Hamiltonian analysis of fermions coupled to the Holst action

TL;DR

This paper develops three Lorentz-invariant Hamiltonian formulations for fermions coupled to the Holst action, including gauge fixing and transformations, providing real phase-space variables and connections for quantum gravity research.

Contribution

It introduces new Hamiltonian formulations with real variables and symplectomorphisms for fermions coupled to the Holst action, enhancing the understanding of Lorentz invariance in quantum gravity.

Findings

Three Lorentz-invariant Hamiltonian formulations achieved.

Two formulations involve half-densitized fermion fields.

A symplectomorphism relates to the Palatini action for specific parameters.

Abstract

We report three manifestly Lorentz-invariant Hamiltonian formulations of minimally and nonminimally coupled fermion fields to the Holst action. These formulations are achieved by making a suitable parametrization of both the tetrad and the Lorentz connection, which allows us to integrate out some auxiliary fields without spoiling the local Lorentz symmetry. They have the peculiarity that their noncanonical symplectic structures as well as the phase-space variables for the gravitational sector are real. Moreover, two of these Hamiltonian formulations involve half-densitized fermion fields. We also impose the time gauge on these formulations, which leads to real connections for the gravitational configuration variables. Finally, we perform a symplectomorphism in one of the manifestly Lorentz-invariant Hamiltonian formulations and analyze the resulting formulation, which becomes the Hamiltonian formulation of fermion fields minimally coupled to the Palatini action for particular values of the coupling parameters.