Holomorphic Quantization on the Torus and Finite Quantum Mechanics

Details Holomorphic Quantization on the Torus and Finite Quantum Mechanics Holomorphic Quantization on the Torus and Finite Quantum Mechanics

1995-09-18

arxiv.org/abs/hep-th/9509098v2

J.Phys.A29:6737,1996

Physics

High Energy Physics

High Energy Physics - Theory

12 pages LaTeX (needs amssymb.sty). Version as will appear in J. Phys. A

Scientific paper

10.1088/0305-4470/29/21/010

We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

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