COORDINATE AND MOMENTUM REPRESENTATIONS OF THE Q-DEFORMED OSCILLATOR AND THEIR INTERPRETATION

A derivation of the coordinate representation of the q-deformed oscillator is given. The Hilbert space associated with it is defined on a finite coordinate space y: (-pi/2s,pi/2s), where s is related to the deformation q via q = exp(-s2), with the respective ''coordinate'' and ''momentum'' operators given by X(s) = y - A(s)(y), P(s) = -id/dy - iA(s)(y), where A(s)(y) = 1/2d lnsigma(s)(y)/dy, sigma(s), being a measure in this space. The commutation rule obeyed by them is X(s)P(s)-P(s)X(s) = iHBAR(s)(y) = i{1-d A(s)/dy}. After constructing a suitable representation in the momentum space, we also set up the Wigner distribution function in this phase space. A brief discussion of these results and their interpretation are given.