p-ADIC and adelic harmonic oscillator with a time-dependent frequency

Abstract

The classical and quantum formalism for a p-adic and adelic harmonic oscillator with a time-dependent frequency is developed, and general formulas are obtained for the main theoretical quantities. In particular, the p-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. A spatial discreteness and a p-adic quantum mechanical phase are noted.