Canonical quantization of general relativity in discrete space-times

TL;DR

This paper develops a canonical quantization method for discrete general relativity that preserves diffeomorphism invariance and automatically resolves the constraint algebra, demonstrated with a cosmological model showing quantum singularity removal.

Contribution

It introduces a novel canonical formalism for lattice general relativity that maintains gauge invariance and simplifies the constraint algebra, applicable to other field theories.

Findings

Successful quantization of discrete Lorentzian gravity.

Automatic resolution of the constraint algebra.

Quantum elimination of the big bang singularity.

Abstract

It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. Additionally, the traditional lattice field theory approach consists in fixing the gauge in a Euclidean action, which does not appear appropriate for general relativity. We analyze discrete lattice general relativity and develop a canonical formalism that allows to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a ``dynamical gauge'' fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. Among other issues the problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories like BF theory. We discuss a simple cosmological application that exhibits the quantum elimination of the singularity at the big bang.