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

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

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• Poincare-Plebanski formulation of GR and dual simplicity constraints
• Revisiting the solution of the second-class constraints of the Holst action
• Barbero's formulation from a $BF$-type action with the Immirzi parameter
• Perturbative BF theory
• Polysymplectic formulation for BF gravity with Immirzi parameter
• Canonical analysis with no second-class constraints of $BF$ gravity with Immirzi parameter