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 ℂⁿ. An explicit transformation relates the variables and theq-difference operators on ℂⁿ 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 ℂⁿ. 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 ℂⁿ is presented.