Quantization of kinematics and dynamics on with difference operators and a relatedq-deformation of the Witt algebra

Abstract

Motivated by the fact that momentum observables which are modelled in a classical theory as difference quotients of functions are directly accessible to measurements, whereas differentials are mathematical idealizations, which are obtained from difference quotients via a limiting process, we propose a quantization method in which momentum operators are given in terms of q-derivatives, i.e. a particular type of difference quotient, which is particularly suitable on . The quantization scheme is obtained from Borel quantization, in which momentum operators are given as differential operators, via a q-deformation of the kinematical algebra. It will be applied to a system localized and moving on the N-point discretization of and leads to a discrete, nonlinear Schrodinger equation. In the limit , i.e. the continuous idealization, we find evolution equations which are special cases of the nonlinear Schrodinger equation derived from Borel quantization on , which is based on the undeformed kinematical algebra. It turns out that with this procedure both the real and imaginary part of the nonlinearity can be derived, which without deformation was only possible for the imaginary part. Hence, one can learn on the situation in the continuous case if it is viewed as a limit of the (q-deformed) case.