Spin foam models for quantum gravity from lattice path integrals

TL;DR

This paper derives spin foam models for quantum gravity from lattice path integrals, connecting BF theory and Regge calculus, and explores their geometric and dynamical properties.

Contribution

It introduces a new derivation of spin foam models from lattice path integrals incorporating a measure for consistent simplex gluing.

Findings

Models reduce to known spin foam models with specific gluing choices

On-shell action becomes independent of the Immirzi parameter

Stationarity implies flat spacetime geometry

Abstract

Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations which satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models correspond to a special choice of gluing. In this case, the equations of motion describe genuine geometries, where the constraints of area-angle Regge calculus are satisfied. Furthermore, the Immirzi parameter drops out of the on-shell action, and stationarity with respect to area variations requires spacetime geometry to be flat.