The third volume of Steven Weinberg's * Quantum Theory of Fields * is devoted to a detailed introduction to supersymmetry, which unites fermions and bosons in common multiplets, and to supersymmetric field theories. This is an interesting choice of subject, because, although supersymmetry is an area of enormous interest in theoretical particle physics, there is no experimental evidence whatsoever that it is a symmetry of nature -in sharp contrast to the standard model of particle physics that is based on quantum-field theories with a local-gauge symmetry.

Although the standard model gives a successful description of the interactions of the fundamental constituents of matter, it has an uncomfortably large number of parameters that have to be determined experimentally, as well as leaving many fundamental questions unanswered. Most attempts to reduce the number of free parameters and to answer the remaining questions are based on the "grand unification" of the strong, weak and electromagnetic interactions (and eventually gravity). Such a unification could occur only at energy scales that are many orders of magnitude larger than those we are able to study directly in accelerator experiments. In such unified theories, the "light" and "heavy" particles interact, and it becomes difficult to prevent these interactions generating masses of the order of the unification scale for some of the particles we study experimentally and that we therefore know to be light. This is the "hierarchy problem", which traditionally can be solved only by * unnaturally * fine-tuning the strengths of some of the interactions to about 25 decimal places. In supersymmetric unified field theories, there are cancellations between bosonic and fermionic contributions to the interactions between the heavy and light particles, and hence the possibility of avoiding the hierarchy problem. This is perhaps the main reason why Weinberg is "reasonably confident that supersymmetry will be found to be relevant to the real world, and perhaps soon".

Weinberg begins with an historical introduction. He discusses the nonrelativistic quark model of the 1960s in which Murray Gell-Mann and George Zweig's SU(3) symmetry relating the three quarks that were known at the time (the up, down and strange quarks) was combined with a spin symmetry into a larger SU(6) symmetry. SU(3) multiplets of different spins were combined into multiplets of SU(6). Attempts to generalise these ideas to relativistic theories came to an end with the Coleman-Mandula no-go theorem, which states that the most general symmetry in a relativistic field theory consists of the direct product of an internal symmetry and the Poincaré group (translations and Lorentz transformations). Supersymmetry circumvents the theorem because its generators satisfy a super-algebra (which includes anticommutation relations) and not a Lie algebra (with commutation relations only). This opens the possibility of constructing a large class of relativistic theories. The author recalls, however, that the motivation in the original papers on supersymmetry was not directly related to circumventing the no-go theorem.

Thereafter comes a pedagogical discussion of the basics of supersymmetry starting with a presentation of supersymmetry algebras and the corresponding particle spectra. In the simplest case, for each (spin-1/2) quark and lepton of the standard model, there are corresponding spin-0 bosons, the "squarks" and "sleptons", and for each of the spin-1 force carriers (the photon, gluons and W-and Z-bosons) there are corresponding spin-1/2 "gauginos". In theories of gravity, the force carrier, the spin-2 graviton, is accompanied by spin-3/2 "gravitinos". With the presence of more supersymmetries (ie theories with "extended supersymmetry"), the spectrum is even richer. As none of these supersymmetric partners of the known quarks, leptons and gauge bosons has been discovered, supersymmetry, if it plays a role in particle physics, must be a broken symmetry. The breaking of supersymmetry is subsequently discussed in considerable detail.

Weinberg then teaches how to construct supersymmetric field theories. He introduces superfields, containing fermion and boson fields, which depend not only on the four space-time coordinates but also on fermionic coordinates. Weinberg devotes a chapter to the construction of supersymmetric gauge theories, ie of theories with both supersymmetry and gauge invariance. Another chapter reviews supersymmetric extensions of the standard model. For such theories to be viable, the supersymmetry must be broken. There are various approaches to this, which are reviewed, with a discussion of the perturbative and non-perturbative aspects.

The cancellation between bosonic and fermionic contributions to transition amplitudes in supersymmetric theories means that the ultraviolet divergences are generally less severe in supersymmetric theories (and, with sufficiently many supersymmetries, may be absent altogether). The corresponding non-renormalisation theorems are reviewed using both conventional techniques and with supergraphs. Such cancellations will hopefully be useful in constructing a consistent quantum theory of gravity, and there follows a discussion of "supergravity".

There is a growing community of theoretical physicists that is trying to construct a "theory of everything". Many of these attempts are based on theories of strings or other extended objects that generally include supersymmetry. The book's final chapter presents a review of supersymmetry algebras in higher dimensions.

Weinberg's volume will be popular with postgraduate workers in theoretical physics and a standard reference for experienced researchers. If some of the supersymmetric partners of the quarks, leptons and gauge bosons are discovered at the Large Hadron Collider at Cern or at another particle accelerator, then this book will be found not on the shelves but on the desks of particle physicists.

Chris Sachrajda is professor of physics, University of Southampton.

## The Quantum Theory of Fields, Volume Three: Supersymmetry

Author - Steven Weinberg

ISBN - 0 521 66000 9

Publisher - Cambridge University Press

Price - £35.00

Pages - 419

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