Magic of tiny good vibrations

String theory can answer questions that the standard model cannot. The latter is a quantum field theory of the elementary particles and their strong, weak and electromagnetic interactions, that has survived unscathed after more than 25 years of increasingly rigorous experimental testing. For all we know it might be the theory, rather than merely a very successful model that economically accounts for the current, low-energy data.

However, it leaves many unanswered questions. Why are the fundamental interactions described by non-Abelian gauge theories? Why do the particular gauge groups, SU(3) for strong interactions and SU(2)xU(1) for electroweak theory, arise; and why do their coupling strengths have the measured values? Why does only the observed matter - electrons, muons, quarks etc - occur; why is it arranged in three "families", each with identical properties except for their masses; and why do the particles have their measured masses? Why are there three space and one time dimensions; and why does the standard model not include gravitational interactions?

Only the last of these questions has to be answered. For the remainder, one can just accept that this is the way the world is: the 20 or so undetermined parameters of the standard model just happen to have the values they do, values that allow the existence of life and hence particle theorists to ask such questions! In contrast, string theory has no adjustable parameters, it is unique, and can, in principle, answer such questions. It gives a consistent quantum theory of gravitation that reduces to Einstein's general theory of relativity at low energy. It requires a definite number, ten, of space-time dimensions, and the field equations have solutions with four large flat and six small curved dimensions, with four-dimensional physics that resembles the standard model. These features are undoubtedly part of the allure of the topic for the bright young graduate students who flock to the subject each year. Joseph Polchinski's book is aimed at them and at all who want to understand in detail the recent exciting developments in the subject.

The basic postulate of string theory is that the elementary ingredients of the universe are not point-like particles as assumed by the standard model.

Rather they are tiny, one-dimensional, vibrating filaments, which may be open or closed, with a length far smaller (by a factor of 1016) than can be probed by current high-energy physics experiments. The patterns of the string's vibrations appear to us as the masses and charges of the familiar elementary particles. All attempts to include a quantum theory of gravitation in the standard model founder on the short-distance divergences, infinities, which afflict the theory, and which stem from the assumed point-like nature of the particles and their interactions. This suggests that the problem might be alleviated by smearing the interaction over a finite region of space-time.

Strings do this, but so do membranes or three-dimensional objects. However, the latter two introduce further divergences from their internal degrees of freedom. Only strings, it seems, keep both the space-time and internal divergences under control. The theory is formulated geometrically,the action being the area of the two-dimensional "world-sheet" swept out by the string as it moves in space-time, just as the action of a point particle is given by the length of its world-line. Amazingly, this is sufficient to ensure that the theory contains general relativity, and that it is a finite quantum theory of gravitation, free of the divergences endemic in theories involving point-like particles.

Throughout, Polchinski emphasises such world-sheet symmetries and their consequences for space-time symmetries. We are a long way from understanding the full symmetry of string theory and workers in string theory often feel that the string is smarter than they are, with consistent results sometimes being derived from seemingly quite different aspects of the theory. For example, in the light-cone quantisation of string theory the requirement of ten dimensions emerges from the requirement of Lorentz invariance in the higher-dimensional space-time, whereas in the Polyakov path integral approach, the (same) requirement derives from the requirement of Weyl invariance on the world-sheet at the quantum level. If the extra dimensions required by string theory really exist, then six of them must be curled up on small scales just as all of the dimensions were in the early universe.

An important symmetry, which has played a major role in the recent exciting developments in string theory, is "T-duality". It arises, for example, when one spatial coordinate is curled up, "compactified", on a circle of radius R. It turns out that closed string theory compactified in this way is exactly equivalent to the theory compactified on a circle of radius R-1. This is T-duality. In particular, the theory with R very small, so this dimension has shrunk almost to zero, is indistinguishable from the one with R very large, in which the dimension is almost infinite, a truly stringy symmetry quite unlike the familiar symmetries of point-like theories.

A theory with open strings necessarily involves closed strings, although not vice versa and when T-duality is applied to open string theory it shows that a theory compactified on a very small circle is equivalent to one on a very large circle, but with the open strings ending on fixed hyper-planes. These hyper-planes turn out to be dynamical objects in their own right. They and their generalisations are called D-branes or Dirichlet branes, and it was Polchinski who recognised their importance in 1995.

Thus, although we start a theory involving just strings, we are forced to consider other extended objects. Evidence of another, even more extraordinary duality, S-duality, involving the string coupling strength g,emerges when we study the properties of these branes. We can show that the weakly-coupled open-string theory, ie gopen is small, is equivalent to one of the closed string theories with strong coupling, ie gclosed is large.

In other words, two theories formerly regarded as quite distinct are in fact different limits of a single theory. Discoveries such as this have led to the conclusion that string theory is unique, and that it is no longer a theory only of strings. There is much still to be done and further insights into the magic of string theory to be found. For example, although the theory is unique, it has an enormous number of classical solutions, and so far there is no a priori understanding of why the one that leads to the standard model, or its supersymmetric extension, should be selected.

It is important that youthful enthusiasm is not unduly dampened by the hard slog needed to get graduate students from their BSc or MPhys degree-level studies, which these days might include a smattering of quantum field theory, to the string theory frontline. Polchinski is fully alive to this imperative, recalling the "student's pleasure in finding a text that made a difficult subject accessible". A major contributor to the exciting developments that have revolutionised our understanding of string theory during the past four years, he is also an exemplary teacher, as Steven Weinberg attests in his foreword. He has produced an outstanding two-volume text, with numerous exercises accompanying each chapter.

It amply fulfils the need to inspire future string theorists on their long slog and is destined to become a classic. It is the first to include the recent revolutionary developments into the mainstream of the exposition. All of these appeared first on the web, so it is natural that the book comes complete with its own erratum website ("Joe's big book of string"). It is a truly exciting enterprise and one hugely served by this magnificent book.

David Bailin is professor of theoretical physics, University of Sussex.