Analyticity with respect to Momentum Transfer and Dispersion Relations

Abstract

The amplitudes of processes in relativistic quantum theory possess certain analytic properties with respect to the momentum variables. These properties establish relations between the experimentally observed quantities and are therefore of great importance for the theory of interactions of elementary particles as compared with the significance of analyticity in quantum mechanics. Thus pole-type singularities indicate the presence of bound states in a given channel (more generally, localization of singularities of amplitudes is defined by physical parameters such as the masses of the particles and resonances, the energies of the bound states, reaction threshholds and so on). The dispersion relations derived from analyticity express the amplitude of an elastic process in terms of its imaginary part (which in forward scattering is expressed, according to the optical theorem, in terms of the total cross section). One of the specific peculiarities of the relativistic S-matrix (in contrast to the ordinary quantum mechanical one) is the crossing property which compares the value of the amplitude of the process (in the physical region) with the amplitude of some other (crossing-) process at non physical points; an understanding of relations of this kind can only be achieved by going over to the complex domain and by analytic continuation of the physical amplitude.