Reformulation of the Hamiltonian constraint of four dimensional gravity for arbitrary values of the Immirzi parameter via the affine group formalism

One of the virtues of the Ashtekar variables is the simplification of the initial value constraints for gravity. In the case of self-dual variables this entails a complexification of the phase space which comes at the expense of having to implement reality conditions in the Lorentzian signature case. A reformulation of the theory in terms of real variables eliminates this difficulty, albeit at the expense of having to deal with a more complicated Hamiltonian constraint. The set of available gravitational theories classically equivalent to Einstein’s is parametrized by a parameter $\beta$, known as the Immirzi parameter. We rephrase the Hamiltonian constraint into the form of an affine Lie algebra for arbitrary $\beta$, and perform a quantization.