Induction of Representations in Deformation Quantization

Bursztyn, Henrique; Waldmann, Stefan

doi:10.1142/9789812775061_0004

Cited by 2 publications

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“…Remark 3.2 In the approach of [5,[7][8][9] one main point was to replace the real numbers Ê by an arbitrary ordered ring R and by the ring extension C = R(i) with i 2 = −1. This allows to include also the formal star product algebras from deformation quantization into the game.…”

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confidence: 99%

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The Covariant Picard Groupoid in Differential Geometry

Waldmann

2006

Int. J. Geom. Methods Mod. Phys.

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Section: X B • Ymentioning

confidence: 99%

“…From the differential geometric point of view, it is a natural question whether all these techniques as developed in [5,[7][8][9]31] can be made compatible with a certain given symmetry of the underlying manifold. On the purely algebraic level, a fairly general notion of 'symmetry' is that of a Hopf algebra action of a given Hopf algebra.…”

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confidence: 99%

The Covariant Picard Groupoid in Differential Geometry

Waldmann

2006

Int. J. Geom. Methods Mod. Phys.

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“…7] and the lecture notes [144]. Beside studying the existence of positive functionals as deformations of classical ones [37,43] one can establish a notion of strong Morita equivalence yielding the equivalence of categories of * -representations [32][33][34]36,39,41,42,[44][45][46] culminating in a complete and geometrically simple classification of star products up to Morita equivalence in [40,49], first in the symplectic and then in the general Poisson case. Conversely, the investigations of the Picard group of star products triggered an analogous program also for the semi-classical counterpart, the Picard group(oid) in Poisson geometry [35,47,48].…”

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Convergence of star products: From examples to a general framework

Waldmann

2019

EMS Surv. Math. Sci.

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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...

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Induction of Representations in Deformation Quantization

Abstract: We discuss the procedure of Rieffel induction of representations in the framework of formal deformation quantization of Poisson manifolds. We focus on the central role played by algebraic notions of complete positivity.

Cited by 2 publications

References 19 publications

The Covariant Picard Groupoid in Differential Geometry

The Covariant Picard Groupoid in Differential Geometry

Convergence of star products: From examples to a general framework

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