A Dynamical System with Q-deformed Phase Space Represented in Ordinary Variable Spaces

Dynamical systems associated with a q-deformed two dimensional phase space are studied as effective dynamical systems described by ordinary variables. In quantum theory, the momentum operator in such a deformed phase space becomes a difference operator instead of the differential operator. Then, using the path integral representation for such a dynamical system, we derive an effective short-time action, which contains interaction terms even for a free particle with q-deformed phase space. Analysis is also made on the eigenvalue problem for a particle with q-deformed phase space confined in a compact space. Under some boundary conditions of the compact space, there arises fairly different structures from $q=1$ case in the energy spectrum of the particle and in the corresponding eigenspace .