The Birkhoff Theorem in the Quantum Theory of Two-Dimensional Dilaton Gravity

TL;DR

This paper investigates how the classical Birkhoff theorem extends into the quantum realm within the CGHS model of two-dimensional dilaton gravity, demonstrating that quantum states preserve classical metric properties.

Contribution

It explicitly constructs the quantum Hilbert space for the CGHS model and proves a quantum version of the Birkhoff theorem, showing metric properties are maintained at the quantum level.

Findings

Quantum states satisfy classical metric conditions

Mass operator eigenvalues label physical states

Hilbert space matches static quantum mechanical treatment

Abstract

In classical two-dimensional pure dilaton gravity, and in particular in spherically symmetric pure gravity in d dimensions, the generalized Birkhoff theorem states that, for a suitable choice of coordinates, the metric coefficients are only functions of a single coordinate. It is interesting to see how this result is recovered in quantum theory by the explicit construction of the Hilbert space. We examine the CGHS model, enforce the set of auxiliary conditions that select physical states a` la Gupta-Bleuler, and prove that the matrix elements of the metric and of the dilaton field obey the classical requirement. We introduce the mass operator and show that its eigenvalue is the only gauge invariant label of states. Thus the Hilbert space is equivalent to that obtained by quantum mechanical treatment of the static case. This is the quantum form of the Birkhoff theorem for this model.