Quantum{like Behaviour and Semiclassical Collective Dynamics for Charged Beams Inferred from Classical Stochastic Uctuations

In this paper we simulate semiclassical corrections to classical dynamics by suitable classical stochastic uctuations, then we apply it to the speciic instance of charged beams in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diiusion process, the Nelson process, that is time{reversal invariant. Its diiusion coeecient p N c represents a semiclassical unit of emit-tance (here N is the number of particles in the beam, and c is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum{like models of beam dynamics proposed in recent years. We also brieey touch upon applications of the hydrodynamic formalisms.