1994/01/25 by Robert Henderson, R. J. Henderson, S. G. Rajeev · 4 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Diffeomorphism #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Metric tensor #Noncommutative and Quantum Gravity Theories #Phase space #Physics #Quantum #Quantum gravity #Quantum mechanics #Wave function #gr-qc #hep-th
paper · pdf · doi:10.1088/0264-9381/11/7/006
published in Classical and Quantum Gravity 11(7), 1631-1651 (IOP Publishing) · 35 pages, Tex
arxiv created 1994/01/25 · openalex publication_date 1994/07/01 · arxiv updated 2010/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 Schrödinger 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 . The resulting metric on the configuration space is analysed and found to have singularities. The second-quantized Schrödinger equation is derived, some exact solutions are found, and a generic wavefunction behaviour near one of the metric singularities is described. Finally, some directions for further study are indicated, including an analogue of the Yamabe problem of differential geometry.