Causality and dispersion relations for fixed momentum transfer

Abstract

In order to investigate which physical assumptions are relevant to the validity of dispersion relations for fixed momentum transfer, a simple case is treated: the scattering of a classical scalar field by an arbitrary spherically symmetric scatterer of finite radius. It is sufficient to assume: (a) restrictions, due to causality, on the propagation of signals with sharp fronts; (b) conditions on the behaviour of the phase-shifts in the low-frequency and high-angular-momentum limits. To relate the scattering amplitude for fixed momentum transfer with the principle of strict causality, a new representation for this amplitude, in terms of the scattered wave at finite distances from the scatterer, is introduced. The dispersion relations are rigorously derived from the basic assumptions. The results are partially extended to the scattering of Schrödinger particles. An explicit example (totally reflecting sphere) is treated as an illustration.