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Quasi-local probability averaging in the context of cutoff regularization

A. V. Ivanov, I. V. Korenev

Abstract

In this paper, we study the properties of averaged fundamental solutions of a special type for Laplace operators in the Euclidean space of an arbitrary dimension. We consider a class of kernels suitable for probabilistic averaging, and propose new representations for the deformed fundamental solutions and their values at zero. In addition, we give examples related to specific quantum field models in the context of studying renormalization properties.

Quasi-local probability averaging in the context of cutoff regularization

Abstract

In this paper, we study the properties of averaged fundamental solutions of a special type for Laplace operators in the Euclidean space of an arbitrary dimension. We consider a class of kernels suitable for probabilistic averaging, and propose new representations for the deformed fundamental solutions and their values at zero. In addition, we give examples related to specific quantum field models in the context of studying renormalization properties.

Paper Structure

This paper contains 8 sections, 4 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main properties
  3. Proofs
  4. New examples
  5. The case $t=s=1/2$.
  6. The case $n=3$.
  7. The case $n=2$.
  8. Conclusion

Key Result

Theorem 1

Let $n\in\mathbb{N}$ and $x,y,z\in\mathbb{R}^n$. Let us introduce notation for the absolute values of $r=|x|$, $s=|y|$, and $t=|z|$. The symbol $\mathrm{S}^{n-1}$ denotes the unit $(n-1)$-dimensional sphere centered at the origin. Consider an integral of the form where $\hat{y}=y/|y|$, $\hat{z}=z/|z|$, and also $\mathrm{d}^{n-1}\sigma(\hat{y})$ denotes the standard measure on the sphere, which is

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1