Condensation of gauge interacting mass-less fermions

A single mass-less fermionic field with an abelian U(1) gauge interaction (electrodynamics of a mass-less Dirac fermion) is studied by a variational method. Even without the insertion of any extra interaction the vacuum is shown to be unstable towards a particle-antiparticle condensate. The single particle excitations do acquire a mass and behave as massive Fermi particles. An explicit low-energy gap equation has been derived and numerically solved. Some consequences of condensation and mass generation are discussed in the framework of the standard model.