Metaplectic-c Quantomorphisms

TL;DR

This paper extends the concept of quantomorphisms to metaplectic-c prequantizations, providing a new framework that generalizes classical prequantization and preserves the Poisson algebra structure.

Contribution

It defines metaplectic-c quantomorphisms and infinitesimal quantomorphisms, showing they form a structure isomorphic to the Poisson algebra, thus generalizing classical quantization methods.

Findings

Metaplectic-c quantomorphisms preserve all prequantization structures.

The space of infinitesimal metaplectic-c quantomorphisms is isomorphic to the Poisson algebra.

The definition includes an extra condition due to the complexity of metaplectic-c structures.

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 $C^\infty(M)$.