  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.
