Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity

TL;DR

This paper introduces a new 4d gravity model within group field theory that incorporates the Immirzi parameter, unifies spin foam and non-commutative path integral approaches, and addresses issues in existing models.

Contribution

It defines a novel 4d gravity model with the Immirzi parameter, connecting spin foam and non-commutative formulations, and relaxes rationality constraints present in previous models.

Findings

Reproduces Barrett-Crane amplitudes as gamma approaches infinity

Does not require rationality condition for the Immirzi parameter

Provides explicit non-commutative BF path integral formulation

Abstract

In a recent work, a dual formulation of group field theories as non-commutative quantum field theories has been proposed, providing an exact duality between spin foam models and non-commutative simplicial path integrals for constrained BF theories. In light of this new framework, we define a model for 4d gravity which includes the Immirzi parameter gamma. It reproduces the Barrett-Crane amplitudes when gamma goes to infinity, but differs from existing models otherwise; in particular it does not require any rationality condition for gamma. We formulate the amplitudes both as BF simplicial path integrals with explicit non-commutative B variables, and in spin foam form in terms of Wigner 15j-symbols. Finally, we briefly discuss the correlation between neighboring simplices, often argued to be a problematic feature, for example, in the Barrett-Crane model.