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

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

1995-01-18

arxiv.org/abs/gr-qc/9501020v2

Phys.Rev.D53:3147-3155,1996

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

The proof for the nonexistence of the one- and two-dimensional representations of PSL(2,Z) in the relevant Hilbert space, has

Scientific paper

10.1103/PhysRevD.53.3147

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.

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