Husain-Kuchar model as a constrained BF theory

Abstract

The Husain-Kuchar theory is a four-dimensional background-independent model that has long been viewed as a useful model for addressing several conceptual and technical problems appearing in the quantization of general relativity mainly in the loop quantum gravity approach. The model was defined at Lagrangian level in terms of a su(2)-valued connection one-form $A$ coupled through its curvature to a su(2)-valued one-form field $e$. We address here the problem of writing a Lagrangian formulation for the Husain-Kuchar model as a constrained BF theory motivated by the fact that spin foam models for quantum gravity are related to action principles of the BF type. The Lagrangian action principle for the Husain-Kuchar model reported here differs from a previous one found by Barbero et al in that this description involves a single constrained BF theory rather than two interacting BF theories. It is, essentially, the Plebanski action with the condition on the trace of the Lagrange multipliers removed. Moreover, it can be stated that the relationship between our BF-like action and the original one for the Husain-Kuchar model is the same relationship that exists between the Plebanski action and the self-dual Palatini action for complex general relativity, first because the solution to the constraint on the two-forms $Σ^i$ coming from the BF-like action leads to the Husain-Kuchar action, and second because the Hamiltonian analysis of the Husain-Kuchar model is straightforward starting from the BF-like action principle.