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Paper   IPM / Physic / 7866
School of Physics
Title:   Dynamical Symmetry Group and Quantum Splittings for A Free Particle on the Group Manifold AdS3
Author(s):
 1 A.R. Chenaghlou 2 H. Fakhri
Status:   Published
Journal: J. Math. Phys.
Vol.:  46
Year:  2005
Pages:   083501
Supported by:  IPM
Abstract:
It is shown that the set of all quantum states corresponding to the motion of a free particle on the group manifold AdS3 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 AdS3 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.