QUANTUM MECHANICS WITH p-ADIC NUMBERS

We discuss unitary realizations of the Heisenberg group and the linear canonical transformations over a complex Hilbert space but with dynamical variables on a p-adic field Q p . For all p, an appropriate choice of phase turns the realization of the linear canonical transformation into a representation up to a sign of SL (2, Q p ). We give the spectra of the subgroups corresponding to the free particle and the harmonic oscillator. We discuss briefly the possibility of an adelic interpretation.