Coherent States of the SU(N) groups

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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 CPⁿ⁻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 CPⁿ⁻¹ phase space. The logarithm of the modulus of the CS overlapping, being interpreted as a symmetric in the space, gives the Fubini-Study metric in CPⁿ⁻¹. The CS constructed are useful for the quasi-classical analysis of the quantum equations of the SU(N) gauge symmetric theories.