The Deformation Quantizations of the Hyperbolic Plane

Bieliavsky, Pierre; Detournay, Stéphane; Spindel, Philippe

doi:10.1007/s00220-008-0697-9

Cited by 33 publications

(28 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It took some considerable effort to find integral formulas beyond the abelian case. Here Bieliavsky and Gayral found a vast generalization of Rieffel's original ideas and proposed universal deformation formulas in [18], see also the earlier works [12,13,15,16,19,20]. These ideas lead ultimately to a quantization of Riemann surfaces of higher genus [14].…”

Section: The Quest For Convergencementioning

confidence: 99%

Convergence of star products: From examples to a general framework

Waldmann

2019

EMS Surv. Math. Sci.

View full text Add to dashboard Cite

We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincaré disc. * Awarded with the Prix du Concour annuel 2018: On demande une contribution à la construction d'un cadre convergent pour la quantification par déformation of the Académie royale des Sciences, des Lettres et des Beaux-Arts de Belgique.In the by now classical paper [3], Bayen, Frønsdal, Flato, Lichnerowicz, and Sternheimer introduced the notion of a formal star product on a Poisson manifold as a general method to pass from a classical mechanical system encoded by the Poisson manifold to its corresponding quantum system. The main difference compared to other, and more ad-hoc, quantization schemes is the emphasis on the role of the observable algebra. The observable algebra is constructed not as particular operators on a Hilbert space as this is usually done. Instead, one stays with the same vector space of smooth functions on the Poisson manifold and just changes the commutative pointwise product into a new noncommutative product using Planck's constant as a deformation parameter. This way one tries to model the quantum mechanical commutation relations.While in [3] it is shown that this leads to the same results as expected in quantum mechanics for those systems where alternative quantization schemes are available, the proposed scheme of deformation quantization has several conceptual advantages: the first is of course its vast generality concerning the formulation. While other quantization schemes make much more use of specific features of the classical system, deformation quantization can be seen as almost universal in the sense that its requirements are virtually minimal. Only a Poisson algebra of classical observables is needed to formulate the program of deformation quantization. Of course, the hard part of the work consists then in actually proving the existence (and possible classifications) of star products, but in any case, the conceptual framework is fixed from the beginning. A second advantage is that the physical interpretation of the observables is fixed from the beginning: the observables simply stay the same elements of the same underlying vector space. Thus it is immediately clear which quantum observable is the Hamiltonian, the momentum etc. since they are the same as classical. It is only the product law which changes, the correspondence of classical and quantum observables is implemented trivially. A third advantage of this approach to focus on the algebra first is shared also by other formulations of quantum theory, most notably by axiomatic quantum field theory, see e.g. [100]: having the focus on the...

show abstract

Section: The Quest For Convergencementioning

confidence: 99%

Convergence of star products: From examples to a general framework

Waldmann

2019

EMS Surv. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…Rieffel [31] built the deformation of Abelian Lie groups and the associated universal deformation formula (UDF). This was also recently extended to (non-Abelian) Kählerian Lie groups [32,33] and to the case of SL(2, R) [34] and of SU (1, n) [35]. Star-exponentials [36,37] associated to these deformations were computed in the non-formal setting in [38,39] and such deformations were linked to Hilbert algebras and multipliers in [40].…”

Section: Application To Non-formal Deformation Quantizationmentioning

confidence: 99%

Noncommutative supergeometry and quantum supergroups

Goursac

2015

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

This is a review of concepts of noncommutative supergeometry -namely Hilbert superspace, C*-superalgebra, quantum supergroup -and corresponding results. In particular, we present applications of noncommutative supergeometry in harmonic analysis of Lie supergroups, non-formal deformation quantization of supermanifolds, quantum field theory on noncommutative spaces; and we give explicit examples as deformation of flat superspaces, noncommutative supertori, solvable topological quantum supergroups.

show abstract

“…The strategy above has been carried out successfully in some concrete examples and special classes of integrable Poisson manifolds ( [2,3,13,12,23,25,26,27]). However, with this approach (as opposed to deformation quantization for instance), non integrable Poisson manifolds are discarded from the start, and functoriality issues, due to the ill-behaved composition of canonical relations, have to be faced.…”

Section: Functorial Quantizationmentioning

confidence: 99%