Quantization: Towards a comparison between methods

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. 27, 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. 40, 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: 38, 215 (1983); Lett. Math. Phys. 7, 181 (1983); C. Moreno, Lett. Math. Phys. 11, 361 (1986); 12, 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.

Generalized Berezin-Toeplitz quantization of Kähler supermanifolds

“…in [16] and more recently in [17]. Generalized Berezin and Berezin-Toeplitz quantizations were defined and analyzed in [10] and include interesting new possibilities, such as Berezin-Bergman quantization.…”

“…Such an operator has the block diagonal structure A = A + ⊕ A − (with A ± ∈ GL(E ± )) where A ± are ( , ) ± -Hermitian and A + is positive definite. The supercoherent states with respect to the new product ( , ) ′ , which in turn induces the pairing (( , )) ′ x , are given by: 16) while the new supercoherent projectors take the form:…”

“…If twisted chiral superfields are included as well, there is an analogous relation to generalized complex geometry. One can furthermore add superpotential terms of the form 16) which have to be accompanied by their complex conjugate. Here, W andŴ are polynomials in the chiral and twisted chiral superfields, restricted by renormalizability of the theory.…”

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

“…Equation (5.2) is related to a formula due to Tuynman [13]. Namely, for any smooth function f ∈ C ∞ (M),…”

Quantization Methods: A Guide for Physicists and Analysts

“…This link between geometric and Berezin quantization was discovered by Tuynman [242] [243], who showed that on a compact Kähler manifold (as well as in some other situations) the operators Q f of the geometric quantization coincide with the Toeplitz operators T f +h∆f , where ∆ is the Laplace-Beltrami operator. Later on this connection was examined in detail in a series of papers by Cahen [55] and Cahen, Gutt and Rawnsley [56] (parts I and II of [56] deal with compact manifolds, part III with the unit disc, and part IV with homogeneous spaces).…”