THE Q-DIFFERENCE OPERATOR, THE QUANTUM HYPERPLANE, HILBERT-SPACES OF ANALYTIC-FUNCTIONS AND Q-OSCILLATORS

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.