Operator Spin Foam Models

TL;DR

This paper introduces operator spin foam models as a systematic framework for spin foams, unifying various approaches including quantum gravity models like EPRL and FK, through a general and consistent formalism.

Contribution

It develops a comprehensive operator-based formalism for spin foams, enabling systematic construction and analysis of models relevant to quantum gravity.

Findings

Defined operator spin foams with group representations and operators.

Established a symmetry-based classification of natural models.

Connected the framework to existing quantum gravity models like EPRL and FK.

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.