Asymptotic principal values and regularization methods for correlation functions with reflective boundary conditions

TL;DR

This paper introduces asymptotic principal values to rigorously handle singular integrals in correlation functions with reflective boundaries, providing a foundation for developing and analyzing regularization methods.

Contribution

It develops a new concept of asymptotic principal values and proves related theorems, facilitating the analysis of regularization methods for singular integrals in quantum correlation functions.

Findings

Theorems on asymptotic principal values are established.

The concept aids in comparing and understanding regularization methods.

A natural regularization method derived from asymptotic principal values is proposed.

Abstract

We introduce a concept of asymptotic principal values which enables us to handle rigorously singular integrals of higher-order poles encountered in the computation of various quantities based on correlation functions of a vacuum. Several theorems on asymptotic principal values are proved and they are expected to become bases for investigating and developing some class of regularization methods for singular integrals. We make use of these theorems for analyzing mutual relations between some regularization methods, including a method naturally derived from asymptotic principal values. It turns out that the concept of asymptotic principal values and the theorems for them are quite useful in this type of analysis, providing a suitable language to describe what is discarded and what is retained in each regularization method.