Dynamics of a BrokenSUNSymmetry for the Oscillator

All states of a one-dimensional harmonic oscillator are represented by a special unitary representation of the noncompact 2+1 Lorentz group. The direct product of $N$ such representations leads to a degeneracy which is represented by the group $S{U}_N$, whereas all states of the $N$-dimensional oscillator are represented by the noncompact unitary group $NU_{N+1\mathrm{}}^{}{}_{}{}^N$ whose Casimir operator determines the energy spectrum. The anharmonic oscillator is represented by a broken symmetry and mass-splitting formulas are obtained. It is shown how new quantum numbers arise from the direct products of basic dynamical groups, corresponding to composite structures.