# Integrating Gauge Fields in the $\zeta$-formulation of Feynman's path integral

## TL;DR

This paper extends the Fourier integral operator $ta$-function regularization method for Feynman's path integral to include gauge fields, providing new insights into quantum field theory calculations.

## Contribution

It introduces a framework for integrating gauge fields into the $ta$-function regularization of Feynman's path integral, filling a gap in previous work.

## Key findings

- Successfully applied the method to well-known quantum field examples
- Demonstrated the regularization of vacuum expectation values with gauge fields
- Enhanced understanding of gauge field contributions in path integral formalism

## Abstract

In recent work by the authors, a connection between Feynman's path integral and Fourier integral operator $\zeta$-functions has been established as a means of regularizing the vacuum expectation values in quantum field theories. However, most explicit examples using this regularization technique to date, do not consider gauge fields in detail. Here, we address this gap by looking at some well-known physical examples of quantum fields from the Fourier integral operator $\zeta$-function point of view.