The Regularized Hadamard Expansion

TL;DR

This paper introduces a local regularized Hadamard expansion for two-point distributions in four-dimensional space-time, providing explicit insights into the effects of ultraviolet regularization on global dynamics.

Contribution

It develops an iterative method to compute the regularization functions via transport equations and proves that the regularized structure is preserved during Cauchy evolution.

Findings

Regularized Hadamard expansion explicitly describes global regularization effects.

Transport equations enable iterative computation of regularization functions.

Cauchy evolution preserves the regularized Hadamard structure.

Abstract

A local expansion is proposed for two-point distributions involving an ultraviolet regularization in a four-dimensional globally hyperbolic space-time. The regularization is described by an infinite number of functions which can be computed iteratively by solving transport equations along null geodesics. We show that the Cauchy evolution preserves the regularized Hadamard structure. The resulting regularized Hadamard expansion gives detailed and explicit information on the global dynamics of the regularization effects.