Canonical formulation of Poincare BFCG theory and its quantization

TL;DR

This paper develops a canonical formulation and quantization approach for Poincare BFCG theory, establishing its equivalence to Poincare BF theory and exploring the construction of spin-foam states as a categorical extension of loop quantum gravity.

Contribution

It provides the first canonical formulation of Poincare BFCG theory, relates it to BF theory via a canonical transformation, and discusses quantization and spin-foam state construction.

Findings

Canonical formulation in terms of 2-connection and conjugate momenta.

Equivalence between Poincare BFCG and BF actions established.

Outline for constructing spin-foam states in categorical quantum gravity.

Abstract

We find the canonical formulation of the Poincare BFCG theory in terms of the spatial 2-connection and its canonically conjugate momenta. We show that the Poincare BFCG action is dynamically equivalent to the BF action for the Poincare group and we find the canonical transformation relating the two. We study the canonical quantization of the Poincare BFCG theory by passing to the Poincare-connection basis. The quantization in the 2-connection basis can be then achieved by performing a Fourier transform. We also briefly discuss how to approach the problem of constructing a basis of spin-foam states, which are the categorical generalization of the spin-network states from Loop Quantum Gravity.