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

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons61279

Guedes,  Carlos
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons20698

Oriti,  Daniele
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons26303

Raasakka,  Matti
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Fulltext (public)

1301.7750
(Preprint), 470KB

JoMP_4818638.pdf
(Any fulltext), 551KB

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Citation

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.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000E-B4C1-2
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.