Theq-difference operator, the quantum hyperplane, Hilbert spaces of analytic functions andq-oscillators

Abstract

It is shown that the differential calculus of Wess and zumino for the quantum hyperplane is intimately related to theq-difference operator acting on then-dimensional complex space ℂ n . An explicit transformation relates the variables and theq-difference operators on ℂ n to the variables and the quantum derivatives on the quantum hyperplane. For real values of the quantum parameterq, 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 then dimensional harmonic oscillator invariant under the unitary quantum groupU 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.