Long Range Dynamics and Spontaneous Symmetry Breaking in Many-Body Systems

Abstract

The algebraic approach to quantum mechanics is used to describe the quantum properties of many-body systems, in particular when the dynamics involves long range interactions. In contrast with the short range case, in the presence of long range interactions the time evolution of essentially localized observables involves variables at infinity; this means that infinitely delocalized “classical-like” observables have to be introduced for a complete description of the system. They play a crucial rôle in the phenomenon of spontaneous symmetry breaking (generalized Goldstone’s theorem). In particular, the occurrence of an energy gap (at zero momentum) for the generalized Goldstone’s boson spectrum is related to a non-trivial “classical” motion of the variables at infinity, induced by the effective dynamics (in a factorial representation). As explicit examples we discuss the Heisenberg model in the molecular field approximation, the BCS model for superconductivity, and the electron gas in uniform background as a model of metals.