Barbero-Immirzi parameter in Regge calculus

TL;DR

This paper explores the role of the Barbero-Immirzi parameter within Regge calculus formulated with area tensors and connections, analyzing its impact on the quantum measure and revealing exponential decay related to geometric scales.

Contribution

It extends Regge calculus by incorporating the Holst action with the Barbero-Immirzi parameter, analyzing its influence on the quantum measure and geometric cut-off scales.

Findings

Quantum measure decreases exponentially with areas.

Typical cut-off scales are proportional to $4\\pi G$ and $4\pi G\gamma$.

Results highlight the geometric significance of the Barbero-Immirzi parameter.

Abstract

We consider Regge calculus in the representation in terms of area tensors and self- and antiselfdual connections generalised to the case of Holst action that is standard Einstein action in the tetrad-connection variables plus topological (on equations of motion for connections) term with coefficient $1/\gamma$, $\gamma$ is Barbero-Immirzi parameter. The quantum measure is shown to exponentially decrease with areas with typical cut-off scales $4\pi G$ and $4\pi G\gamma$ in spacelike and timelike regions, respectively ($G$ is the Newton constant).