Dynamics of Dollard asymptotic variables. Asymptotic fields in Coulomb scattering

Generalizing Dollard's strategy, we investigate the structure of the scattering theory associated to any large time reference dynamics $U_D(t)$ allowing for the existence of M{\o}ller operators. We show that (for each scattering channel) $U_D(t)$ uniquely identifies, for $t \to \pm \infty$, {\em asymptotic dynamics} $U_\pm(t)$; they are unitary {\em groups} acting on the scattering spaces, satisfy the M{\o}ller interpolation formulas and are interpolated by the $S$-matrix. In view of the application to field theory models, we extend the result to the adiabatic procedure. In the Heisenberg picture, asymptotic variables are obtained as LSZ-like limits of Heisenberg variables; their time evolution is induced by $U_\pm(t)$, which replace the usual free asymptotic dynamics. On the asymptotic states, (for each channel) the Hamiltonian can by written in terms of the asymptotic variables as $H = H_\pm (q_{out/in}, p_{out/in})$, $H_\pm (q,p)$ the generator of the asymptotic dynamics. As an application, we obtain the asymptotic fields $\psi_{out/in}$ in repulsive Coulomb scattering by an LSZ modified formula; in this case, $U_\pm(t)= U_0(t)$, so that $\psi_{out/in}$ are \emph{free} canonical fields and $H = H_0(\psi_{out/in})$.Comment: 34 pages, with minor improvements in the text and correction of misprint

Gauge Invariance and Symmetry Breaking by Topology and Energy Gap

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra $\mathcal A$, with a non-trivial center $\mathcal Z$, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of $\mathcal A$. In particular, this gives rise to a gauge symmetry described by the action of $\mathcal Z$. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries which do not commute with the topological invariants represented by elements of $\mathcal Z$ are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.Comment: 23 page

Charge density and electric charge in quantum electrodynamics

The convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (non-local) charged states in Quantum Electrodynamics (QED). Delicate, but physically relevant, mathematical points like the domain dependence of local charges as quadratic forms and the time smearing needed for strong convergence of integrals of charge densities are analyzed. The results are applied to QED and the choice of time smearing is shown to be crucial for the removal of vacuum polarization effects responible for the time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman-Gupta-Bleuler gauge as limits of local states vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification by a gauge term of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman-Gupta-Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with well defined correlation functions with local fields