Quantization of Kähler manifolds. I: Geometric interpretation of Berezin’s quantization. (English) Zbl 0719.53044

Rawnsley, J.; Cahen, M.; Gutt, S.

J. Geom. Phys. 7, No. 1, 45-62 (1990).

Summary: We give a geometric interpretation of Berezin’s symbolic calculus on Kähler manifolds in the framework of geometric quantization. Berezin’s covariant symbols are defined in terms of coherent states and we study a function \(\omega\) on the manifold which is the central object of the theory. When this function is constant Berezin’s rule coincides with the prescription of geometric quantization for the quantizable functions. It is defined on a larger class of functions. We show in the compact homogeneous case how to extend Berezin’s procedure to a dense subspace of the algebra of smooth functions.

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53D50	Geometric quantization

Keywords:

Berezin’s symbolic calculus; Kähler manifolds; geometric quantization; algebra of smooth functions

References:

[1]	Bayen, F., Quantization by deformations, Ann. phys., 111, 61, (1978), and al. · Zbl 0377.53024
[2]	Berezin, F., Quantization, Math. USSR izvestija, 8, 1109, (1974) · Zbl 0312.53049
[3]	Hartshorne, R., Algebraic geometry, (), 432
[4]	Kostant, B., Quantization and unitary representations, () · Zbl 0249.53016
[5]	Moreno, C., * products on some Kähler manifolds, Let. math. phys., Invariant * products and representations of compact semi simple Lie groups, Lett. math. phys., 12, 217, (1986)
[6]	Rawnsley, J., A nonunitary pairing of polarisations for the Kepler problem, Trans. amer. math. soc., 250, 167, (1979) · Zbl 0422.58019
[7]	Rawnsley, J., Coherent states and Kähler manifolds, Quat. J. math. Oxford, 28, 2, 403, (1977) · Zbl 0387.58002
[8]	Souriau, J., Structure des systèmes dynamiques, (1970), Dunod · Zbl 0186.58001