Defining integrals over connections in the discretized gravitational functional integral

TL;DR

This paper develops a method to define and evaluate integrals over connection variables in the discrete gravitational path integral, ensuring well-defined behavior in Minkowski space and analyzing divergence contributions.

Contribution

It introduces a way to define and compute moments of the connection integrals directly in Minkowski space, extending previous Euclidean-based approaches.

Findings

Distribution decays exponentially at large areas in physical region

Divergences only affect non-physical complex plane regions

Basic integrals over connections are explicitly evaluated

Abstract

Integration over connection type variables in the path integral for the discrete form of the first order formulation of general relativity theory is studied. The result (a generalized function of the rest of variables of the type of tetrad or elementary areas) can be defined through its moments, i. e. integrals of it with the area tensor monomials. In our previous paper these moments have been defined by deforming integration contours in the complex plane as if we had passed to an Euclidean-like region. In the present paper we define and evaluate the moments in the genuine Minkowsky region. The distribution of interest resulting from these moments in this non-positively defined region contains the divergences. We prove that the latter contribute only to the singular (\dfun like) part of this distribution with support in the non-physical region of the complex plane of area tensors while in the physical region this distribution (usual function) confirms that defined in our previous paper which decays exponentially at large areas. Besides that, we evaluate the basic integrals over which the integral over connections in the general path integral can be expanded.