By utilization of three elementary vector operators as position, angular momentum and their cross product, a simple realization of gl(2,c) Lie algebra on sphere are constructed. The coherent states based on this algebra can then be constructed by standard manner.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces.
In this paper we study the generalized coherent states (g.c.s.) in n -dimensional space. Their annihilation operators consist of linear combinations of the annihilation operators a l , and their coherent states are the product of the simple coherent states. All the known annihilation operators which have been studied as yet, and also the magnetic ones, are special cases of these operators. The new operators A k + A m form the Lie algebra U ( n ) of the Lie group U ( n ). Finally, instead of the operators a l and a l + we can use the new operators, which depend on a complex parameter.
The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, i.e. the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski et al ( 1996 J. Phys. A: Math. Gen. 29 4149 ), we deal with a deformation of the classical phase space related to quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.