QUANTUM ELECTRODYNAMICS OF PARTICLES ON A PLANE AND THE CHERN-SIMONS THEORY

We study the electrodynamics of generic charged particles (bosons, fer mions, relativistic or not) constrained to move on an infinite plane. An effective gauge theory in (2 + 1)-dimensional space-time which desc ribes the real electromagnetic interaction of these particles is obtai ned. The relationship between this effective theory with the Chem-Simo ns theory is explored. It is shown that the QED lagrangian per se prod uces the Chern-Simons constraint relating the current to the effective gauge field in 2 + 1 dimensions. It is also shown that the geometry o f the system unavoidably induces a contribution from the topological t heta-term that generates an explicit Chem-Simons term for the effectiv e (2 + 1)-dimensional gauge field as well as a minimal coupling of the matter to it. The possible relation of the effective three-dimensiona l theory with the bosonization of the Dirac fermion field in 2 + 1 dim ensions is briefly discussed as well as the potential applications in condensed matter systems.