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

TL;DR

This paper reformulates gravity with fermions as an SU(2) gauge theory, revealing how the Immirzi parameter acts as a coupling constant in the presence of fermions, and clarifying the phase space structure.

Contribution

It provides a Hamiltonian formulation of gravity with fermions as an SU(2) gauge theory, including the role of the Immirzi parameter as a coupling constant.

Findings

Phase space structure differs from background independent Lorentz gauge theory.

SU(2) connections can be defined with proper phase space coordinates.

Immirzi parameter acts as a coupling constant for fermion-gravity interaction.

Abstract

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.