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

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.


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