No abstract

Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, more than fifty years ago, of an "obstruction" to quantization. Their "no-go theorems" assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space R 2n in a physically meaningful way. Similar obstructions have been recently found for S 2 and T * S 1 , buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so-it has just been proven that there are no obstructions to quantizing either T 2 or T * R+. In this paper we work towards delineating the circumstances under which such obstructions will appear, and understanding the mechanisms which produce them. Our objectives are to conjecture-and in some cases prove-generalized Groenewold-Van Hove theorems, and to determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of symplectic manifolds and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory, here formulated in terms of "basic algebras of observables." We then review in detail the known results for R 2n , S 2 , T * S 1 , T 2 , and T * R+, as well as recent theoretical work. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert spacebased quantization techniques.

Abstract. Pukanszky's condition is a condition used in obtaining representations from coadjoint orbits. In order to obtain more geometric insight in this condition, we relate it to symplectic induction. It turns out to be equivalent to the condition that the orbit in question is a symplectic subbundle of a modified cotangent bundle. §1 Introduction One of the original goals of geometric quantization was to obtain a general method of constructing (irreducible) representations of Lie groups out of their coadjoint orbits. The idea was to generalize the Borel-Weil-Bott theorem for compact groups and Kirillov's results for nilpotent groups. Since then geometric quantization has led a somewhat dual life. On the one hand in representation theory where it is called the orbit method (see [Gu] for a relatively recent review). On the other hand in physics where it serves as a procedure that starts with a symplectic manifold (a classical theory) and creates a Hilbert space and a representation of the Poisson algebra as operators on it (the quantum theory).Recent results in quantum reduction theory [DEGST] allow us to show rigorously in some particular cases that geometric quantization intertwines the procedures of symplectic induction and unitary induction. Since the latter is one of the ingredients in the orbit method, this gives a geometrical insight into the "classical" part of the orbit method. In particular, it allows us to give a geometrical interpretation of Pukanszky's condition on a polarization which is completely different from the well known interpretation that says that the coadjoint orbit contains an affine plane. In fact, we prove (Proposition 3.9) that Pukanszky's condition is equivalent to the statement that the coadjoint orbit in question is, in a noncanonical way, symplectomorphic to a symplectic subbundle of a modified cotangent bundle (where modified means that the canonical symplectic form on the cotangent bundle is modified by adding a closed 2-form on the base space). For real polarizations these results have also been obtained with completely different methods in [KP]. Again for real polarizations we obtain as a corollary that Pukanszky's condition links the two dual lives of geometric quantization.Typeset by A M S-T E X

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R d . However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R m|n as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C*-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative Quantum Field Theory.

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.