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.