The q-difference operator, the quantum hyperplane, Hilbert spaces of analytic functions and q-oscillators
Abstract
It is shown that the differential calculus of Wess and zumino for the quantum hyperplane is intimately related to the q-difference operator acting on the n-dimensional complex space ℂ n . An explicit transformation relates the variables and the q-difference operators on ℂ n to the variables and the quantum derivatives on the quantum hyperplane. For real values of the quantum parameter q, the consideration of the variables and the derivatives as hermitean conjugates yields a quantum deformation of the Bargmann-Segal Hilbert space of analytic functions on ℂ n . Physically such a system can be interpreted as the quantum deformation of the n dimensional harmonic oscillator invariant under the unitary quantum group U q ( n) with energy eigenvalues proportional to the basic integers. Finally, a construction of the variables and quantum derivatives on the quantum hyperplane in terms of variables and ordinary derivatives on ℂ n is presented.
- Publication:
-
Zeitschrift fur Physik C Particles and Fields
- Pub Date:
- December 1991
- DOI:
- 10.1007/BF01565589
- Bibcode:
- 1991ZPhyC..51..627A