Isomonodromic Quantization of Dimensionally Reduced Gravity

TL;DR

This paper develops an isomonodromic quantization approach for a simplified model of quantum gravity derived from Einstein's theory, revealing a well-defined Hilbert space and explicit quantum states, with implications for string theory.

Contribution

It introduces a novel isomonodromic quantization method for dimensionally reduced gravity, connecting it to integrable systems and explicitly constructing quantum states within this framework.

Findings

Physical states form a well-defined Hilbert space.

Quantum states are explicitly constructed using $SL(2,R)$ representations.

Constraints related to the coset space require solutions from principal series representations.

Abstract

We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable ``midi-superspace" version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation of variables in the isomonodromic sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are thereby reduced to the Knizhnik-Zamolodchikov equations for $SL(2,R)$. The physical states are shown to live in a well defined Hilbert space and are manifestly invariant under the full diffeomorphism group. An infinite set of independent observables \`a la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of $SL(2,R)$, we construct explicit quantum states. However, satisfying the additional constraints associated with the coset space $SL(2,R)/SO(2)$ requires solutions based on the principal series representations, which are not yet known. We briefly discuss the possible implications of our results for string theory.