By Pierre Bieliavsky, Axel de Goursac, Florian Spinnler (auth.), Piotr Kielanowski, S. Twareque Ali, Alexander Odesskii, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov (eds.)

The Białowieża workshops on Geometric tools in Physics, occurring within the detailed atmosphere of the Białowieża normal woodland in Poland, are one of the vital conferences within the box. each year a few eighty to a hundred members either from arithmetic and physics sign up for to debate new advancements and to replace rules. the present quantity used to be produced at the social gathering of the XXXI assembly in 2012. For the 1st time the workshop used to be through a college on Geometry and Physics, which consisted of complicated lectures for graduate scholars and younger researchers. chosen audio system of the workshop have been requested to give a contribution, and extra overview articles have been additional. the choice indicates that regardless of its now lengthy culture the workshop continues to be constantly on the leading edge of ongoing study. The XXXI workshop had as a distinct subject the works of the overdue Boris Vasilievich Fedosov (1938–2011) who's top recognized for an easy and intensely ordinary development of a deformation quantization for any symplectic manifold, and for his contributions to index theory.​

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Ciafaloni, G. : (1989), Can spacetime be probed below the string size Phys. Lett. B 216 41. J. : (1995) Models of neutrino masses and mixings, Int. J. Mod. Phys. A10, 145. C. : (2002), Deformation quantization in the teaching of quantum mechanics, American Journal of Physics, 70 (5) pp. 537–547, May. [8] Quesne C. : Lorentz-covariant deformed algebra with minimal length, [arXiv:hep-th/0612093]. [9] Lay Nam Chang, Dlordje Minic, Naotoshi Okamura and Tatsu Takeuchi: Exact solutions of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations, [arXiv:hep-th/0111181].

Global Anal. Geom. 13 (1995) 155–168. [7] K. Habermann, Basic Properties of Symplectic Dirac Operators, Comm. in Math. Phys. 184 (1997) 629–652. [8] Katharina Habermann and Lutz Habermann, Introduction to Symplectic Dirac Operators, Lecture Notes in Mathematics 1887, Springer-Verlag, 2006. [9] B. Kostant, Symplectic Spinors. Symposia Mathematica, vol. XIV, pp. 139–152, Cambridge University Press, 1974. L. H. S. vol. 81, no. 410, 1989. Michel Cahen Universit´e Libre de Bruxelles Campus Plaine, CP 218, Bvd.

Namely, we have Theorem 3. 1. For 0 < ???? ≤ 2, (ℰ???? , ∗Λ ) is a Fr´echet algebra. Moreover, for any Λ′ having the ′ ′ same skew symmetric part as Λ, ????ΛΛ = exp( ????ℏ 4 ∂????∂) with ???? = Λ − Λ is well defined intertwiner from (ℰ???? , ∗Λ ) to (ℰ???? , ∗Λ′ ). 2. For ???? > 2, the multiplication ∗Λ : ℰ???? × ℰ????′ → ℰ???? is well defined for ????′ such that 1???? + ????1′ = 2, and (ℰ???? , ∗Λ ) is a ℰ????′ -bimodule. Star Product 43 4. Star exponentials Exponentials are very important in analysis, however they are hard to deal with in star product algebras in general.

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