A perturbation theoretic approach to the analytic properties of collision amplitudes

Abstract

In recent years the quantum theory of fields has become increasingly more complicated, and, despite the remarkable successes of the theory, it is widely held that, without some new simplifying principle, there is little future for this theory as an effective tool in the study of elementary particle interactions. At the present time the best hope of building a new and powerful theory seems to lie in the development of the method of dispersion relations. The basic idea of dispersion relations is that the quantum mechanical amplitudes, which describe physical processes, are the boundary values of functions of one, or several, complex variables, regular apart from poles in a suitably cut space: an early indication of this notion was given in 1955 by Chew and Low who discovered that, in the static model for elastic scattering of a pion on a nucleon, one was dealing with an amplitude, meromorphic in the energy plane except for branch cuts lying along the real axis. Dispersion relations are based on very broad general principles such as covariance, spectral conditions, and locality. In the so- called axiomatic approach to dispersion theory one attempts to deduce from these basic principles, in a rigorous fashion, the analytic properties of the collision amplitudes as functions of complex invariants. Then one is able to exploit a knowledge of the location and nature of the singularities of an amplitude to derive a useful integral representation for it using Cauchy's integral theorem. These representations are called "dispersion relations" or "spectral representations."

As we have already emphasised, the axiomatic approach is exceedingly laborious and very few dispersion relations can actually be proved. On the other hand, dispersion relations written down on the evidence of perturbation theory (and to a lesser extent on the evidence supplied by potential theory in non-relativistic quantum mechanics) have met with unprecedented success. We are forced to the stage of asking ourselves whether it is worth while continuing with the old axiomatic approach -- indeed, the status and consistency of the axioms of field theory have been questioned by some writers. In perturbation theory which can be derived non- rigorously from the axioms, we have explicit expressions for each term in the series for an amplitude, and if the method can be applied exhaustively to various interactions, one might hope to utilise the perturbation theoretic predictions to postulate the analytic structure of a new theory. The situation towards which we are moving is, in fact, a new starting point for the theory of elementary particle interactions based on a few simple new axioms, one of which would relate to analytic structure: the others would probably be concerned with unitarity and crossing symmetries. We would then be in a position to discard the old axioms and start afresh. Such a scheme has been proposed by Chew.

This, then, is the background against which the author has undertaken to write a thesis entitled ".t. perturbation theoretic approach to the analytic properties of collision amplitudes." The dissertation is a chronological account of the author's studies of perturbative methods, and falls, broadly speaking, into two distinct parts. The first part, which serves as an introduction to the subject, describes properties of the Landau curves which, as will be discussed later, are loci, in the multidimensional complex space of the invariants, corresponding to points of possible singularity of a collision amplitude. At the time of commencing the present work (Summer 1960) the author had participated in the Scottish Universities' Summer School which was devoted to the subject of dispersion relations: also several papers, in preprint form, dealing with the elastic scattering of two particles in perturbation theory, had recently come to hand. These two circumstances inevitably coloured the author's entry into the subject, and some of his earlier work consisted in the elaboration of points raised in the literature, or else arose, directly or indirectly, from discussions of then current topics with Dr. G.R. Screaton, the author's research supervisor, himself actively engaged in work on perturbation theory.

The second section gives a detailed account of an independent investigation carried out by the author into the problem of finding an integral representation for production processes in perturbation theory. The type of process considered is characterized by the inelastic scattering of a pion on a nucleon:

π + N —» π + π + N

The outcome of this investigation was quite surprising for it brought to light a new problem hitherto unencountered in simple elastic scattering processes,x namely the occurrence of complex branch points which prevent the writing of conventional single dispersion relations for any physical values of the fixed invariants. One is thus faced with the problem of fitting this new feature into the existing scheme, and, in general, the matter appears to be an exceedingly complicated one: the urgency of coping with complex branch cuts becomes even more intense because of the recent discovery by Polkinghorne that complex singularities can arise as a direct consequence of the unitarity assumption. Unitarity is one of the basic physical principles of the theory.