Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools

We provide a brief overview of the basic tools and concepts of quantum field theoretical scattering theory. This article is commissioned by the Encyclopedia of Mathematical Physics, edited by J.-P. Françoise, G. Naber and T.S. Tsun, to be published by the Elsevier publishing house. 1 Physical Motivation and Mathematical Setting The primary connection of relativistic quantum field theory to experimental physics is through scattering theory, i.e. the theory of the collision of elementary (or compound) particles. It is therefore a central topic in quantum field theory and has attracted the attention of leading mathematical physicists. Although a great deal of progress has been made in the mathematically rigorous understanding of the subject, there are important matters which are still unclear, some of which will be indicated below. In the paradigmatic scattering experiment, several particles, which are initially sufficiently distant from each other that the idealization that they are not mutually interacting is physically reasonable, approach each other and interact (collide) in a region of microscopic extent. The products of this collision then fly apart until they are sufficiently well separated that the approximation of noninteraction is again reasonable. The initial and final states of the objects in the scattering experiment are therefore to be modeled by states of noninteracting, i.e. free, fields, which are mathematically represented on Fock space. Typically, what is measured in such experiments is the probability distribution (cross section) for the transitions from a specified state of the incoming particles to a specified state of the outgoing particles. It should be mentioned that until the late 1950’s, the scattering theory of relativistic quantum particles relied upon ideas from nonrelativistic quantum mechanical scattering theory (interaction representation, adiabatic limit, etc), which were invalid in the relativistic context. Only with the advent of axiomatic quantum field theory did it become