Spin Foam Models of Matter Coupled to Gravity

TL;DR

This work constructs a class of spin foam models that couple matter to 4D gravity by assigning unitary irreps to the gravitational sector and finite-dimensional irreps to matter within a shared group $G$. It provides both a spin-network formulation and a quantum field theory formulation with spin-network fields, and develops explicit Euclidean and Lorentzian constructions, including fermionic matter and generalized interactions. The amplitudes reduce to integrals over homogeneous spaces in the Euclidean case and to matrix-spherical-function propagators in the Lorentzian case, offering a concrete framework to study matter–gravity coupling in spin foams. The approach lays groundwork for analyzing finiteness, semiclassical behavior, and a spin-foam gauge structure, with future work aimed at continuum limits and more realistic matter sectors.

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.

Figures (9)

• Figure I: A spin foam Feynman diagram describing propagation of a matter field, which is represented by a dotted line.
• Figure II: The simplest spin net graph which gives a 7-valent vertex for a spin foam Feynman diagram.
• Figure III: A nontrivial spin net graph which gives a matter propagator for a spin foam Feynman diagram.
• Figure IV: An example of a more complex spin net graph determining the 7-valent spin foam vertex.
• Figure V: A more complex spin net graph determining the matter spin foam propagator.