Two-Dimensional Massless Quantum Electrodynamics in the Landau-Gauge Formalism and the Higgs Mechanism

Abstract

The Schwinger model is considered in the LandaJl-gauge formalism of quantum electro­ dynamics. This model can be solved exactly on the assumption of no radiative corrections to the anomaly. It is found that the photon obtains a non-zero mass through the Higgs mechanism. In this case, the would-be Nambu-Goldstone boson is an associated boson which is constructed from a pair of two-component massless fermions. This would-be Nambu-Goldstone boson appears as a result of the spontaneous breaking of the gauge invariance of the first kind, and it becomes unphysical through the Higgs mechanism. However, as all the fermions themselves decouple from photons, they cannot appear as real particles in our world. § 1 .. Introduction Recently several ideas are proposed to explain why the constituent part.icles do not appear as real particles. The most fantastic idea is infrared shielding, i.e.; if there were an extremely long range force, it would bind the constituents and they could appear not as free particles but only as' bound states. In this case, we can hope that the Yang-Mills fields play a role of a long-:range force and also that the symmetry breaking occurs because such a dynami<;al system is very unstable owing to the existence of long range correlations. Then we expect that the· gauge fields get· a mass through the Higgs mechanism and the constituent particles cannot app.ear since the Nambu-Goldstone boson becomes unphysical through the Higgs mechanism. In order to analyze the conjecture, we consider the two-dimensional massless QED (the Schwinger model). We can solve this model exactly and show explic­ itly that the gauge invariance of the first kind is broken spontaneously, and the Nambu-Goldstone boson appears as a bound state of the massless free fermions. This paper is organized as follows: In § 2, we· consider the Schwinger model in the Landau gauge. In § 3, we solve the Dirac equation for the electron iri the gauge field. In § 4, we reconstruct the electromagnetic current from the fermion wave functions obtained in § 3. .This technique is due to Lowenstein. In § 5, we construct the associated boson from the massless free fermions. This is just the Nambu-Goldstone boson and appears as a bound state of the fermions. In § 6, we discuss the construction of the Hilbert space and the representation