Constraint Analysis and Quantization of Anomalous 2-D Thomas-Whitehead Gravity

TL;DR

This paper explores the constraint structure and quantization of a generalized 2D gravity model, the Thomas-Whitehead gravity, which extends the Polyakov action by making the coadjoint element a dynamical field, thus resolving the issue of vanishing Hamiltonians.

Contribution

It introduces a dynamical extension of 2D gravity via the Thomas-Whitehead formalism and analyzes its constraints and quantization in different metric formalisms.

Findings

Adding dynamics to the diffeomorphism field removes vanishing Hamiltonians.

Constraint analysis is consistent across different gauges and backgrounds.

The extended model provides a framework for non-trivial Hamiltonian dynamics in 2D gravity.

Abstract

The two-dimensional effective Polyakov action is often realized as the anomalous contributions of string theories and fermions coupled to gravity in two-dimensions. However, as a result of the reparameterization invariance, one finds that the effective action produces vanishing Hamiltonians as constraints even in disparate gauges such as the dynamical light-cone and the ADM formalism of the metric. On the other hand, two-dimensional gravitational theories naturally arise as geometric actions on the coadjoint orbits of the Virasoro algebra. The Thomas-Whitehead gravity formalism extends the effective Polyakov action in such a way that the defining coadjoint element for the orbit becomes a dynamical field, viz the diffeomorphism field. In this work, we examine the constraint analysis and quantization of the Hamiltonian in the context of Thomas-Whitehead gravity using both the dynamical light-cone and the ADM formalisms of the metric. Constraint analysis is then repeated in a Minkowski background and with a dynamical action for the diffeomorphisms field arising from the Thomas-Whitehead action. Adding dynamics to the diffeomorphism field subsequently removes the vanishing Hamiltonians.