Hamiltonian analysis of SO(4,1) constrained BF theory

TL;DR

This paper performs a canonical analysis of SO(4,1) constrained BF theory, revealing its equivalence to vacuum Einstein equations and its relation to Holst formulation of gravity, including topological terms and the Immirzi parameter.

Contribution

It demonstrates the canonical structure of SO(4,1) constrained BF theory and its connection to Holst gravity, including topological invariants and the Immirzi parameter.

Findings

Equivalence to vacuum Einstein equations

Contains Holst term with Immirzi parameter

Constraint structure matches Holst gravity

Abstract

In this paper we discuss canonical analysis of SO(4,1) constrained BF theory. The action of this theory contains topological terms appended by a term that breaks the gauge symmetry down to the Lorentz subgroup SO(3,1). The equations of motion of this theory turn out to be the vacuum Einstein equations. By solving the B field equations one finds that the action of this theory contains not only the standard Einstein-Cartan term, but also the Holst term proportional to the inverse of the Immirzi parameter, as well as a combination of topological invariants. We show that the structure of the constraints of a SO(4,1) constrained BF theory is exactly that of gravity in Holst formulation. We also briefly discuss quantization of the theory.