general relativity, sought Thus, while the treatment is clearly original theoretical and applied papers out Penrose for additional evening lectures rigorous, the style is not terse. There are as well as overviews covering a large during the conference itself. Spinorial remarks to assist the reader to see why one number of topics, but with particular techniques soon began to appear in is doing what one is doing and where one is focus on the following: laser·selective chemistry; reactive scattering and numerous research articles, particularly in headed, and beautiful figures to help state-to-state dynamics; multiphoton the subject of gravitational radiation. visualization of many constructions. dissociation, ionization and The field next received a boost from the Consequently, the book will serve at least absorption phenomena; laser-assisted detailed article by Felix Pirani in the three purposes: it can be used as a text in a collisions and reactions in l.H ilks and on proceedings of the 1964 Brandeis summer "special topics" course in general relativity surfaces; primary processes in photo• school, which made the subject accessible or mathematical physics; it can be read chemistry and chemiluminescence; to beginning research students. profitably by individual graduate students molecular relaxation processes; Penrose himself had given a series of and beginning researchers wanting to dephasing. energy redistribution and lectures on spinors somewhat earlier and become acquainted with the spino rial ultrafast dynamics; new laser the present monograph grew out of notes description of the world on their own; and techniques; application of lasers in biolOgy. taken by Wolfgang Rindler. Although the it will be a very valuable source for existence of these notes was well known in advanced researchers. Here, at last, one For speedy publication, LASER the relativity community, they did not can find, in one place, the numerous CHEMISTRY also publishes 'rapid receive wide circulation and their spinorial formulae - not to mention communications' in camera·ready appearance in book form, polished and up­ constructions, theorems and their proofs form. dated, has been eagerly awaited for more - that researchers in general relativity so Subscription Information: than 15 years! During the hiatus the field often need, present~d in the most general 6 issues per volume ISSN: 0278-6273 has grown and with it the size of the book. context. Current volume: Volume 6, 1984- 1985 In addition, the book has several novel Institutional rate: $ I 46/volume It is now a work in two parts. The second Individual rate: $74lvolume volume, to be published at the end of this features which will make it attractive to a (Individual rates available only to those who year, will contain material on null infinity wider variety of readers. One is the subscribe directly from the publisher and pay and twistor theory, as well as certain recent systematic development and use of the through personal cheques or credit cards.) developments such as the proof of the abstract index notation. This notation was SE~DPORPREESAMPLECOPY positive energy theorems and the definition invented by Penrose in order to treat tensor To order, send cheque or credit card of quasi-local conserved quantities. fields in an intrinsic, coordinate-free way information to: Roughly, one can think of spinors as and yet retain the flexibility offered by the 1.._ Harwood square roots of vectors; it takes the product presence of indices, which is so useful in ... Academic of a spinor with its complex conjugate to long calculations. In this notation, indices ap Publishers make a vector. It is therefore only natural serve simply as abstract markers which tell Marketing Dept. 50 West 23 Street, to regard spinors as the primitive objects the nature of the tensor considered (for New York. NY 10010, USA or I Bedford from which vectors and tensors can be example its contravariant versus covariant Street. London WC2E 9HD, UK built. In this, though, there is a price to be character) and do not take on numerical

© 1985 Nature Publishing Group 601 ------_____ BOOK~EVVS,------NA~~~-V~O~L~"3~13~1~4~Am==RU~AR~Y~I~= values. Mathematicians should find this British English from North America. The notation most interesting. Another feature Pronouncing on the popular myth that the English spoken in is that 2-component spinors are used the United States is antiquated and there­ throughout the book; the simplicity and language fore somehow "purer" than that spoken utility of these objects is thus brought Joanna Czechowska in Britain, he largely dispels; in the home and high energy physicists may well seventeenth century words such as fertile. succumb to their charm. Also, the leisure and dormitory were pronounced in definition of a manifold in terms of its ring The English Language. By Robert Burchfield. both the British English and American of smooth functions should please many a English manner. mathematical physicist. While this Oxford University Press: 1985. Pp.194. £9.50, $19.95. This is a brief but clear and immensely approach is not new, to my knowledge readable book. Fifteen-hundred years of this is the first time that it appears in a English has been condensed into some monograph accessible to physicists. DURING the fifth century AD the Roman 200 pages of text without losing sight There are, however, a few omissions that establishment moved out of the British Isles of how rich, diverse and complex our lang­ I personally regret. The first is the and Germanic tribes from northern uage is. 0 geometrical interpretation of Lie Germany and southern Denmark moved in. derivatives, and of curvature tensors. The descendant of the language they spoke Joanna Czechowska is on the staff of Nature. While the "algebraic or axiomatic is now the mother tongue of more than 300 approach" to derivative operators used million people worldwide and the second here is excellent, in that it brings