A lattice quantum gravity model with surface-like excitations in 4-dimensional spacetime

TL;DR

This paper introduces a 4D lattice quantum gravity model based on SU(2) Ashtekar variables, representing spacetime as a sum over surface-like excitations, and explores its properties and lower-dimensional reductions.

Contribution

It constructs a novel 4D lattice quantum gravity model using spin foam concepts and defines its action and path integral in a discrete setting.

Findings

Path integral expressed as sum over surface-like excitations

3D version reduces to lattice BF theory

Expectation values computed in 3D case

Abstract

A lattice quantum gravity model in 4 dimensional Riemannian spacetime is constructed based on the SU(2) Ashtekar formulation of general relativity. This model can be understood as one of the family of models sometimes called ``spin foam models.'' A version of the action of general relativity in continuum is introduced and its lattice version is defined. A dimensionless ``(inverse) coupling'' constant is defined so that the value of the action of the model is finite per lattice point. The path integral of the model is expanded in the characters and shown to be written as a sum over surface-like excitations in spacetime. A 3 dimensional version of the model exists and can be reduced to lattice BF theory. The expectation values of some quantities are computed in 3 dimensions and the meanings of the results are discussed. Although the model is studied on a hyper cubic lattice for simplicity, it can be generalized to a randomly triangulated lattice with small modifications.