Isomonodromic quantization of dimensionally reduced gravity Original Research Article

We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable “midisuperspace” 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 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, the problem of satisfying the additional constraints associated with the coset space SL(2,R)/SO(2) remains open. We briefly discuss the possible implications of our results for string theory.