Vacuum Polarization Renormalization and the Geometric Phase

TL;DR

This paper introduces a causal method to define the phase of the quantum scattering matrix for fermions in external Yang-Mills fields, utilizing renormalization and geometric concepts within the framework of the restricted unitary group.

Contribution

It provides a novel, causal definition of the scattering matrix phase using renormalization and geometric phase concepts in the context of external gauge fields.

Findings

Defined the phase of the scattering matrix via parallel transport.

Connected the phase definition to the central extension of the restricted unitary group.

Applied the renormalization method to quantum scattering in gauge fields.

Abstract

As an application of the renormalization method introduced by the second author we give a causal definition of the phase of the quantum scattering matrix for fermions in external Yang-Mills potentials. The phase is defined using parallel transport along the path of renormalized time evolution operators. The time evolution operators are elements of the restricted unitary group $U_{res}$ of Pressley and Segal. The central extension of $U_{res}$ plays a central role.