Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension

The first part of this work focuses on the canonical group quantization approach applied to non-commutative spaces, namely plane R2 and two-torus T2. Canonical group quantization is a quantization approach that adopts the group structure that respects the global symmetries of the phase space as a...

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Main Author:	Umar, Mohd Faudzi
Format:	Thesis
Language:	English
Published:	2020
Subjects:	Heisenberg uncertainty principle Noncommutative function spaces
Online Access:	http://psasir.upm.edu.my/id/eprint/98672/1/IPM%202021%205%20-%20IR.pdf
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id	my-upm-ir.98672
record_format	uketd_dc
spelling	my-upm-ir.986722022-09-08T06:48:15Z Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension 2020-11 Umar, Mohd Faudzi The first part of this work focuses on the canonical group quantization approach applied to non-commutative spaces, namely plane R2 and two-torus T2. Canonical group quantization is a quantization approach that adopts the group structure that respects the global symmetries of the phase space as a main ingredient. This is followed by finding its unitary irreducible representations. The use of noncommutative space is motivated by the idea of quantum substructure of space leading to nontrivial modification of the quantization. Extending to noncommuting phase space includes noncommuting momenta that arises naturally in magnetic background as in Landau problem. The approach taken is to modify the symplectic structures corresponding to the noncommutative plane, noncommutative phase space and noncommutative torus and obtain their canonical groups. In all cases, the canonical group is found to be central extensions of the Heisenberg group. Next to consider is to generalize the approach to twisted phase spaces where it employs the technique of Drinfeld twist on the Hopf algebra of the system. The result illustrates that a tool from the deformation quantization can be used in canonical group quantization where the deformed Heisenberg group H2 q is obtained and its representation stays consistent with the discussion in the literature. In the second part, the two-parameter deformations of quantum group for Heisenberg group and Euclidean group are studied. Both can be achieved through the contraction procedure on SU(2)q;p quantum group. The study also continues to develop (q; p)-extended Heisenberg quantum group from the previous result. As conclusion, it is shown that the extensions of Heisenberg group arise from quantizing noncommutative plane, noncommutative phase space, noncommutative two-torus, and twisted phase space. The work on two-parameter deformation of quantum group also further shows generalizations of the extension of Heisenberg group. Heisenberg uncertainty principle Noncommutative function spaces 2020-11 Thesis http://psasir.upm.edu.my/id/eprint/98672/ http://psasir.upm.edu.my/id/eprint/98672/1/IPM%202021%205%20-%20IR.pdf text en public doctoral Universiti Putra Malaysia Heisenberg uncertainty principle Noncommutative function spaces Zainuddin, Hishamuddin
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
advisor	Zainuddin, Hishamuddin
topic	Heisenberg uncertainty principle Noncommutative function spaces
spellingShingle	Heisenberg uncertainty principle Noncommutative function spaces Umar, Mohd Faudzi Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
description	The first part of this work focuses on the canonical group quantization approach applied to non-commutative spaces, namely plane R2 and two-torus T2. Canonical group quantization is a quantization approach that adopts the group structure that respects the global symmetries of the phase space as a main ingredient. This is followed by finding its unitary irreducible representations. The use of noncommutative space is motivated by the idea of quantum substructure of space leading to nontrivial modification of the quantization. Extending to noncommuting phase space includes noncommuting momenta that arises naturally in magnetic background as in Landau problem. The approach taken is to modify the symplectic structures corresponding to the noncommutative plane, noncommutative phase space and noncommutative torus and obtain their canonical groups. In all cases, the canonical group is found to be central extensions of the Heisenberg group. Next to consider is to generalize the approach to twisted phase spaces where it employs the technique of Drinfeld twist on the Hopf algebra of the system. The result illustrates that a tool from the deformation quantization can be used in canonical group quantization where the deformed Heisenberg group H2 q is obtained and its representation stays consistent with the discussion in the literature. In the second part, the two-parameter deformations of quantum group for Heisenberg group and Euclidean group are studied. Both can be achieved through the contraction procedure on SU(2)q;p quantum group. The study also continues to develop (q; p)-extended Heisenberg quantum group from the previous result. As conclusion, it is shown that the extensions of Heisenberg group arise from quantizing noncommutative plane, noncommutative phase space, noncommutative two-torus, and twisted phase space. The work on two-parameter deformation of quantum group also further shows generalizations of the extension of Heisenberg group.
format	Thesis
qualification_level	Doctorate
author	Umar, Mohd Faudzi
author_facet	Umar, Mohd Faudzi
author_sort	Umar, Mohd Faudzi
title	Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
title_short	Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
title_full	Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
title_fullStr	Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
title_full_unstemmed	Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
title_sort	deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension
granting_institution	Universiti Putra Malaysia
publishDate	2020
url	http://psasir.upm.edu.my/id/eprint/98672/1/IPM%202021%205%20-%20IR.pdf
_version_	1747813884783230976

Deformed heisenberg group for a particle on noncommutative spaces via canonical group quantization and extension

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