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

TL;DR

This work constructs a four-dimensional lattice quantum gravity model based on SU(2) Ashtekar variables and the Samuel-Jacobson-Smolin action, defining a finite per-site lattice action and a gauge-invariant path integral that expands into surface-like excitations. The path integral is reformulated via a SU(2) character expansion, revealing a 2D surface complex built from faces carrying spins and edges carrying intertwiners, i.e., a spin-foam–style description with local degrees of freedom. A three-dimensional version is derived that reduces to lattice BF theory, providing a nontrivial consistency check and illustrating the absence of local propagating degrees of freedom in 3D. In 4D, the framework aims to capture the physical degrees of freedom of general relativity through surface excitations, with vanishing expectations for basic observables under appropriate gauge choices and the constraint-surface behavior, and it is extensible to randomly triangulated lattices and triangulation-sum approaches for enhanced robustness.

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.