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Zbl 1172.53056
Léandre, Rémi
Deformation quantization in infinite dimensional analysis. (English)
[A] Blath, Jochen (ed.) et al., Trends in stochastic analysis. Papers dedicated to Professor Heinrich von Weizsäcker on the occasion of his 60th birthday. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 353, 303-325 (2009). ISBN 978-0-521-71821-9/pbk

This paper reviews different aspects of deformation quantization. Let us recall that the simplest case of a star-product is the Moyal star-product on $\Bbb R^n$. Several works used this star-product in order to define a star-product on any symplectic manifold $M$ (for instance, the standard Fedosov quantization). With Kontsevich's formality, the existence has been generalized to any Poisson manifold. In this work, the author replaces the algebra $C(M)$ by a Frechet algebra. The notion of equivalences of two star-products can be extended to this situation. Considering specific cases of functions, the author studies to types of deformations: continuous deformations and differential deformations. The algebra of functionals related to some infinite-dimensional manifolds and the quantization of the free path space of a manifold are finally discussed.
[Angela Gammella-Mathieu (Metz)]

MSC 2000:: *53D55 Deformation quantization, star products
60H07 Stochastic calculus of variations and the Malliavin calculus

Keywords: deformation quantization; star product; Poisson manifold; symplectic manifold; Frechet algebra; infinite dimensional analysis

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