2 Fock Representation

Abstract

The general lesson from the GNS theorem is that a state Ω on the algebra of observables, namely a set of expectations, defines a realization of the system in terms of a Hilbert space $\mathcal{H}_{\Omega}$ of states with a reference vector Ψ _Ω which represents Ω as a cyclic vector (so that all the other vectors of $\mathcal{H}_{\Omega}$ can be obtained by applying the observables to Ψ _Ω ). In this sense, a state identifies the family of states related to it by observables, equivalently accessible from it by means of physically realizable operations. Thus, one may say that $\mathcal{H}_{\Omega}$ describes a closed world, or phase, to which Ω belongs. An interesting physical and mathematical question is how many closed worlds or phases are associated to a quantum system. In the mathematical language this amounts to investigating how many inequivalent (physically acceptable) representations of the observable algebra which defines the system exist.