Integration over connections in the discretized gravitational functional integrals

TL;DR

This paper investigates the integration over connection variables in discretized gravitational path integrals, revealing a distribution with both singular and regular parts that suppresses large areas, supporting the consistency of discrete gravity models.

Contribution

It introduces a method to define the gravitational path integral as a generalized function, addressing issues with non-compact gauge groups and exponential growth in the measure.

Findings

Distribution has delta-function-like support in nonphysical regions

Regular part of the distribution decays exponentially at large areas

Large edge lengths are suppressed, supporting discrete gravity consistency

Abstract

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas) i. e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular ($\delta$-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some another features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.