Scattering matrix in external field problems

TL;DR

This paper develops a general framework for analyzing second quantized scattering operators for fermions in external fields, introducing a new method for proving their existence and exploring gauge transformation effects, with applications to Yang--Mills and gravitational fields.

Contribution

It presents a novel, abstract method for proving the existence of scattering operators applicable to various external fields, and derives phase change formulas under gauge transformations.

Findings

New method for proving existence of scattering operators

Derivation of phase change under gauge transformations

Connection between anomalies and second quantization

Abstract

We discuss several aspects of second quantized scattering operators $\hat S$ for fermions in external time dependent fields. We derive our results on a general, abstract level having in mind as a main application potentials of the Yang--Mills type and in various dimensions. We present a new and powerful method for proving existence of $\hat S$ which is also applicable to other situations like external gravitational fields. We also give two complementary derivations of the change of phase of the scattering matrix under generalized gauge transformations which can be used whenever our method of proving existence of $\hat S$ applies. The first is based on a causality argument i.e.\ $\hat S$ (including phase) is determined from a time evolution, and the second exploits the geometry of certain infinite-dimensional group extensions associated with the second quantization of 1-particle operators. As a special case we obtain a Hamiltonian derivation of the the axial Fermion-Yang-Mills anomaly and the Schwinger terms related to it via the descent equations, which is on the same footing and traces them back to a common root.