The fermion propagator and the 4-fermion Green function in the massless QED 2 are explicitly found with topological effects taken into account. The corrections due to instanton sectors k = ±1, contributing to the propagator, are shown to be just the homogenous terms admitted by the Dyson-Schwinger equation for S. In the case of the 4-fermion function also sectors k = ±2 are included into consideration. The quark condensates are then calculated and are shown to satisfy cluster property. The θ-dependence exhibited by the Green functions corresponds to and may be removed by performing certain chiral gauge transformation.

W e ex tend our pre viou s analysi s of the motion of vortex lines in wave me chanics to the case of more ela borate vortex patterns and to a rotatin g harmonic trap .

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Electromagnetic waves with phase defects in the form of vortex lines combined with a constant magnetic field are shown to pin down cyclotron orbits ͑Landau orbits in the quantum mechanical setting͒ of charged particles at the location of the vortex. This effect manifests itself in classical theory as a trapping of trajectories and in quantum theory as a Gaussian shape of the localized wave functions. Analytic solutions of the Lorentz equation in the classical case and of the Schrödinger or Dirac equations in the quantum case are exhibited that give precise criteria for the localization of the orbits. There is a range of parameters where the localization is destroyed by the parametric resonance. Pinning of orbits allows for their controlled positioning: They can be transported by the motion of the vortex lines.

The propagation of a light ray in thin layer (film) within geometrical optics is considered. It is assumed that the ray is captured inside the layer due to reflecting walls or total internal reflection (in the case of a dielectric layer). It has been found that for a very thin film (the length scale is imposed by the curvature of the surface at a given point) the equations describing the trajectory of the light beam are reduced to the equation of a geodesic on the limiting curved surface. There have also been found corrections to the trajectory equation resulting from the finite thickness of the film. Numerical calculations performed for a couple of exemplary curved layers (cone, sphere, torus and catenoid) confirm that for thin layers the light ray which is repeatedly reflected, propagates along the curve close to the geodesic but as the layer thickness increases, these trajectories move away from each other. Because the trajectory equations are complicated non-linear differential equations, their solutions show some chaotic features. Small changes in the initial conditions result in remarkably different trajectories. These chaotic properties become less significant the thinner the layer under consideration.