Planar limit for SU(N) symmetric quantum dynamical systems

Abstract

The spectrum of a quantum mechanical Hamiltonian with SU(N) symmetry is studied in the limit N→∞. A complete description is given for the states which transform according to (i) the scalar representation (singlet states) and (ii) the adjoint representation of SU(N) (’’adjoint’’ states). The eigenvalues of the singlet states are equally spaced with a finite gap ω (g) as N→∞. The spectrum of the adjoint states is equally spaced asymptotically for large excitations with the same gap ω (g). The first excitation of the adjoint states is lower than the first singlet excitation. An accidental degeneracy appears which is removed by 1/N corrections. We compute explicitly this splitting for the singlet states. In this formulation 1/N² plays a role similar to h/ so that all computed quantities are related to the corresponding classical system. All the quantities which enter in the calculation of the spectrum are analytic near g=0 with the same radius of convergence.