Dirac Geometric Approach for the Unimodular Holst Action

Abstract

We perform a Hamiltonian analysis of unimodular gravity in its first-order formulation, specifically a modification of the Holst action. In order to simplify the analysis, prior studies on this theory have introduced (for several reasons) additional elements, such as parametrization, complex fields, or considering the Barbero–Immirzi parameter as imaginary. We show that, by using a geometric implementation of the Dirac algorithm, a comprehensive analysis of the theory can be conducted without relying on these additional ingredients. The resulting theory reproduces the behavior of metric unimodular gravity.