Schrödinger dynamics and physical folia of infinite mean-field quantum systems

Abstract

This work deals with the general (i.e. non-equilibrium) Schrödinger dynamics of infinite mean-field quantum systems. It is shown how this dynamics is related to the Hamiltonian flowϕ ^Q _t on the “classical phase-space”$E \subseteqq \mathbb{R}^L$ recently defined by Bona [10] to describe the time evolution of classical (macroscopic) observables of the system. These connections allow us to clarify the structure of the set of all physical folia, a notion introduced by Sewell for the dynamical description of infinite systems in cases where this description is representation-dependent. They also yield a result showing that the Heisenberg picture is the more general approach to such descriptions in the sense that there are more representations in which a Heisenberg dynamics can be defined than ones which allow for the definition of a Schrödinger dynamics. Finally, our theory makes it possible to construct many explicit examples of physical folia; in this connection it is shown that there can be overcountably many inequivalent representations with the same macroscopic dynamical structure.