Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non--L2 ground state. As a prototype the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite volume measures.Comment: 15 pages, Plain Te

Localization and symmetries

The violation of the Noether relation between symmetries and charges is reduced to the time dependence of the charge associated to a conserved current. For the U(1) gauge symmetry a non-perturbative control of the charge commutators is obtained by an analysis of the Coulomb charged fields. From this, in the unbroken case we obtain a correct expression for the electric charge on the Coulomb states, its superselection and the presence of massless vector bosons; in the broken case, we obtain a general non-perturbative version of the Higgs phenomenon, i.e. the absence of massless Goldstone bosons and of massless vector bosons. The conservation of the (gauge dependent) current associated to the U(1) axial symmetry in QCD is shown to be compatible with the time dependence of the corresponding charge commutators and a non-vanishing eta' mass, as a consequence of the non locality of the (conserved) current.Comment: Invited contribution to ``The Quantum Universe'', dedicated to G. Ghirardi for his 70th birthda