Regge calculus and Ashtekar variables

TL;DR

This paper explores the formulation of Regge calculus using Ashtekar variables, focusing on the kinematic structure of the initial value problem and the role of reality conditions, aiming to simplify the constraints of the theory.

Contribution

It introduces a selfdual variable approach to discretized spacetime in Regge calculus, clarifying the initial value problem and constraints in this framework.

Findings

Clarified the role of reality conditions for Ashtekar variables in discrete spacetime.

Presented a simplified form of vector and scalar constraints.

Analyzed the kinematic structure of the initial value problem in discretized gravity.

Abstract

Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way.