QUANTUM-MECHANICS WITH Q-DEFORMED COMMUTATORS AND PERIODIC VARIABLES

A q-deformed commutator for arbitrary q is derived from a variable with a periodic boundary condition such as an azimuthal angle psi (0 less-than-or-equal-to psi < 2pi). A Hamiltonian can be written down in an Hermitian form for q = e(alpha) or q = e(ialpha) with alpha is-an-element-of R, and its eigenfunctions and eigenvalues are obtained. Algebraic structures, W1+infinity and U(q)(sl2), of this model and introductions of gauge interactions are discussed. Extensions to man variables and some elementary examples are presented.