  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.
