The Hamiltonian formulation of the Holst action in presence of a massless
fermion field with a non-minimal Lagrangian is performed without any
restriction on the local Lorentz frame. It is outlined that the phase space
structure does not resemble that one 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 non-minimal parameter the
kinematics coincides with that one 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.Comment: 5 pages, accepted for publication in Phys. Rev.
