Along the lines of a previous work, the geometrical structure of Hibert bundles describing extended quantum free particles is repeated with Galilei external and internal independent symmetries. Then, in order to introduce the interaction, this structure is extended by replacing configuration and momentum spaces by the socelled spaces of trajectories and extended velocity boosts, respectively. These provide representations giving the probability amplitudes for the particle to follow certain trajectories. The interaction can be introduced in the transformation law from functions on (...) the space of trajectories (free dynamics) to functions on spacetime (intracting dynamics). This transformation law, which makes use of a universal distribution, is seen as a functional in our work according to a quantum functional theory which generalizes the ideas of de Broglie. Intertwining of induced representations gives the free propagator in the space of trajectories and, henceforth, the propagator with interaction in space-time for the extended particle. (shrink)