Field Theories in 2+1 Dimensions
Abstract
Field theories in 2+1 dimensions provide a simple framework for the study of complex 3+1 dimensional phenomena. Here we begin with a study of chiral symmetry breaking. The theory considered is 2+1 dimensional QED with N 4component fermions. We employ an expansion in 1/N which alleviates infrared divergence problems in this theory. We then solve a truncated version of the DysonSchwinger gap equation, both analytically and numerically. We find a critical value of N, such that for N > N_{c } there is no chiral symmetry breaking and for N < N_{c} there is. We also demonstrate numerically the universal nature of this criticality. It is important to show that this critical behavior remains when higher order terms are included. We therefore compute all of the 1/N^2 corrections to the kernel of the gap equation, and show that our conclusions remain unchanged. We also demonstrate a form of gauge invariance for N_{c}. Another question we address is whether or not this theory will spontaneously break parity. We study the gap equation with N 2component fermions. In this case any fermion mass would violate parity. We show that a mass can only be spontaneously induced in such a way as to preserve parity overall. We finally study nonAbelian theories in 2+1 dimensions. We consider an arbitrary YangMills theory with N _{f} 4component fermions. We show that it does not suffer the usual infrared divergence problems if a 1/N_{f} expansion is used. Also, in this expansion the beta function will be zero. In analogy with the Abelian case we demonstrate the existence of an N _{c}. Where, if N_ {f} > N_{c} the fermions remain massless, and if N_{f} < N_{c} a mass is induced. From these results we argue that this theory will not confine for N_{f} > N_{c} , but will confine for N_{f } < N_{c}. The confinement scale will be set by the induced fermion mass.
 Publication:

Ph.D. Thesis
 Pub Date:
 1989
 Bibcode:
 1989PhDT........45N
 Keywords:

 GAUGE THEORY;
 QED;
 Physics: Elementary Particles and High Energy