Dortmund 2006 – wissenschaftliches Programm

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 7: Symmetrien, integrable Systeme und Nichtkomm. Geometrie

MP 7.1: Vortrag

Mittwoch, 29. März 2006, 17:00–17:30, P1-02-111

Matrix-differential-operator symmetries of the Dirac equation with external electromagnetic fields — •Stephan Balliel-Zakowicz — Departement of Mathematics, ETH Zurich, 8092 Zurich, Switzerland — Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, Heinrich-Buff-Ring 16, 35392 Giessen

Symmetries of the Dirac equation with external electromagnetic fields are considered. The focus is set on symmetry generators in the form of first-order matrix-differential operators (MDOs), of which the generators of Lie point symmetries are an important but not exhaustive subclass. These MDOs generate one-parameter groups of transformations leaving the space of solutions of the Dirac equation invariant and may thus be employed to find new solutions from known ones. Given an arbitrary external electromagnetic field, the general condition for the existence of a first-order symmetry MDO and–if one exists–its structure are expressed in a covariant form.

The results are then applied to the case of an external plane electromagnetic wave, for which all first-order symmetry MDOs are presented explicitly. The action of some of these operators on a wavefunction can be calculated analytically. This transformation involves derivatives as well as an integration over the wavefunction, which is in stark contrast to the much simpler action generated by Lie point symmetries. As has been long known, the external plane-wave field admits the analytical solutions found by and named after Volkov [1]. These solutions show a rather trivial transformation behavior under the above symmetry generators.

[1] D. M. Volkov, Z. Phys. 94, 250 (1935).