Finite two-dimensional oscillator: I. The cartesian model

A finite two-dimensional oscillator is built as the direct product of two f inite one-dimensional oscillators, using the dynamical Lie algebra su(2)(x) circle times su(2)(y). The position space in this model is a square grid o f points. While the ordinary 'Continuous' two-dimensional quantum oscillato r has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)(x) circle times u(1)(y), because it lacks rotations in the positi on (and momentum) plane. We show how to 'import' an SO (2) group of rotatio ns from the continuum model that transforms unitarily the finite wavefuncti ons on the fixed square grid. We thus propose a finite analogue for fractio nal U(2) Fourier transforms.