Large Diffeomorphisms in (2+1)-Quantum Gravity on the Torus

Abstract

The issue of how to deal with the modular transformations -- large diffeomorphisms -- in (2+1)-quantum gravity on the torus is discussed. I study the Chern-Simons/connection representation and show that the behavior of the modular transformations on the reduced configuration space is so bad that it is possible to rule out all finite dimensional unitary representations of the modular group on the Hilbert space of $L^2$-functions on the reduced configuration space. Furthermore, by assuming piecewise continuity for a dense subset of the vectors in any Hilbert space based on the space of complex valued functions on the reduced configuration space, it is shown that finite dimensional representations are excluded no matter what inner-product we define in this vector space. A brief discussion of the loop- and ADM-representations is also included.

The proof for the nonexistence of the one- and two-dimensional representations of PSL(2,Z) in the relevant Hilbert space, has been extended to cover all finite dimensional unitary representations. The notation is slightly improved and a few references are added.

References in corpus (14)

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Cited by in corpus (11)

• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• On the relation between 2+1 Einstein gravity and Chern Simons theory
• A Hexagon Model for 3D Lorentzian Quantum Cosmology
• Lectures in (2+1)-Dimensional Gravity
• Diffeomorphism invariant subspaces in Witten's 2+1 quantum gravity on $\mathbf{R}\times T^2$
• Combinatorial quantisation of the Euclidean torus universe
• The Quantum Modular Group in (2+1)-Dimensional Gravity
• QUANTUM ASPECTS OF 2+1 GRAVITY
• Mapping class group and U(1) Chern-Simons theory on closed orientable surfaces
• 2+1 gravity, chaos and time machines
• Topological Order of Quantum Gravity in $Ad{S}_3$ Spacetime