Running of Newton's constant and non integer powers of the d'Alembertian

TL;DR

This paper explores nonlocal modifications of Einstein's equations involving nonanalytic functions of the d'Alembertian, such as fractional powers and logarithms, to incorporate the running of Newton's constant.

Contribution

It defines and analyzes nonlocal operators involving noninteger powers and logarithms of the d'Alembertian using two-point functions of massive fields.

Findings

Defined nonlocal operators in terms of two-point functions.

Analyzed properties of these operators in flat and Robertson-Walker spacetimes.

Provided specific calculations demonstrating their behavior.

Abstract

The running of Newton's constant can be taken into account by considering covariant, non local generalizations of the field equations of general relativity. These generalizations involve nonanalytic functions of the d'Alembertian, as $(-\Box)^{-\alpha}$, with $\alpha$ a non integer number, and $\ln[-\Box]$. In this paper we define these non local operators in terms of the usual two point function of a massive field. We analyze some of their properties, and present specific calculations in flat and Robertson Walker spacetimes.