Asymptotic Principal Values and Regularization Methods for Correlation Functions with Reflective Boundary Conditions

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 classes 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. 1. Introduction Physics of quantum vacuum fluctuations is one of the intriguing research topics expected to be developed through the interplay between theories, experiments, and practical applications. Investigations of quantum vacuum fluctuations even stimulate the border area between physics and mathematics. As a typical example of this sort, we often encounter singular integrals in computing several quantities based on correlation functions of a vacuum in question. The occurrence of singularity or divergence is often a signal of surpassing the border of validity of a model by too much extrapolation. Furthermore, it could originate from deeper physical processes for which satisfactory consistent mathematics is still unavailable. How to handle singular integrals can be then a challenging topic, requiring both mathematical analysis and physical considerations. Faced with singular integrals, we need to resort to some regularization method to get a finite result. The aim of this paper is to give an organized mathematical basis underlying some typical regularization methods and make clear their mutual relations. We introduce below a concept of asymptotic principal values which can be a key tool to analyze some classes of regularization methods. We then prove several theorems on the asymptotic principal values useful for studying regularization procedures. There are still various uncertainties to clear up in regularization methods, reflecting our lack of mathematical basis for handling infinities. In this situation, we cannot expect any universal regularization method, but we need to customize the method by try and error depending on the problem in question. It is far from the aim of this paper to judge which method is better than the