Dynamical symmetry group and quantum splittings for a free particle on the group manifold AdS{sub 3}
- Physics Department, Faculty of Science, Sahand University of Technology, P.O. Box 51335-1996, Tabriz, Iran and Department of Mathematical Sciences, University of Durham, Durham DH1 3LE (United Kingdom)
It is shown that the set of all quantum states corresponding to the motion of a free particle on the group manifold AdS{sub 3} as the bases with two different labels, constitute a Hilbert space. The second label is bounded by the first one however, the first label is semibounded. The Casimir operator corresponding to the simultaneous and agreeable shifting generators of both labels along with the Cartan subalgebra generator describe the Hamiltonian of a free particle on AdS{sub 3} with dynamical symmetry group U(1,1) and infinite-fold degeneracy for the energy spectrum. The Hilbert space for the Lie algebra of the dynamical symmetry group is a reducible representation space. But the Hilbert subspaces constructed by all the bases which have a given constant value for the difference of two their labels, constitute an irreducible representation for it. It is also shown that the irreducible representation subspaces of the Lie algebras u(1,1) and u(2) are separately spanned by the bases which have the same value for the second and first labels, respectively. These two bunches of Hilbert subspaces present two different types of quantum splittings on the Hilbert space.
- OSTI ID:
- 20699348
- Journal Information:
- Journal of Mathematical Physics, Vol. 46, Issue 8; Other Information: DOI: 10.1063/1.1982767; (c) 2005 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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