Reality Conditions for Spin Foams

TL;DR

This paper develops reality conditions for spin foam models, linking bivector scalar products to real area metrics, and shows how classical real general relativity emerges from complex models via these constraints.

Contribution

It introduces the area metric reality constraint for spin foams and demonstrates how classical real general relativity can be derived from complex models using these conditions.

Findings

Reality conditions relate bivectors to real area metrics.

Classical real GR emerges from complex models with imposed reality constraints.

Spin foam models for all signatures are projections of the SO(4,C) model.

Abstract

An idea of reality conditions in the context of spin foams (Barrett-Crane models) is developed. The square of areas are the most elementary observables in the case of spin foams. This observation implies that simplest reality conditions in the context of the Barrett-Crane models is that the all possible scalar products of the bivectors associated to the triangles of a four simplex be real. The continuum generalization of this is the area metric reality constraint: the area metric is real iff a non-degenerate metric is real or imaginary. Classical real general relativity (all signatures) can be extracted from complex general relativity by imposing the area metric reality constraint. The Plebanski theory can be modified by adding a Lagrange multiplier to impose the area metric reality condition to derive classical real general relativity. I discuss the SO(4,C) BF model and SO(4,C) Barrett-Crane model. It appears that the spin foam models in 4D for all the signatures are the projections of the SO(4,C) spin foam model using the reality constraints on the bivectors.