This commented bibliography gives an overview on the results and techniques in deformation quantization. We have tried to sort the definitions, results, techniques, questions etc. into some categories listed below. It is certainly a very personal picture and we have certainly not mentioned all relevant articles for one of these topics but a rather small selection. If you believe that some important topic or work is missing please do not hesitate to point out the approriate references to us.
Deformation quantization was born as an attempt to interpret the quantization of a classical system as an associative deformation (i.e. star-products) of the algebra of classical observables in the direction of the Poisson bracket. This idea was behind the mind of many mathematical physicists and physicists (e.g. Weyl [weyl:1931a], Wigner [wigner:1932a]) as illustrated by the historical developments which led to deformation quantization.
In flat phase space R2n, one has the Weyl correspondence [weyl:1931a] which associate to a classical function f the operator W(f)= F acting on the Hilbert space L2(Rn). Shortly after Weyl, Wigner [wigner:1932a] introduced his famous map that "dequantizes" an operator F and the phase-space approach to quantum mechanics. The Wigner map associates to an operator its symbol by a trace formula. In a quantum statistical mechanics context, Moyal [moyal:1949a] has derived an expression of the classical function [f, g]* associated to the commutator of operators [F, G]: the well-known Moyal bracket. Actually, Groenewold [groenewold:1946a] did find a few years before, the classical function f * g mapped to the product of operators F G through the Weyl correspondence. In this paper was introduced for the first time the explicit expression of the Moyal product as an exponential on the Poisson bracket on R2n, i.e., the Moyal product.
Flato, Lichnerowicz, and Sternheimer [flato.lichnerowicz.sternheimer:1974a, 1975a, 1976a] introduced and studied 1-differentiable deformations of the Poisson bracket (formal Poisson brackets) on a symplectic manifold and then proposed the current interpretation of quantization as a deformation of the classical theory. Vey [vey:1975a] studied important cohomological problems related to deformations, obtained new results on the existence of star-products on symplectic manifolds, and also rediscovered Moyal bracket.
Deformation quantization was comprehensively presented in the twin papers [bayen.et.al:1978a]. One can find there the mathematical framework and tools of the theory and its physical interpretation. The deformation quantization of standard examples are worked out in details: the harmonic oscillator, the hydrogen atom, and angular momentum. The basics of star-representation theory are also introduced in these papers.
Soon after setting up the framework of deformation quantization it became clear that the general question of existence of star products and their classification was a non-trivial problem. First results on both questions where found in the early eighties culminating with the general existence theorem of DeWilde and Lecomte [dewilde.lecomte:1983b] on symplectic manifolds in 1983. Soon later Fedosov gave his geometric construction of star products on symplectic manifolds in 1985 which was unfortunately unnoticed till the early nineties, see his book for a detailed description [fedosov:1996a]. A third proof for the symplectic case was given by Omori, Maeda and Yoshioka [omori.maeda.yoshioka:1991a] in 1991.
The classification problem was unsolved for longer time though it was known very early that if the second deRham cohomology of the symplectic manifold is trivial then all star products are equivalent. In 1995 Nest and Tsygan showed that every star product on a symplectic manifold is equivalent to a Fedosov star product and that the later are classified by formal series in the second deRham cohomology [nest.tsygan:1995a]. Many other approaches to this classification followed, e.g. by Bertelson, Cahen and Gutt, Deligne, Weinstein and Xu leading to what is now called the characteristic class of a star product on a symplectic manifold. For an overview see e.g. Gutt's review [gutt:2000a].
The general Poisson case proved to be a much harder problem and it took until 1997 when Kontsevich proved the general existence theorem for arbitrary Poisson manifolds including also a classification of the star products up to equivalence by formal deformations of the Poisson structure modulo formal diffeomorphisms. In fact he proved his much more profund formality conjecture, see e.g. [kontsevich:1999a] and [dito.sternheimer:2002a] for a recent review. Since then the formality theorem found various applications not only in deformation quantization.
Strict Quantization has been introduced by Rieffel as a functional framework for convergent deformation quantization of C* algebra [rieffel:1989a]. It therefore deals mainly with bounded operator algebras. Roughly, a strict deformation quantization of a (Poisson) C* algebra A is the data of a continuous field of C* algebras over an interval containing zero. The 0-fiber of the field should coincide with the given A and the definition is designed to implement Dirac's condition in a natural way. Many versions of strict quantization can be found in the literature: see e.g. [landsman:1998a] and references therein. Strict Quantization is closely related to quantum groups in their C* version [nagy:1992a] as well as Noncommutative Geometry [connes:1994a, connes.landi:2001a]. A powerful tool for producing non-commutative spaces in this context is Rieffel's deformation method for actions of Rd [rieffel:1993a]. The key point there is to interpret the Moyal-Weyl product formula on R2n as a universal deformation formula for actions of R2n on any C* algebra. Such universal formulas exists for (some) non-Abelian Lie groups in the formal setting [giaquinto.zhang:1998a] as well as in the strict one [bieliavsky:2002a].
