Lorentz-covariant Hamiltonian analysis of BF gravity with the Immirzi parameter

TL;DR

This paper conducts a Lorentz-covariant Hamiltonian analysis of two BF gravity formulations with the Immirzi parameter, revealing their similar constraint structures and algebraic relations at the Hamiltonian level.

Contribution

It extends the understanding of the relationship between two BF gravity actions with the Immirzi parameter by analyzing their Hamiltonian constraint structures and algebraic equivalences.

Findings

Both actions have the same number of first- and second-class constraints.

The constraint algebras are similar and can be transformed into each other via variable changes.

The Immirzi parameter's inclusion affects the constraint structure but not the overall algebraic form.

Abstract

We perform the Lorentz-covariant Hamiltonian analysis of two Lagrangian action principles that describe general relativity as a constrained BF theory and that include the Immirzi parameter. The relation between these two Lagrangian actions has been already studied through a map among the fields involved. The main difference between these is the way the Immirzi parameter is included, since in one of them the Immirzi parameter is included explicitly in the BF terms, whereas in the other (the CMPR action) it is in the constraint on the B fields. In this work we continue the analysis of their relationship but at the Hamiltonian level. Particularly, we are interested in seeing how the above difference appears in the constraint structure of both action principles. We find that they both possess the same number of first-class and second-class constraints and satisfy a very similar (off-shell) Poisson-bracket algebra on account of the type of canonical variables employed. The two algebras can be transformed into each other by making a suitable change of variables