# Simplicial Palatini action

## TL;DR

This paper extends the simplicial Palatini formulation of general relativity by including parity violation and Barbero-Immirzi parameters, analyzing path integrals and measures to explore finiteness in a discrete quantum gravity setting.

## Contribution

It introduces parity violation and Barbero-Immirzi parameters into the simplicial Palatini action and studies their effects on path integrals and measures for quantum gravity.

## Key findings

- Path integral results align with orthogonal connection cases in certain limits.
- Measures on lengths/areas suggest elementary lengths are bounded away from zero.
- Potential for finite perturbative diagrams due to measure properties.

## Abstract

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables classically gives exactly the Regge action. This paper continues our previous work. Now we include the parity violation term and the analogue of the Barbero-Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over connection for the analogous {\it orthogonal} connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as finiteness of a theory on a lattice with an analogous probability distribution of spacings.