Point-splitting regularization of the stress tensor of a coupling scalar field in de Sitter space

TL;DR

This paper applies point-splitting regularization to the vacuum stress tensor of a scalar field in de Sitter space, revealing conditions under which the regularized tensor is finite and consistent with inflation, and comparing it to adiabatic regularization.

Contribution

It demonstrates the effectiveness of second-order point-splitting regularization for certain couplings and clarifies issues with divergences and path dependence, aligning results with adiabatic regularization.

Findings

For minimal coupling, the stress tensor is finite and positive, acting as a cosmological constant.

For non-zero coupling, additional treatments are needed to remove divergences and path dependencies.

Regularized stress tensor matches that from adiabatic regularization across cases.

Abstract

We perform the point-splitting regularization on the vacuum stress tensor of a coupling scalar field in de Sitter space under the guidance from the adiabatically regularized Green's function. For the massive scalar field with the minimal coupling $\xi=0$, the 2nd order point-splitting regularization yields a finite vacuum stress tensor with a positive, constant energy density, which can be identified as the cosmological constant that drives de Sitter inflation. For the coupling $\xi\ne 0$, we find that, even if the regularized Green's function is continuous, UV and IR convergent, the point-splitting regularization does not automatically lead to an appropriate stress tensor. The coupling $\xi R$ causes log divergent terms, as well as higher-order finite terms which depend upon the path of the coincidence limit. After removing these unwanted terms by extra treatments, the 2nd-order regularization for small couplings $\xi \in(0,\frac{1}{7.04})$, and respectively the 0th-order regularization for the conformal coupling $\xi=\frac16$, yield a finite, constant vacuum stress tensor, in analogy to the case $\xi=0$. For the massless field with $\xi=0$ or $\xi=\frac16$, the point-splitting regularization yields a vanishing vacuum stress tensor, and there is no conformal trace anomaly for $\xi=\frac16$. If the 4th-order regularization were taken, the regularized energy density for general $\xi$ would be negative, which is inconsistent with the de Sitter inflation, and the regularized Green's function would be singular at the zero mass, which is unphysical. In all these cases, the stress tensor from the point-splitting regularization is equal to that from the adiabatic one.