COHERENT STATES OF SU (L, 1) GROUPS

This work can be considered as a continuation of our previous study, i n which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we ex tend that approach to any SU(l, 1) group and construct explicitly corr esponding cs. The cs are parametrized by dots of a coset space, which is, in that particular case, the open complex ball CD(l). This space t ogether with the projective space CP(l), which parametrizes the cs of the SU(l + 1) group, exhaust all complex spaces of constant curvature. Thus, both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. T his is why the cs of the SU(N) and SU(l, 1) groups are of importance i n connection with quantization theory. The constructed cs form an over completed system in the representation space and, as quantum states po ssessing a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is investigated in terms of symbols of operators; the limit of the so called star com mutator of the symbols generates the Poisson bracket in CD(l), the lat ter plays the role of the phase space for the corresponding classical mechanics.