Quantum Field Theory of Spin Networks

TL;DR

The paper develops a quantum field theory of spin networks by recasting state-sum models of gravity as a field theory on $G^D$, where group Fourier modes become creation/annihilation operators and spin networks emerge as operator constructions in a Fock space. Transition amplitudes between boundary spin networks are computed as matrix elements of a discrete covariant time evolution operator, with time measured by the number of $D$-simplices, yielding finite sums of Feynman diagrams and a natural notion of discrete universes. The approach is implemented explicitly for $D=2$ (Ponzano–Regge-like structure), $D=3$ (Boulatov model and Ponzano–Regge amplitudes), and $D=4$ (Ooguri/BF theory with Barrett–Crane constraints), including BC-related finite formulations via quantum groups or modified actions. The framework unifies canonical and covariant perspectives, highlights the role of regularization, and points toward a third-quantization picture of gravity with potential matter coupling and semiclassical regimes amenable to analysis.

Abstract

We study the transition amplitudes in state-sum models of quantum gravity in D=2,3,4 spacetime dimensions by using the field theory over a Lie group formulation. By promoting the group theory Fourier modes into creation and annihilation operators we construct a Fock space for the quantum field theory whose Feynman diagrams give the transition amplitudes. By making products of the Fourier modes we construct operators and states representing the spin networks associated to triangulations of spatial boundaries of a triangulated spacetime manifold. The corresponding spin network amplitudes give the state-sum amplitudes for triangulated manifolds with boundaries. We also show that one can introduce a discrete time evolution operator, where the time is given by the number of D-simplices in the triangulation, or equivalently by the number of the vertices in the Feynman diagram. The corresponding transition amplitude is a finite sum of Feynman diagrams, and in this way one avoids the problem of infinite amplitudes caused by summing over all possible triangulations.