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Metaplectic-c Quantomorphisms
Jennifer Vaughan Department of Mathematics, University of Toronto, Canada
Abstract:
In the classical Kostant–Souriau prequantization procedure, the Poisson algebra of a symplectic manifold $(M,\omega)$ is realized as the space of infinitesimal quantomorphisms of the prequantization circle bundle. Robinson and Rawnsley developed an alternative to the Kostant–Souriau quantization process in which the prequantization circle bundle and metaplectic structure for $(M,\omega)$ are replaced by a metaplectic-c prequantization. They proved that metaplectic-c quantization can be applied to a larger class of manifolds than the classical recipe. This paper presents a definition for a metaplectic-c quantomorphism, which is a diffeomorphism of metaplectic-c prequantizations that preserves all of their structures. Since the structure of a metaplectic-c prequantization is more complicated than that of a circle bundle, we find that the definition must include an extra condition that does not have an analogue in the Kostant–Souriau case. We then define an infinitesimal quantomorphism to be a vector field whose flow consists of metaplectic-c quantomorphisms, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen for the infinitesimal quantomorphisms of a prequantization circle bundle. In particular, this space is isomorphic to the Poisson algebra $C^\infty(M)$.
Keywords:
geometric quantization; metaplectic-c prequantization; quantomorphism.
Received: December 13, 2014; in final form March 16, 2015; Published online March 24, 2015
Citation:
Jennifer Vaughan, “Metaplectic-c Quantomorphisms”, SIGMA, 11 (2015), 025, 15 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1006 https://www.mathnet.ru/eng/sigma/v11/p25
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Abstract page: | 505 | Full-text PDF : | 33 | References: | 25 |
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