# Detailed discussions and calculations of quantum Regge calculus of Einstein-Cartan theory

## TL;DR

This paper develops a detailed framework for quantum Einstein-Cartan theory using Regge calculus, discretizing space-time with simplicial complexes and holonomy fields, enabling analytical and numerical studies of quantum gravity.

## Contribution

It introduces a regularized Einstein-Cartan action on a simplicial complex that respects diffeomorphism and Lorentz invariance, bridging discrete and continuum quantum gravity.

## Key findings

- The regularized action approaches the continuum Einstein-Cartan theory.
- The mean-field approximation indicates the 4-simplex size exceeds the Planck length.
- Provides a basis for future analytical and numerical quantum gravity research.

## Abstract

This article presents detailed discussions and calculations of the recent paper "Quantum Regge calculus of Einstein-Cartan theory" in Phys. Lett. B682 (2009) 300. The Euclidean space-time is discretized by a four-dimensional simplicial complex. We adopt basic tetrad and spin-connection fields to describe the simplicial complex. By introducing diffeomorphism and local Lorentz invariant holonomy fields, we construct a regularized Einstein-Cartan theory for studying the quantum dynamics of the simplicial complex and fermion fields. This regularized Einstein-Cartan action is shown to properly approach to its continuum counterpart in the continuum limit. Based on the local Lorentz invariance, we derive the dynamical equations satisfied by invariant holonomy fields. In the mean-field approximation, we show that the averaged size of 4-simplex, the element of the simplicial complex, is larger than the Planck length. This formulation provides a theoretical framework for analytical calculations and numerical simulations to study the quantum Einstein-Cartan theory.