Coherent states associated with the groups SU(N) and SU(N, 1)

Alexander V. Gorokhov, V. A. Mikhailov

V. P. Karasev, Leonid A. Shelepin

Manu Mathur, Indrakshi Raychowdhury

We exploit the SU( N ) irreducible Schwinger boson to construct SU( N ) coherent states. This construction of an SU( N ) coherent state is analogous to the construction of the simplest Heisenberg–Weyl coherent states. The coherent states belonging to irreducible representations of SU( N ) are labeled by the eigenvalues of ( N − 1) SU( N ) Casimir operators and characterized by ( N − 1) complex orthonormal vectors describing the SU( N ) group manifold.

Masashi Ban

Manu Mathur, Hamdi Mani

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

D M Gitman, A L Shelepin

This work can be considered as a continuation of our previous study, in which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l, 1) group and construct explicitly corresponding 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 together 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. This is why the cs of the SU(N) and SU(l, 1) groups are of importance in connection with quantization theory. The constructed cs form an overcompleted system in the representation space and, as quantum states possessing 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 commutator of the symbols generates the Poisson bracket in CD l , the latter plays the role of the phase space for the corresponding classical mechanics.