Quantum Mechanics of Infinite Degrees of Freedom

Quantum systems with infinite degrees of freedom exhibit properties, like relaxation to equilibrium, phase-transitions and the existence of inequivalent representations of the CAR and CCR, that can satisfactorily be dealt with by means of the methods and techniques of algebraic quantum statistical mechanics [80, 81, 130]. The point of departure from standard quantum mechanics, is that in an infinite dimensional context one is usually provided with the algebraic properties of the relevant observables, but, in general, not with an a priori given representation on a Hilbert space; the latter rather depends on to the physical properties of the systems under consideration [324, 342, 343].