Quantization of symplectic tori in a real polarization

Details Quantization of symplectic tori in a real polarization Quantization of symplectic tori in a real polarization

1996-09-30

arxiv.org/abs/dg-ga/9609012v1

Physics

High Energy Physics

High Energy Physics - Theory

65 pages, AMSLaTeX version 1.2

Scientific paper

10.1063/1.531970

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.

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