Quantization and Constraint Analysis of the Diffeomorphism Field in Two-Dimensional Anomalous Gravity

The geometric action of the Virosaro algebra recovers the effective gravitational action in two dimensions with a new contribution. A similar analysis of the Kac-Moody algebra finds the partially gauge-fixed Wess-Zumino-Witten action, where the new contribution is identified as the gauge field. Motivating the treatment of the new term in the effective gravitational action as a potentially dynamical field. Utilizing the invariance of geodesics to projective transformations of the connection and the construction of Projective Thomas-Whitehead gravity, the new field is identified as the diffeomorphism field. Where the diffeomorphism field is a piece of a larger connection. We can identify the effective gravitational action as now containing a dynamical dark energy term from geometry, and not matter. An analysis of the constraints and the canonical quantization of the effective gravitational action in the dynamical light-cone metric and in the ADM formalism is done. The Hamiltonian constraints are recovered, however, using the canonical commutation relations of the metric fields and their corresponding momenta, we are now able to solve the constraints for the diffeomorphism field in terms of quantum operators. In the light-cone metric, the diffeomorphism field will be seen to be the number operator with an extra overall κ dependence. In the ADM formalism, the Hamiltonian can again be found constrained to vanish, but now through constraints applied to the diffeomorphism field. Attempting to include dynamics of the diffeomorphism field through the Projective Gauss-Bonnet action finds classical wave-like solutions.