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  Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups

Guedes, C., Oriti, D., & Raasakka, M. (2013). Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups. Journal of Mathematical Physics, 54: 083508. doi:10.1063/1.4818638.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000E-B4C1-2 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0015-0FDB-9
Genre: Journal Article

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1301.7750 (Preprint), 470KB
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 Creators:
Guedes, Carlos1, Author              
Oriti, Daniele1, Author              
Raasakka, Matti1, Author              
Affiliations:
1Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_67201              

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Free keywords: Mathematical Physics, math-ph,General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th,Mathematics, Mathematical Physics, math.MP
 Abstract: The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations and non-commutative plane waves.

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 Dates: 2013-01-3120132013
 Publication Status: Published in print
 Pages: 35 pages
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 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 1301.7750
DOI: 10.1063/1.4818638
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Title: Journal of Mathematical Physics
Source Genre: Journal
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Publ. Info: Woodbury, N.Y. [etc.] : American Institute of Physics
Pages: - Volume / Issue: 54 Sequence Number: 083508 Start / End Page: - Identifier: ISSN: 0022-2488
CoNE: /journals/resource/954922836227