Symplectic Structure of 2D Dilaton Gravity

TL;DR

This paper investigates the symplectic structure of 2D dilaton gravity, revealing the phase space structure and quantum Hilbert space in the compact case, while highlighting boundary ambiguities in the non-compact case.

Contribution

It provides a detailed analysis of the symplectic form and phase space of 2D dilaton gravity, including quantum aspects and boundary issues.

Findings

Reduced phase space is a two-dimensional cotangent bundle in the absence of matter.

Quantum Hilbert space is explicitly determined for the compact case.

Boundary ambiguities prevent a well-defined symplectic form in the non-compact case.

Abstract

We analyze the symplectic structure of two-dimensional dilaton gravity by evaluating the symplectic form on the space of classical solutions. The case when the spatial manifold is compact is studied in detail. When the matter is absent we find that the reduced phase space is a two-dimensional cotangent bundle and determine the Hilbert space of the quantum theory. In the non-compact case the symplectic form is not well defined due to an unresolved ambiguity in the choice of the boundary terms.