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 TMF, 2014, Volume 181, Number 3, Pages 568–596 (Mi tmf8745)

Star products on symplectic vector spaces: Convergence, representations, and extensions

M. A. Soloviev

Lebedev Physical Institute, RAS, Moscow, Russia

Abstract: We briefly survey the general scheme of deformation quantization on symplectic vector spaces and analyze its functional analytic aspects. We treat different star products in a unified way by systematically using an appropriate space of analytic test functions for which the series expansions of the star products in powers of the deformation parameter converge absolutely. The star products are extendable by continuity to larger functional classes. The uniqueness of the extension is guaranteed by suitable density theorems. We show that the maximal star product algebra with the absolute convergence property, consisting of entire functions of an order at most $2$ and minimal type, is nuclear. We obtain an integral representation for the star product corresponding to the Cahill–Glauber $s$-ordering, which connects the normal, symmetric, and antinormal orderings continuously as $s$ varies from $1$ to $-1$. We exactly characterize those extensions of the Wick and anti-Wick correspondences that are in line with the known extension of the Weyl correspondence to tempered distributions.

Keywords: deformation quantization, Weyl correspondence, Wick symbol, anti-Wick symbol, star-product algebra, noncommutative quantum field theory

DOI: https://doi.org/10.4213/tmf8745

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English version:
Theoretical and Mathematical Physics, 2014, 181:3, 1612–1637

Bibliographic databases:

Citation: M. A. Soloviev, “Star products on symplectic vector spaces: Convergence, representations, and extensions”, TMF, 181:3 (2014), 568–596; Theoret. and Math. Phys., 181:3 (2014), 1612–1637

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8745
• https://doi.org/10.4213/tmf8745
• http://mi.mathnet.ru/eng/tmf/v181/i3/p568

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. P. Adam, V. A. Andreev, A. Isar, V. I. Man'ko, M. A. Man'ko, “Star product, discrete Wigner functions, and spin-system tomograms”, Theoret. and Math. Phys., 186:3 (2016), 346–364
2. M. A. Soloviev, “Weyl correspondence for a charged particle in the field of a magnetic monopole”, Theoret. and Math. Phys., 187:2 (2016), 782–795
3. M. A. Soloviev, “Dirac's magnetic monopole and the Kontsevich star product”, J. Phys. A-Math. Theor., 51:9 (2018), 095205
4. M. A. Soloviev, “Spaces of Type $S$ as Topological Algebras under Twisted Convolution and Star Product”, Proc. Steklov Inst. Math., 306 (2019), 220–241
5. M. A. Soloviev, “Spaces of type $S$ and deformation quantization”, Theoret. and Math. Phys., 201:3 (2019), 1682–1700
6. M. A. Soloviev, “Characterization of the Moyal Multiplier Algebras for the Generalized Spaces of Type $S$”, Proc. Steklov Inst. Math., 309 (2020), 271–283
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