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arXiv:hep-th/9406170 · Journal of Mathematical Physics

From geometric quantization to Moyal quantization

José M. Gracia-Bondía, Joseph C. Várilly

Published Jun 1, 1995• 22 citations
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Abstract

It is shown how the Moyal product of phase-space functions, and the Weyl correspondence between symbols and operator kernels, may be obtained directly using the procedures of geometric quantization, applied to the symplectic groupoid constructed by ‘‘doubling’’ the phase space.
DOI
10.1063/1.531059

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Quantization of R2R^2R2 and S1×S1S^1 \times S^1S1×S1 phase spaces are explicitly carried out tweaking the techniques of geometric quantization. Crucial is a combined use of left and right invariant vector fields. Canonical bases, operators and their actions are explicitly presented. Arguments of Dirac and also Wu and Yang for monopoles are applied for obtaining the quantization of the phase space area. Equivalence of states in the infinite dimensional prequantum Hilbert space resulting in a physical Hilbert space of dimension equal to the phase space area in units of the Planck constant is demonstrated. These techniques can be applied to any manifold with a symplectic structure.