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The SO-q(N,R) symmetric harmonic oscillator on the quantum Euclidean space R-g(N) and its Hilbert space structure

May, 1992
58 pages
Published in:
  • Int.J.Mod.Phys.A 8 (1993) 4679-4729
e-Print:
DOI:
Report number:
  • SISSA-102-92-EP
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Abstract:
We show that the isotropic harmonic oscillator in the ordinary euclidean space R N{\bf R}~N (N≥3N\ge 3) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), SOq(N,R)SO_q(N,{\bf R}). The q-deformation is the consequence of replacing R N{\bf R}~N by R Nq{\bf R}~N_q (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over R Nq{\bf R}~N_q, which we use for the definition of the scalar product of states.
  • model: oscillator
  • quantum group: SO(N)
  • quantum space: Euclidean
  • quantum mechanics
  • linear space: Hilbert space