Relating Spin Foams and Canonical Quantum Gravity: (n-1)+1 formulation of nD spin foams

TL;DR

This paper establishes a formal connection between spin foam models of gravity and their canonical continuum formulations by deriving transition amplitudes and exploring their triangulation independence in n-dimensional BF theory.

Contribution

It introduces a foliation-based reformulation of nD spin foam models and explicitly computes triangulation-independent transition amplitudes for 2D BF theory.

Findings

Transition amplitudes in 2D BF theory are triangulation independent.

Spin network functionals naturally emerge on foliations.

The formal relationship between spin foams and canonical quantum gravity is clarified.

Abstract

In this article we lay foundations for a formal relationship of spin foam models of gravity and BF theory to their continuum canonical formulations. First the derivation of the spin foam model of the BF theory from the discrete BF theory action in n dimensions is reviewed briefly. By foliating the underlying n dimensional simplicial manifold using n-1 dimensional simplicial hypersurfaces, the spin foam model is reformulated. Then it is shown that spin network functionals arise naturally on the foliations. The graphs of these spin network functionals are dual to the triangulations of the foliating hypersurfaces. Quantum Transition amplitudes are defined. I calculate the transition amplitudes related to 2D BF theory explicitly and show that these amplitudes are triangulation independent. The application to the spin foam models of gravity is discussed briefly.