Scattering theory without large-distance asymptotics in arbitrary dimensions

TL;DR

This paper develops a scattering theory applicable in any number of dimensions without relying on large-distance asymptotics, preserving distance information and aiding quantum field theory dimensional renormalization.

Contribution

It introduces an arbitrary-dimensional scattering framework that defines wave functions, boundary conditions, cross sections, and phase shifts without large-distance asymptotics.

Findings

Established scattering wave functions without large-distance asymptotics

Derived scattering boundary conditions and phase shifts in arbitrary dimensions

Discussed applications to one- and two-dimensional scatterings

Abstract

In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes a large-distance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, cross sections, and phase shifts are given without large-distance asymptotics. The importance of an arbitrary-dimensional scattering theory is that the dimensional renormalization procedure in quantum field theory needs an arbitrary-dimensional result. Moreover, we give a discussion of one- and two-dimensional scatterings.