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. 
R.J. Henderson and S.G. Rajeev