Gravity in the presence of fermions as an SU(2) gauge theory

The Hamiltonian formulation of the Holst action in the presence of a massless fermion field with a nonminimal Lagrangian is performed without any restriction on the local Lorentz frame. It is outlined that the phase-space structure does not resemble that of a background-independent Lorentz gauge theory, as some additional constraints are present. Proper phase-space coordinates are introduced, such that SU(2) connections can be defined, and the vanishing of conjugate momenta to boost variables is predicted. Finally, it is demonstrated that for a particular value of the nonminimal parameter, the kinematics coincides with that of a background-independent SU(2) gauge theory, and the Immirzi parameter becomes the coupling constant of such an interaction between fermions and the gravitational field.