Attributing sense to some integrals in Regge calculus

TL;DR

This paper investigates the integration over connection variables in Regge calculus's path integral, demonstrating how to define and compute the resulting distribution through moments, which decay exponentially at large areas, providing a way to handle nonabsolutely convergent integrals in quantum gravity.

Contribution

It introduces a method to attribute sense to integrals over connection variables in Regge calculus by computing moments and reconstructing the distribution, aiding in defining the quantum gravity path integral.

Findings

The moments of the distribution are finite except on a discrete set of points.

The distribution decays exponentially at large areas.

A method to define nonabsolutely convergent path integrals in 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 (now usual multiple integral) as function of area tensors even in larger region considered as independent variables. To find this function (or distribution), we compute its moments, i. e. integrals with monomials over area tensors. Calculation proceeds through intermediate appearance of $\delta$-functions and integrating them out. Up to a singular part with support on some discrete set of physically unattainable points, the function of interest has finite moments. This function in physical region should therefore exponentially decay 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.