[en] The problem of direct interaction in relativistic particle mechanics has been extensively studied and a variety of models has been proposed avoiding the conclusions of the so-called no-interaction theorems. In this thesis the authors studied scattering in the relativistic two-body problem. He uses the results to analyze gauge invariance in Hamiltonian constraint models and the uniqueness of the symplectic structure in manifestly covariant relativistic particle mechanics. A general geometric framework that underlies approaches to relativistic particle mechanics is presented and the kinematic properties of the scattering transformation, i.e., those properties that arise solely from the invariance of the theory under the Poincare group are studied. The second part of the analysis of the relativistic two-body scattering problem is devoted to the dynamical properties of the scattering process. Using general geometric arguments, gauge invariance of the scattering transformation in the Todorov-Komar Hamiltonian constraint model is proved. Finally, quantization of the models is discussed

[en] The non-relativistic quantum mechanics of a test particle in an external gravitational field is formulated in accordance with the equivalence principle. More precisely, the basic tenets of quantum mechanics, as captured in the path integral formalism, are combined with the equivalence principle to define a new quantum propagator for the non-relativistic motion of a test particle. The definition is given in terms of the Newtonian connection describing the field, extended to a Bargmann connection on a Bargmann bundle over spacetime. It is proven that, in an appropriate gauge (i.e. an appropriate section of the Bargmann bundle), the new propagator equals the familiar Feynman propagator and therefore gives the correct quantum dynamics. It is shown that external electromagnetic fields can be incorporated into the formalism. (Author)

[en] Hamiltonian particle mechanics is formulated on a space-time (M,g). The restrictions imposed by the isometry group G of g on the possible dynamics are analyzed; a no-interaction theorem is obtained on a large class of homogeneous space-times. The results are in particular applied to the de Sitter space-time: here it is proven that a G-invariant dynamics can describe geodesic particle motion only

[en] In the presence of competing relativistic formalisms for interacting particle dynamics, a model-independent axiomatic approach is proposed for the study of the following asymptotic aspects of relativistic classical particle dynamics: the definition of the scattering operator, scattering angle and timedelay, and the specification of a general functional interdependence between the objects so defined. (orig.)

[en] We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of ''chaoticity.'' The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple. (orig.)

[en] We show for a large class of random Schroedinger operators H_ω on l²(Z^ν) and on L²(R^ν) that dynamical localization holds, i.e. that, with probability one, for a suitable energy interval I. Here ψ is a function of sufficiently rapid decrease, ψ_t=e^-iH_ωtψ and P_I(H_ω) is the spectral projector of H_ω corresponding to the interval I. The result is obtained through the control of the decay of the eigenfunctions of H_ω and covers, in the discrete case, the Anderson tight-binding model with Bernoulli potential (dimension ν=1) or singular potential (ν>1), and in the continuous case Anderson as well as random Landau Hamiltonians. (orig.)

[en] We study numerically and theoretically the d-dimensional Hamiltonian motion of fast particles through a periodic field of scatterers, modelled by bounded, localized (time-dependent) potentials that we refer to as (in)elastic non-dissipative Lorentz gases. We illustrate the wide applicability of a random walk picture previously developed for a field of scatterers with random spatial and/or time dependence by applying it to four other models. First, for a periodic array of spherical scatterers in d ≥ 2, with a smooth (quasi-)periodic time dependence, we show Fermi acceleration: the ensemble averaged kinetic energy (||p(t)||²) grows as t^2/5. Nevertheless, the mean squared displacement (||q(t)||²) ∼ t² behaves ballistically. These are the same growth exponents as for random time-dependent scatterers. Second, we show that in the soft elastic and periodic Lorentz gas, where the particles' energy is conserved, the motion is diffusive, as in the standard hard Lorentz gas, but with a diffusion constant that grows as ||p₀||⁵, rather than only as ||p₀||. Third, we note the above models can also be viewed as pulsed rotors: the latter are therefore unstable in dimension d ≥ 2. Fourth, we consider kicked rotors, and prove them, for sufficiently strong kicks, to be unstable in all dimensions with (||p(t)||²) ∼ t and (||q(t)||²) ∼ t³. Finally, we analyse the singular case d = 1, where (||p(t)||²) remains bounded in time for time-dependent non-random potentials, whereas it grows at the same rate as above in the random case.