There are several refined notions of star products. First a star product is called differentiable if the cochains are bidifferential operators. Some authors even take this as part of the definition. If the bidifferential operator in order r of the formal parameter is even a bidifferential operator of differentiation order r in each argument then the star product is called of Vey type. The star product is called Hermitian (or symmetric or involutive) if the complex conjugation is a *-involution, i.e. a involutive anti-linear anti-automorphism. Finally, the star product is called of Weyl type if the cochains in order r have the parity (-1)r and are real if one expands in powers of i times Planck's constant h. Of course the Weyl-Moyal star product has this property. Connes, Flato and Sternheimer introducted in the symplectic case the notion of a (strongly) closed star product [connes.flato.sternheimer:1992a] Here the star product is called (strongly) closed if the integration with respect to the Liouville measure is a trace for the star product. The closed star products are important for all kind of index theorems, see e.g. the work of Omori, Maeda and Yoshioka for an existence proof [omori.maeda.yoshioka:1992a]. The notion of closed star products is also meaningful in the Poisson case after one has specified a volume form [felder.shoikhet:2000a].
While the above notions are meaningful on any Poisson manifold the following notions need some more specific geometry. On a cotangent bundle a star product is called of standard order type if the first argument is differentiated only in momentum direction, which is a global characterization on a cotangent bundle. Pflaum and Bordemann, Neumaier and Waldmann have shown that such star products always exist on cotangent bundles [bordemann.neumaier.waldmann:1998a, pflaum:1998b] For a Kähler manifold a star product has the separation of variable property (also called Wick type or anti-Wick type property) if one argument is differentiated only in holomorphic the other argument only in anti-holomorphic directions. Karabegov showed that such star products always exist [karabegov:1996a], see also the work of Bordemann and Waldmann [bordemann.waldmann:1997a] where a modified Fedosov proceedure was used.
Phase space reduction provides a powerful tool to handle classical systems with symmetries. Hence it is of major importance to understand its quantum counterpart, in particular as this is needed for all physical systems with gauge symmetries.
Fundamental for phase space reduction is the notion of a moment map. On the level of star products the quantized version is a quantum moment map which now encodes the symmetry. The probably first appearence of quantum moment maps can be found in the simple example of the (quantized) reduction from Cn+1 to CPn by Bordemann et al. [bordemann.brischle.emmrich.waldmann:1996a]. A systematic study of quantum moment maps was started by Xu in [xu:1998a].
Having a quantum moment map one can try to perform the reduction also for the star products. This was done by Fedosov [fedosov:1998a] in the framework of his star products. Bordemann, Herbig and Waldmann used the BRST cohomological approach to reduction in [bordemann.herbig.waldmann:2000a] to extend Fedosov's approach. Here also some counter-examples to the 'quantization commutes with reduction' conjecture are obtained. In the convergent situation Landsman proposed that Rieffel induction is the correct counterpart of phase space reduction on the quantum side, see his monography [landsman:1998a].
Beside the construction of the observable algebra a good notion of states is crucial in order to have a physically relevant formulation of quantum physics. The advantage of deformation quantization is that one can obtain a notion for states as a derived concept almost for free out of the deformed observable algebra structure, i.e. the star product algebra, very similar to the case of C*-algebras. The key observation is that the ring of real formal power series is an ordered ring and hence one can define positive linaer functionals with values in the ring of complex formal power series. This gives indeed a physically relevant notion of states. Moreover, the algebraic analogue of the GNS construction gives a notion of *-representation on a pre Hilbert space over the formal power series ring constructed out of a given positive linear functional. Bordemann and Waldmann have shown that the usual examples like the Schrödinger representation and the Bargmann Fock representation can be obtained this way [bordemann.waldmann:1998a]. Also thermodynamical states have been considered by Basart, Flato, Lichnerowicz, and Sternheimer [basart.flato.lichnerowicz.sternheimer:1984a] and Bordemann, Römer and Waldmann [bordemann.roemer.waldmann:1998a].
This algebraic GNS construction is also the starting point for a general representation theory of the deformed algebras. A recent review on this topic can be found in [waldmann:2002a]. In particular, the notion of strong Morita equivalence was transfered to deformation quantization by Bursztyn and Waldmann and the classification of strongly Morita equivalent star products has been obtained [bursztyn.waldmann:2002a, jurco.schupp.wess:2002a].
Star representation theory is representation theory of Lie groups in the context of star products. It has been initiated in [fronsdal:1978a]. The use of star product methods in representation theory has led to an elegant interpretation of Kirilov's method of orbits in terms of the so-called adapted Fourier transform. This has been particularly efficient for solvable Lie groups yielding, among other things, to a very simple character formula [arnal.cortet:1990a]. The semisimple case has also been investigated in this direction but the theory is far from being completed [cahen:1996a, arnal.cahen.gutt:1989a]. The interplay between non-commutative harmonic analysis and quantization has extensively been studied by Unterberger et al. within the framework of pseudo-differential calculus as well as operator algebras [unterberger:1998a, unterberger.upmeier:1996a]. Let us finally mention the proof of the Kashiwara-Vergne conjecture as a byproduct of Kontsevich's construction [kontsevich:1997a, andler.dvorsky.sahi:2002a].
Last modified: Fri Feb 21 17:34:02 CET 2003