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Star products: a group-theoretical point of view

Paolo Aniello

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Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product, explicit formulas can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Groenewold–Moyal star product. In the second example, the link with wavelet analysis is clarified.


PACS

03.65.Ca Formalism

02.20.Qs General properties, structure, and representation of Lie groups

03.65.Fd Algebraic methods

03.65.Ta Foundations of quantum mechanics; measurement theory

03.65.Wj State reconstruction, quantum tomography

MSC

81P15 Quantum measurement theory

81R50 Quantum groups and related algebraic methods (See also 16W35, 17B37)

81R15 Operator algebra methods (See also 46Lxx, 81T05)

Subjects

Mathematical physics

Quantum information and quantum mechanics

Dates

Issue 47 (27 November 2009)

Received 7 April 2009, in final form 11 September 2009

Published 6 November 2009



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