We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3 + 1 dimensions) amounts to introducing wave functions that depend on elements of the Poincaré group G. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group Π = G × G. All such transformations can be studied by considering a generalized regular representation of G in the space of scalar functions on the group, f (x, z), that depend on the Minkowski space points x ∈ G/Spin(3, 1) as well as on the orientation variables given by the elements z of a matrix Z ∈ Spin(3, 1). In particular, the field f (x, z) is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties. * zα˙b. Field classification is based on the use of the maximal set of commuting operators (two Casimir operators and four left and right generators each, the total being ten, equal to the number of group parameters).

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Coherent states (CS) of the SU (N ) groups are constructed explicitly and their properties are investigated. They represent a nontrivial generalization of the spining CS of the SU (2) group. The CS are parametrized by the points of the coset space, which is, in that particular case, the projective space CP N −1 and plays the role of the phase space of a corresponding classical mechanics. The CS possess of a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is ivestigated in terms of symbols of operators. The role of the Planck constant playes h = P −1 , where P is the signature of the representation. The classical limit of the so called star commutator generates the Poisson bracket in the CP N −1 phase space. The logarithm of the modulus of the CS overlapping, being in-terpreted as a symmetric in the space, gives the Fubini-Study metric in CP N −1 . The CS constructed are useful for the quasi-classical analysis of the quantum equations of the SU (N ) gauge symmetric theories.

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