The coherent states for the quantum particle on the circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra [Ĵ, U ] = U , where U is unitary, which is a direct consequence of the Heisenberg algebra [φ,Ĵ] = i, but it is more adequate for the study of the circlular motion.

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An investigation is made of coherent states that differ from the usual ones in two ways: (a) they are connected with the coset space G/H, where the stability subgroup H may be noncompact; and (b) the notion of an H-invariant ray is replaced by the more general notion of an H-invariant subspace. A general framework is given for vectorlike coherent states with the help of the nonlinear realization technique as well as with the rigged Hilbert space theory. The vectorlike coherent states are found for the Poincaré group and the hyperbolic coherent states are found for the SU(1,1) group.

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Abstract. Quantum multiparameter deformation of real Clifford algebras is proposed. The corresponding irreducible representations are found.