Time evolution automorphisms in generalized mean-field theories

Abstract

A general class of time evolutionsΤ ^Q of infinite quantum systems is rigorously defined. It generalizes thermodynamic limits of polynomial mean-field evolution of quantum spin lattices, the simplest case of which is the strong coupling version of the quasi spin B.C.S.-model of superconductivity. A distinguished feature of the considered type of time evolution is theΤ ^Q-non-invariance of the usually consideredC ^*-algebraA of quasilocal observables of the infinite system. A largerC ^*-algebraC containingA as a subalgebra is introduced in such a way thatΤ ^Q has a natural extension to a one parameter group^*-automorphisms ofC. The algebraC contains a commutative subalgebra of classical observables (consisting of the intensive observables of the large quantal system determined by a Lie groupG actionσ(G) ε^*-autA) denoted byN which isΤ ^Q invariant and the restriction ofΤ ^Q toN reproduces the classical Hamiltonian flow ϕ^Q corresponding to the chosen classical Hamiltonian functionQ on the classical phase space of the intensive observables. The evolution Τ^Q is determined uniquely by the classical Hamiltonian functionQ as well as by the actionσ(G). Continuity properties ofΤ ^Q are considered and reviewed.