TABLE OF CONTENTS The geometric quantization procedure follows the principle of canonical quantization by replacing a Poisson bracket of smooth functions on a Poisson manifold with a commutator product of operators in a Hilbert space such that Dirac's condition (4.2.1) holds. In the framework of geometric quantization, these operators are constructed by means of a suitable U(1)-connection.
Subsections: A. Prequantization, 296; B. Polarization, 302; C. Quantization, 303.
We refer the reader to [141; 257; 401; 438] for the basics on geometric quantization of symplectic manifolds. This quantization technique has been generalized to Poisson manifolds in terms of contravariant connections [425; 426; 427] (see Section 5.3). Though there is one-to-one correspondence between the (regular) Poisson structures on a smooth manifold and its symplectic foliations, geometric quantization of a Poisson manifold need not imply quantization of its symplectic leaves [427].
Firstly, contravariant connections fail to admit the pull-back operation. Therefore, prequantization of a Poisson manifold does not determine straightforwardly prequantization of its symplectic leaves.
Secondly, polarization of a Poisson manifold is defined in terms of sheaves of functions, and it need not be associated to any distribution. As a consequence, its pull-back onto a leaf is not polarization of a symplectic manifold in general.
Thirdly, a quantum algebra of a Poisson manifold contains the center of a Poisson algebra. However, there are models where quantization of this center has no physical meaning. For instance, the center of the Poisson algebra of a mechanical system with classical parameters...
