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

TL;DR

This work develops a dual non-commutative GFT formalism to quantize Holst-Plebanski gravity, keeping bivector geometry explicit and encoding simplicity constraints via non-commutative delta functions. It defines a 4D gravity model with Immirzi parameter $\gamma$ that reproduces Barrett-Crane amplitudes in the $\gamma\to\infty$ limit while avoiding rationality restrictions on $\gamma$, and presents both BF path-integral and spin-foam representations using projected boundary states. The model introduces an extended GFT with normals to tetrahedra and a simplicity operator $\widehat{S}^\beta$, producing constrained BF amplitudes that depend on $\beta$ (hence on $\gamma$) through edge/face weights but share a common vertex amplitude. It further analyzes limiting cases, the ultralocality issue, and the potential for a covariant continuum limit, positioning the framework as a natural candidate for exploring quantum geometry and the semiclassical emergence of GR from a simplicial, non-commutative setting.

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.

Figures (1)

• Figure 1: GFT propagator and vertex