A convergent star product on the Poincaré disc

Kraus, Daniela; Roth, Oliver; Schötz, Matthias; Waldmann, Stefan

doi:10.1016/j.jfa.2019.02.011

Cited by 14 publications

(37 citation statements)

References 29 publications

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“…For the hyperbolic disc our quantized algebra (A(D n ), * h) agrees with the algebra obtained in [30] while for the 2-sphere, (A(S 2 ), * h) is the algebra considered in [16].…”

Section: Introductionsupporting

confidence: 77%

“…Such an approach needs additional geometric structures to determine such a subalgebra. It has been carried out explicitly for exponential type star products on possibly infinite-dimensional vector spaces [40], for the linear Poisson structure on the dual of a Lie algebra [17] and for the hyperbolic disc D n using an invariant star product obtained with phase space reduction techniques [30]. See also [42] for a survey.…”

Section: Introductionmentioning

confidence: 99%

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Strict quantization of coadjoint orbits

Schmitt¹

2019

Preprint

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We obtain a strict quantization of the holomorphic functions on any semisimple coadjoint orbit of a complex semisimple connected Lie group. By restricting this quantization, we also obtain strict star products on a subalgebra of analytic functions for any semisimple coadjoint orbit of a real semisimple connected Lie group. If this Lie group was also compact, the star product is of Wick type. The main tool to construct our quantization is a construction by Alekseev-Lachowska and an explicit formula for the canonical element of the Shapovalov pairing between generalized Verma modules.

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“…For the hyperbolic disc our quantized algebra (A(D n ), * h) agrees with the algebra obtained in [30] while for the 2-sphere, (A(S 2 ), * h) is the algebra considered in [16].…”

Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Strict quantization of coadjoint orbits

Schmitt¹

2019

Preprint

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“…Several examples have been studied in this area, including also infinite dimensional ones like star products of exponential type in [38] and the Gutt star product on the dual of a Lie algebra in [18]. The hyperbolic disc was treated in [27]. However, the work so far is based on the study of specific examples and a more general approach to these questions has not yet been proposed.…”

Section: Introductionmentioning

confidence: 99%

Comparison and continuity of Wick-type star products on certain coadjoint orbits

2019

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In this paper we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev-Lachowska and Karabegov and we prove that they agree in general. In the case of the 2-sphere we establish the continuity of the star product, thereby allowing for a completion to a FréchetIn this section we collect some known results on the construction of star products on coadjoint orbits which will be used in the sequel. For this purpose, we first recall some basic tools concerning coadjoint orbits which can be found e.g. in [28]. Coadjoint orbits and complex structuresLet G be a real Lie group with Lie algebra g and denote by Ω ξ the coadjoint orbit of G through ξ ∈ g * and by O µ the adjoint orbit of G through µ ∈ g. Recall that Ω ξ has a smooth structure, which makes Ω ξ an immersed submanifold of g * and an embedded submanifold if G is compact. The coadjoint orbits Ω ξ can be endowed with a G-invariant symplectic form ω Ω ξ , called the Kirillov-Kostant-Souriau (KKS) form.Lemma 2.1 Let G be a connected semisimple Lie group. Then:

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“…We will build on this construction of the star product by a phase space reduction form n+1 since this makes the already obtained convergence results for the Wick star product available. This point of view was first taken in [5] and recently substantially extended in [108].…”

Section: The Wick Type Star Product On the Poincaré Discmentioning

confidence: 99%

“…We conclude this section with a few remarks on further properties of the star product ⋆ red . In [108] it was shown that even though the topology is not locally multiplicatively convex for the star product, many interesting transcendental functions can be defined. In particular, the algebra A(D n ) contains many bounded functions even though the span of the Schauder basis only contains constant bounded functions.…”

Section: Lemma 52mentioning

confidence: 99%

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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A convergent star product on the Poincaré disc

Cited by 14 publications

References 29 publications

Strict quantization of coadjoint orbits

Strict quantization of coadjoint orbits

Comparison and continuity of Wick-type star products on certain coadjoint orbits

Convergence of star products: From examples to a general framework

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