Explicit N-Particle Harmonic Oscillator States in Products of SU(1, 1) Representations

In this paper, we present a simple algorithm for the explicit construction of n-particle harmonic oscillator states simultaneously belonging to irreducible representations of Sp(2, R) (or SU(1, 1)) and O(n). For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with n = 2 and n = 3 are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the O(2) structure of Bell states.