Spin Foam Models of Matter Coupled to Gravity

TL;DR

This paper develops spin foam models that incorporate matter fields coupled to four-dimensional quantum gravity, using group representations to describe both sectors and providing a quantum field theory perspective.

Contribution

It introduces a novel class of spin foam models coupling matter to gravity with explicit representation-based amplitudes and a quantum field theory formulation.

Findings

Spin foam amplitudes expressed via spin network pentagrams with matter edges

Matter degrees of freedom modeled by spin network fields with group indices

Proposal for matter representations in Lorentzian gravity based on SO(3) or SO(2)

Abstract

We construct a class of spin foam models describing matter coupled to gravity, such that the gravitational sector is described by the unitary irreducible representations of the appropriate symmetry group, while the matter sector is described by the finite-dimensional irreducible representations of that group. The corresponding spin foam amplitudes in the four-dimensional gravity case are expressed in terms of the spin network amplitudes for pentagrams with additional external and internal matter edges. We also give a quantum field theory formulation of the model, where the matter degrees of freedom are described by spin network fields carrying the indices from the appropriate group representation. In the non-topological Lorentzian gravity case, we argue that the matter representations should be appropriate SO(3) or SO(2) representations contained in a given Lorentz matter representation, depending on whether one wants to describe a massive or a massless matter field. The corresponding spin network amplitudes are given as multiple integrals of propagators which are matrix spherical functions.