A non-perturbative study of fermion propagators and their interactions in gauge theories
In this thesis we study the non-perturbative behaviour of the fermion propagator in an Abelian gauge theory, namely four dimensional, quenched QED –where by quenched we mean that we neglect the effect of the fermion loops in the boson propagator. What is of primary interest is the dynamical generation of mass. In order to carry out this study we need to make use of the Schwinger-Dyson equations, which are the field equations of the theory. For the investigation of the fermion propagator, the form of the three point interaction is of critical importance. We study the usual ansatz, the Ball-Chiu form, for the three point function, that is obtained from the Ward-Takahashi identities, and improve upon it. This is done by making use of the powerful constraints that Multiplicative Renormalizability place upon the theory in the perturbative (high energy) region. We initially study the theory in the massless case, for simplicity, where we find that using our improved ansatz we can obtain an exact, non-perturbative solution for the renormalised wave function. Moving on, we then study the theory in the massive case -where we have a brief interlude to look at the ladder approximation. We solve the theory in the case where there is a finite cutoff and reproduce the well-known critical coupling point. We then consider the case where there is an infinite cutoff, when we find no critical coupling. We discuss and explain the differences. Returning to our improved ansatz for the fermion-boson vertex we solve the renormalised theory for both the wavefunction and mass function and find that there is no critical coupling. In doing this having a form for the fermion-boson vertex that satisfies both the Ward-Takahashi identity and Multiplicative Renormalizability is essential. These studies suggest that full QED may turn out to be a theory without a critical coupling and thus be free of phase changes.