Defining some integrals in Regge calculus

TL;DR

This paper investigates the integration over connection variables in Regge calculus's path integral, demonstrating a method to define a finite, exponentially suppressed integral for gravity by computing moments and restoring the function.

Contribution

It introduces a novel approach to defining the nonabsolutely convergent path integral in Regge calculus through moment calculations and delta-function manipulations.

Findings

The integral over connections is finite and exponentially suppressed at large areas.

Moments of the integral can be computed explicitly, leading to a well-defined function.

The method provides a way to handle non-convergent path integrals in quantum gravity.

Abstract

Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral. To find this function, we compute its moments, i. e. integrals with powers of that variable. Calculation proceeds through intermediate appearance of $\delta$-functions and integrating them out and leads to finite result for any power. The function of interest should therefore be exponentially suppressed at large areas and it really does being restored from moments. This gives for gravity a way of defining such nonabsolutely convergent integral as path integral.