Operator Spin Foam Models

TL;DR

The paper presents operator spin foams as a unified, representation-theoretic framework for 4D spin-foam models, encoding faces by irreps and edges by projection operators. Through a carefully defined equivalence relation and a face amplitude $A_f = \dim \mathcal{H}_f$, it ensures consistent subdivision and glueing, and derives a partition-function formulation ${\cal Z}(\kappa,\rho,P)$. Natural operator spin foam models arise by imposing symmetry constraints on BF spin foams, yielding constrained versions that include Barrett-Crane and EPRL-like intertwiners; the framework clarifies how these models relate and differ, particularly in glueing prescriptions. The formalism provides explicit state-sum expressions and demonstrates how cEPRL emerges as a principled natural model, with boundary amplitudes and normalization ensuring glueing compatibility and connection to standard spin-foam amplitudes.

Abstract

The goal of this paper is to introduce a systematic approach to spin foams. We define operator spin foams, that is foams labelled by group representations and operators, as the main tool. An equivalence relation we impose in the set of the operator spin foams allows to split the faces and the edges of the foams. The consistency with that relation requires introduction of the (familiar for the BF theory) face amplitude. The operator spin foam models are defined quite generally. Imposing a maximal symmetry leads to a family we call natural operator spin foam models. This symmetry, combined with demanding consistency with splitting the edges, determines a complete characterization of a general natural model. It can be obtained by applying arbitrary (quantum) constraints on an arbitrary BF spin foam model. In particular, imposing suitable constraints on Spin(4) BF spin foam model is exactly the way we tend to view 4d quantum gravity, starting with the BC model and continuing with the EPRL or FK models. That makes our framework directly applicable to those models. Specifically, our operator spin foam framework can be translated into the language of spin foams and partition functions. We discuss the examples: BF spin foam model, the BC model, and the model obtained by application of our framework to the EPRL intertwiners.

Figures (11)

• Figure 1: Operator form of Spin-Foam
• Figure 2: The edge Hilbert space ${\cal H}_e$
• Figure 3: Invariance under the face subdivision
• Figure 4: Invariance under the face subdivision
• Figure 5: Invariance under face subdivision