Scattering From a Two Dimensional Array of Flux Tubes: A Study of The Validity of Mean Field Theory

Mean Field Theory has been extensively used in the study of systems of anyons in two spatial dimensions. In this paper we study the physical grounds for the validity of this approximation by considering the Quantum Mechanical scattering of a charged particle from a two dimensional array of magnetic flux tubes. The flux tubes are arranged on a regular lattice which is infinitely long in the ``$y$'' direction but which has a (small) finite number of columns in the ``$x$'' direction. Their physical size is assumed to be infinitesimally small. We develop a method for computing the scattering angle as well as the reflection and transmission coefficients to lowest order in the Aharonov--Bohm interaction. The results of our calculation are compared to the scattering of the same particle from a region of constant magnetic field whose magnitude is equal to the mean field of all the flux tubes. For an incident plane wave, the Mean Field approximation is shown to be valid provided the flux in each tube is much less than a single flux quantum. This is precisely the regime in which Mean Field Theory for anyons is expected to be valid. When the flux per tube becomes of order 1, Mean Field Theory is no longer valid.Comment: 23 pages, University of British Columbia Preprint UBCTP93-01

The Computation of the Magnetic Field of any Axisymmetric Current Distribution—with Magnetospheric Applications

It is shown that the vector potential A of the magnetic field of any axisymmetric electric current distribution can be expressed in the form ∑ A n (r) P 1 n (cos θ). This series is used to compute the field of two model magnetospheric ring currents; the field of one of these was previously determined by double integrations by Akasofu, Cain & Chapman. The calculation of the functions A n (r) does not require double integrations. The two sets of results are in good agreement. The first term in the series for A gives the external magnetic moment of the ring current. The magnetic field energy is calculated for the field as a whole and for each term in the series for A. The field isointensity lines are drawn, and also the field lines for the ring current and for its field combined with that of the geomagnetic dipole. They illustrate the considerable distortion of the field in the magnetosphere during magnetic storms. The series for A may also be helpful in calculating the paths of cosmic rays in the deformed magnetosphere. The numerical convergence of the results is improved by the use of CesÀro summation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72484/1/j.1365-246X.1966.tb03088.x.pd

Spherical Universe topology and the Casimir effect

The mode problem on the factored 3--sphere is applied to field theory calculations for massless fields of spin 0, 1/2 and 1. The degeneracies on the factors, including lens spaces, are neatly derived in a geometric fashion. Vacuum energies are expressed in terms of the polyhedral degrees and equivalent expressions given using the cyclic decomposition of the covering group. Scalar functional determinants are calculated and the spectral asymmetry function treated by the same approach with explicit forms on one-sided lens spaces.Comment: 33 pages, 1 figure. Typos corrected and one reference adde