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

Quantum Mechanics and Stochastic Mechanics for compatible observables at different times

Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.Comment: Plain Te

Classical and Quantum Mechanics from the universal Poisson-Rinehart algebra of a manifold

The Lie and module (Rinehart) algebraic structure of vector fields of compact support over C infinity functions on a (connected) manifold M define a unique universal non-commutative Poisson * algebra. For a compact manifold, a (antihermitian) variable Z, central with respect to both the product and the Lie product, relates commutators and Poisson brackets; in the non-compact case, sequences of locally central variables allow for the addition of an element with the same role. Quotients with respect to the (positive) values taken by Z* Z define classical Poisson algebras and quantum observable algebras, with the Planck constant given by -iZ. Under standard regularity conditions, the corresponding states and Hilbert space representations uniquely give rise to classical and quantum mechanics on M.Comment: Talk given by the first author at the 40th Symposium on Mathematical Physics, Torun, June 25-28, 200

The QED(0+1) model and a possible dynamical solution of the strong CP problem

The QED(0+1) model describing a quantum mechanical particle on a circle with minimal electromagnetic interaction and with a potential -M cos(phi - theta_M), which mimics the massive Schwinger model, is discussed as a prototype of mechanisms and infrared structures of gauge quantum field theories in positive gauges. The functional integral representation displays a complex measure, with a crucial role of the boundary conditions, and the decomposition into theta sectors takes place already in finite volume. In the infinite volume limit, the standard results are reproduced for M=0 (massless fermions), but one meets substantial differences for M not = 0: for generic boundary conditions, independently of the lagrangean angle of the topological term, the infinite volume limit selects the sector with theta = theta_M, and provides a natural "dynamical" solution of the strong CP problem. In comparison with previous approaches, the strategy discussed here allows to exploit the consequences of the theta-dependence of the free energy density, with a unique minimum at theta = theta_M.Comment: 21 pages, Plain Te

Boundary terms and their Hamiltonian dynamics

It is described how the standard Poisson bracket formulas should be modified in order to incorporate integrals of divergences into the Hamiltonian formalism and why this is necessary. Examples from Einstein gravity and Yang-Mills gauge field theory are given.Comment: Talk at 29th Ahrenshoop Symposium in Buckow 1995, 6 pages, espcrc2.sty, twoside.sty, fleqn.sty, amssymb.sty, no figure