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

TL;DR

This paper reformulates the Hamiltonian constraint of four-dimensional gravity using the affine group formalism for any Immirzi parameter value, simplifying the structure and enabling quantization.

Contribution

It introduces a new affine Lie algebra formulation of the Hamiltonian constraint for arbitrary Immirzi parameter, facilitating its quantization.

Findings

Reformulation of the Hamiltonian constraint as an affine Lie algebra.

Quantization of the reformulated constraint.

Applicability to arbitrary Immirzi parameter values.

Abstract

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.