Quantization: Towards a comparison between methods

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Abstract
In this paper it is shown that the procedure of geometric quantiztion applied to Kähler manifolds gives the following result: the Hilbert space H consists, roughly speaking, of holomorphic functions on the phase space M and to each classical observable f (i.e., a real function on M) is associated an operator f on H as follows: first multiply by f+ 1/4 ℏΔdRf (ΔdR being the Laplace–de Rham operator on the Kähler manifold M) and then take the holomorphic part [see G. M. Tuynman, J. Math. Phys. 2 7, 573 (1987)]. This result is correct on compact Kähler manifolds and correct modulo a boundary term ∫Mdα on noncompact Kähler manifolds. In this way these results can be compared with the quantization procedure of Berezin [Math. USSR Izv. 8, 1109 (1974); 9, 341 (1975); Commun. Math. Phys. 4 0, 153 (1975)], which is strongly related to quantization by *‐products [e.g., see C. Moreno and P. Ortega‐Navarro; Amn. Inst. H. Poincaré Sec. A: 3 8, 215 (1983); Lett. Math. Phys. 7, 181 (1983); C. Moreno, Lett. Math. Phys. 1 1, 361 (1986); 1 2, 217 (1986)]. It is shown that on irreducible Hermitian spaces [see S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, Orlando, FL, 1978] the contravariant symbols (in the sense of Berezin) of the operators f as above are given by the functions f+ 1/4 ℏΔdRf. The difference with the quantization result of Berezin is discussed and a change in the geometric quantization scheme is proposed.

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