The "Scattering" matrix, or more briefly, S-matrix, is one of the basic tools in particle physics. In essence, the idea is very simple. When a particle physics experiment is designed the initial particle states, often protons or electrons in a carefully prepared accelerator beam, are allowed to collide with particles in another similar beam travelling in the opposite direction, or with particles trapped in a fixed target. The tracks or other marks of the resulting debris are carefully collected in a set of detectors placed in the vicinity of the interaction region.
The analysis of the debris from the collision often allows a reconstruction of the series of events following the collision and, generally speaking, the result looks very complicated with a great many particles of different types in the final state. The nature and distribution of the final particles provide information concerning the laws governing the fundamental interactions between the initial particles, or their constituents. Given an initial state, the possible outcomes of the collision are determined by the elements of a linear operator - the S matrix - which relates the final states to the initial states. One challenge for theoretical particle physics is to develop ways to compute the elements of the S-matrix in order to be able to predict the possible outcomes of scattering experiments such as those performed at Cern. Another, is to obtain a satisfactory description of the particle states themselves.
Although the basic idea is quite simple, the framework within which the present understanding of fundamental particles and their interactions is cast, quantum field theory, is far from simple. Nevertheless, the "standard model", which achieves a structural unification of three of the four fundamental forces - weak, electromagnetic and strong, is formulated in a manner which allows at least partial calculations of the S-matrix in special circumstances. These have been sufficient, so far, to build up substantial confidence in the model. The picture is far from adequate, however. For example, although there is no evidence to suggest that the electron, its neutrino partner and a number of similar types of particle are anything other than elementary, the same cannot be said of protons, neutrons and a variety of others. On the contrary, these are believed to be permanently bound states of three other, more elementary, "particles" - the quarks. The precise mechanism via which the quantum field theory describing quarks is responsible for the spectrum of bound states such as the proton is not completely settled although there is a heuristic understanding of the phenomenon. The idea is that quarks themselves interact by exchanging other hidden particles called "glue", in much the same way as electrons interact with one another by exchanging photons, except that the glue also interacts with itself. It is the existence of this sticky self-interaction which is responsible for the confinement of both quarks and "glue". If these ideas are correct it will never be possible to split a proton into its constituents although they may be discerned indirectly in scattering experiments. It is not hard to imagine that any attempt to calculate the scattering of bound states is difficult, but it cannot be avoided if one requires a complete description of proton-proton scattering. It is remarkable that although the projectiles in scattering experiments can be such complex objects it is nevertheless possible to calculate enough of the S-matrix to allow a quantitative verification of much of the standard model.
There is another side to this story. As has been mentioned, quantum field theory used as a daily tool is essentially heuristic. The mathematical foundations of the theory are lagging some way behind the frontiers at which it is routinely used. One can ask questions about simpler, and therefore likely to be unrealistic, field theory models which mimic some of the features of the standard model. Often, in a simple toy model, the mathematical structure can be explored and results can be established rigorously which serve to increase the value placed on the heuristic methods which have to be used elsewhere. Sometimes there is a bonus in the sense that a study of a toy model may lead to an application of quantum field theory elsewhere in physics, or even within mathematics. For example, two dimensional quantum field theories have found application in condensed matter physics; three-dimensional quantum field theory has become a useful tool in topology.
Daniel Iagolnitzer's book studies scattering and the S-matrix but not with any realistic calculations or predictions in mind. Rather, it is concerned with those properties of the S-matrix which can be established assuming certain general principles such as special relativity, locality and causality. As such, these properties have relevance to a wide class of quantum field theories.
Besides this, the book describes, within the context of constructive field theory satisfying the Wightman axioms, many of the techniques routinely used by theoretical particle physicists, such as renormalisation, the operator product expansion and the description of bound states. From time to time, the author uses illuminating examples from two-dimensional field theory. The advantage of this axiomatic approach is its rigour although it is not yet sufficiently developed to allow an assault on the confinement and other problems of the standard model.
The book itself is quite demanding and looks unlikely to appeal to beginners. Even those with a reasonably sound knowledge may find it daunting.
The author chooses to spend the first few pages of each chapter (there are four) summarising the topics to follow. This is a good idea in principle, but its value is limited by the rush of ideas and lack of patient explanation.
Interest in the book lies in its comprehensive, albeit condensed, treatment which summarises much of the current state of this particular art. It contains a wealth of detail which ought to be attractive to the experts.
Ed Corrigan is professor in mathematical sciences, University of Durham.
Scattering in Quantum Field Theories: The Axiomatic and Constructive Approaches
Author - Daniel Iagolnitzer
ISBN - 0 691 08589 7
Publisher - Princeton University Press
Price - £37.50
Pages - 290