
SIGMA 11 (2015), 025, 15 pages arXiv:1410.5529
https://doi.org/10.3842/SIGMA.2015.025
Metaplecticc Quantomorphisms
Jennifer Vaughan
Department of Mathematics, University of Toronto, Canada
Received December 13, 2014, in final form March 16, 2015; Published online March 24, 2015
Abstract
In the classical KostantSouriau 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 KostantSouriau quantization process in which the prequantization circle bundle and metaplectic structure for $(M,\omega)$ are replaced by a metaplecticc prequantization. They proved that metaplecticc quantization can be applied to a larger class of manifolds than the classical recipe. This paper presents a definition for a metaplecticc quantomorphism, which is a diffeomorphism of metaplecticc prequantizations that preserves all of their structures. Since the structure of a metaplecticc 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 KostantSouriau case. We then define an infinitesimal quantomorphism to be a vector field whose flow consists of metaplecticc quantomorphisms, and prove that the space of infinitesimal metaplecticc 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 $C^\infty(M)$.
Key words:
geometric quantization; metaplecticc prequantization; quantomorphism.
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