Deformation quantization with separation of variables on a super-Kähler manifold

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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“…21 Karabegov's star product on Pol(Ë λ 2 ) can be written asf Ë =L (h) L (h) −1 (f ) * Gutt,1 L (h) −…”

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confidence: 99%

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

2019

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“…Let ⋆ be a star product with separation of variables on (M, ω −1 ) with classifying form ω. It was shown in [20] that the star product ⋆ and the function H determine a unique global star product with separation of variables * on ΠE which is Z 2 -graded with respect to the standard parity of the functions on ΠE and satisfies the following property. Let U ⊂ M be any contractible coordinate chart, ΠE| U ∼ = U × C 0|d be a trivialization, Φ = ν −1 Φ −1 + Φ 0 + .…”

Section: Deformation Quantization On a Super-kähler Manifoldmentioning

confidence: 99%

“…Given a formal function f ∈ C ∞ (ΠE| U )((ν)), one can describe the operator L f as follows. There exists a unique formal differential operator A on ΠE| U which supercommutes with the operators It was shown in [20] that the star product * has a supertrace given by a canonically normalized formal supertrace density globally defined on ΠE.…”

Section: Deformation Quantization On a Super-kähler Manifoldmentioning

confidence: 99%

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A heat kernel proof of the index theorem for deformation quantization

Karabegov¹

2017

Lett Math Phys

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“…The star products obtained this way turn out to be of Wick type in the sense that one function is differentiated in holomorphic directions while the other is differentiated in anti-holomorphic directions only. Such star products have been shown to exist in general by Karabegov [104], see also [29] for an explicit construction based on Fedosov's approach. Beside Kähler manifolds, cotangent bundles are perhaps the most important classical phase spaces.…”

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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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Deformation quantization with separation of variables on a super-Kähler manifold

Abstract: We construct deformation quantizations with separation of variables on a split super-Kähler manifold and describe their canonical supertrace densities.

Cited by 6 publications

References 18 publications

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

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

A heat kernel proof of the index theorem for deformation quantization

Convergence of star products: From examples to a general framework

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