Quantum Gravity on a Circle and the Diffeomorphism Invariance of the Schrodinger Equation

We study a model for quantum gravity on a circle in which the notion of a classical metric tensor is replaced by a quantum metric with an inhomogeneous transformation law under diffeomorphisms. This transformation law corresponds to the co--adjoint action of the Virasoro algebra, and resembles that of the connection in Yang--Mills theory. The transformation property is motivated by the diffeomorphism invariance of the one dimensional Schr\"odinger equation. The quantum distance measured by the metric corresponds to the phase of a quantum mechanical wavefunction. The dynamics of the quantum gravity theory are specified by postulating a Riemann metric on the space $Q$ of quantum metrics and taking the kinetic energy operator to be the resulting laplacian on the configuration space $Q/\rm Diff_0(S^1)$. The resulting metric on the configuration space is analyzed and found to have singularities. The second--quantized Schr\"odinger equation is derived, some exact solutions are found, and a generic wavefunction behavior near one of the metric singularities is described. Finally some further directions are indicated, including an analogue of the Yamabe problem of differential geometry.