×

zbMATH — the first resource for mathematics

Complete positivity of Rieffel’s deformation quantization by actions of \(\mathbb R^d\). (English) Zbl 1172.53055
The authors consider Rieffel’ s deformation by actions of \(\mathbb R^d\) in general and prove that every state of the underformed algebra can be deformed into a continuous field of states for the field of deformed algebras. Moreover, the authors give an explicit construction including a detailed study of the asymptotics of the deformed states for \(\hbar \rightarrow 0\). It turns out that the asymptotic expansion coincides in a precise sense with the formal deformations obtained in [H. Bursztyn and S. Waldmann, On positive deformations of \(\ast\)-algebras. Conférence Moshé Flato 1999: Quantization, deformations, and symmetries, Dijon, France, September 5–8, 1999. Volume II. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 22, 69–80 (2000; Zbl 0979.53098)].

MSC:
53D55 Deformation quantization, star products
46L87 Noncommutative differential geometry
81R60 Noncommutative geometry in quantum theory
46L65 Quantizations, deformations for selfadjoint operator algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111 (1978), 61-110; Deformation theory and quantization. II. Physical applications. ibid. 111 (1978), 111-151. · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5 · www.emis.de · www.ams.org
[2] S. Beiser, H. Römer, and S. Waldmann, Convergence of the Wick star product. Comm. Math. Phys. 272 (2007), 25-52. · Zbl 1203.53089 · doi:10.1007/s00220-007-0190-x · arxiv:math/0506605
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.