Structure of the ground state in two-dimensional massless quantum electrodynamics

Two-dimensional massless quantum electrodynamics is studied in the transverse gauge. It is shown by means of an operator solution that the gauge ransformations compatible with the condition of transversality decompose into classes characterized by integral topological numbers. The operators of gauge transformations with nonvanishing topological numbers carry a fermion number and chirality. The requirement of gauge invariance fixes the vacuum, which does not have a definite fermion number or chirality; the physical excitations are bosons. The properties of the operators of the gauge transformations are also studied by functional integration. The existence of a fermion number and chirality for the operators of the remaining gauge transformations is due to the existence of zero modes of the Dirac operator in an external electromagnetic field with nontrivial behavior at infinity.