A discretization of Holst's action for general relativity

TL;DR

This paper introduces a new simplicial discretization of Holst's action for general relativity, aiming to facilitate quantum gravity models by describing curvature at the 4-simplex level and enabling matter coupling.

Contribution

It proposes a novel simplicial model for gravity that discretizes Lorentz connection and tetrad fields, with potential advantages for spin foam quantum gravity.

Findings

Curvature degrees of freedom are localized at each 4-simplex.

The model's structure may lead to equivalent quantum theories after quantization.

It provides a framework for matter coupling as a sum over 4-simplices.

Abstract

We present a simplicial model for gravity written in terms of a discretized Lorentz connection and a discretized tetrad field. The continuum limit of its action is Holst's action for general relativity. With the intention of using it to construct spin foam modes for quantum gravity, we write two other equivalent models written in terms of a discretized and constrained $B$ field. The differences between our model and existing models are most likely inessential in the sense that a quantization would lead to equivalent quantum theories in the Wilsonian continuum limit. Nevertheless, we mention two features leading to possible advantages: Curvature degrees of freedom are described at the level of each 4-simplex. Our model offers a picture of bulk geometry leading to actions for matter couplings that split as a sum over 4-simplices.