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

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

1994-01-25

arxiv.org/abs/gr-qc/9401029v1

Class.Quant.Grav.11:1631-1652,1994

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

35 pages, Tex

Scientific paper

10.1088/0264-9381/11/7/006

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.

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