Noncommuting contractions of oscillators with constant force

Abstract

We find two noncommuting contractions of the Lie algebras u(N) and gl(N,� ), realized as the symmetry algebras of N-dimensional isotropic harmonic and repulsive oscillators of spring constant k ∈� , with a constant force of magnitude f . The contraction limit to the symmetry algebra of the Ndimensional free system is (k, f ) → (0, 0). We take two paths in this plane, determined by the order of contraction of the two parameters, and show that they yield two closely related—but distinct—Euclidean-type symmetry algebras for the common contracted system. We also show briefly how the wavefunctions of the one-dimensional harmonic oscillator reduce to plane waves along the above two paths.