to the language of many millions more. Robert forefront the properties that one Burchfield's book documents the history of African past frequently needs in practice, graduate English from the fifth century to the students would have been greatly helped if present day, tracing the changes caused by A.C. Hamilton the geometrical structure involved had the absorption of other languages, the been somewhere explained. Secondly, it influence of English settlers in overseas Southern African Prehistory and would have been helpful if a symbol- the colonies and the increasing introduction of Paleoenvironments. abstract index analogue of the Infeld-Van new words designed to cope with techno­ Edited by Richard o. Klein. der Waerden symbol - representing the logical advances. Balkema: 1984. Pp.404. DfJ.70, isomorphism between hermitean spinors Old English, spoken from the fifth cen­ £17.50, $24. and tensors had been first introduced tury until the Norman Conquest, was an explicitly and then dropped in calculations inflected language belonging to the Ger­ EVER since the discovery of involving a fixed space-time metnc. When manic family. It was written down using australopithecines in the 19205 and 19305 one deals with a fixed conformal class of runic characters, an alphabet consisting of southern Africa has been a focus of metrics, this symbol can be dispensed with linear strokes which could easily be carved attention for the study of the evolution of and suitable spinor indices can be simply onto wood or stone. Even when the Roman man. The archaeological record is one of identified with tensor indices. However, alphabet was adopted, certain runic charac­ the longest in the world. Progress has been sometimes - for instance in certain ters remained for a time to symbolize hampered by the difficulty of dating the perturbation calculations as well as in peculiarly English sounds. The advent of major sites, although, as Rightmire points quantum gravity - one needs to work with Christianity in the sixth century brought out in this book, some control is possible a wider class of metrics. In these cases, it is about the introduction of many Latin through comparison of fossil hominids and important to keep track of how the chosen words, but it was after the Conquest of vertebrate assemblages with the better isomorphism between spinors and tensors 1066 that the language changed dated East African material. The Middle changes with the choice of the metric, dramatically when Norman French words Pleistocene fossil hominid record of the particularly because the metric does not became absorbed, leaving traces which are region remains very poor. determine the isomorphism uniquely. An still discernible. A difficulty confronting prehistorians inexperienced reader may miss this point The introduction of printing in the and palaeoenvironmentalists working in altogether. Finally, I feel that it would have fifteenth century fossilized the spelling of southern Africa is that few people outside been useful to have a section or an English while pronunciation continued to their own disciplines are interested in the appendix summarizing the relation change. This accounts for the considerable details of their working techniques; despite between the 2-component spinors and the discrepancy in the way some English words the widespread popular interest in early 4-component ones. are spelt and the way they are pronounced. man. This book is aimed at the specialist These points by no means detract from British settlers in the United States. and all of the articles are marked by an the stature ofthe work as a whole. In recent Canada, Australia, New Zealand and excellent critical approach to evaluation of developments in classical gravity, there South Africa adapted the language to des­ the evidence. Most contributors do, have been instances of constructions which cribe new features they encountered in the however, attempt to provide a story-line are essentially spinorial in the sense that environment, often by borrowing words and some chapters should be they cannot even be recast in the language used by the native inhabitants, for example understandable to a wider audience. Let us of world-tensors. There are growing skunk, kangaroo, moccasin. hope that this is so. The various branches indications that such constructions may Dr Burchfield's summary of English is of Quaternary science have much to also play an important role in quantum perfectly suited for anyone with some contribute to one another and, outside gravity. Thus, the book has come out just interest in the subject. It is not a specialist scientific circles, the study of human in the wake of the realization that spinors book but it leaves the reader with a sound evolution is an especially important subject may play an even more profound role in overall view of the language without having in the Republic of South Africa. Apartheid theoretical physics than they do today. to wade through the minutiae of syntax (in­ is buttressed partly by theological dogma Because of this, and because Penrose's deed"he castigates "despotic professors of which is at complete variance with estab- ideas have influenced the younger gener­ linguistics [who] have failed to notice that ation of relativists so deeply, it is very likely they have taken the subject beyond the to become a classic. 0 • Recently published by Cambridge Univer­ reach of intelligent laymen"). There is a sity Press in their World Archaeology series is fascinating section on slang - rhyming African Archaeology, by David W. Phillipson, a Abhay Ashtekar is a Professor of Physics at slang (butcher's hook, look; Todd Sloan, concise, non-technical account of the subject. Syracuse University, New York, and at the alone), back-slang (yob) and Citizens' The book will be reviewed in a future issue of University of Paris VI. Band Radio slang which has infiltrated Nature.

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