Power Series Representations for Complex Bosonic Effective Actions. III. Substitution and Fixed Point Equations

TL;DR

This paper advances the mathematical framework for analyzing complex bosonic effective actions by integrating fixed point theorems and substitution techniques to better control functional integrals and critical fields.

Contribution

It introduces a fixed point theorem and substitution formulas tailored for controlling analytic functions and critical fields in functional integral analysis.

Findings

Developed a Banach fixed point theorem for critical field construction.

Created substitution formulas to manage norm changes during field replacements.

Enhanced methods for analyzing complex bosonic effective actions.

Abstract

We have previously developed a polymer-like expansion that applies when the (effective) action in a functional integral is an analytic function of the fields being integrated. Here, we develop methods to aid the application of this technique when the method of steepest descent is used to analyze the functional integral. We develop a version of the Banach fixed point theorem that can be used to construct and control the critical fields, as analytic functions of external fields, and substitution formulae to control the change in norms that occurs when one replaces the integration fields by the sum of the critical fields and the fluctuation fields.