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

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.

Authors:
;  [1];  [1]
  1. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam (Germany)
Publication Date:
OSTI Identifier:
22224162
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 54; Journal Issue: 8; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FOURIER TRANSFORMATION; MAPS; PHASE SPACE; POISSON EQUATION; QUANTIZATION; QUANTUM GRAVITY; SU-2 GROUPS; UNITARY SYMMETRY; WAVE PROPAGATION

Citation Formats

Guedes, Carlos, Oriti, Daniele, Raasakka, Matti, and LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse. Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups. United States: N. p., 2013. Web. doi:10.1063/1.4818638.
Guedes, Carlos, Oriti, Daniele, Raasakka, Matti, & LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse. Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups. United States. doi:10.1063/1.4818638.
Guedes, Carlos, Oriti, Daniele, Raasakka, Matti, and LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse. Thu . "Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups". United States. doi:10.1063/1.4818638.
@article{osti_22224162,
title = {Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups},
author = {Guedes, Carlos and Oriti, Daniele and Raasakka, Matti and LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse},
abstractNote = {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.},
doi = {10.1063/1.4818638},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 8,
volume = 54,
place = {United States},
year = {2013},
month = {8}
}