Gauge theories in three dimensions (= at finite temperature)

Abstract

We have seen that 3-dimensional gauge theories offer much interesting insight into the workings of gauge-invariant quantum field theory. Many further investigations can be envisioned: one wants techniques for studying possible non-analytic terms in gauge invariant amplitudes. Since these can occur only in high orders, our direct resummation method becomes prohibitively lengthy. The approach through dimensional regularization" seems promising. It would be worthwhile to establish in greater detail the relation of this method to condensed matter procedures. 17 The topological mass terms and the associated Chern-Simons structures give evidence that gauge theories in odd-dimensional spaces offer unexpected possibilities. Since odd-dimensional gauge theories are of interest in supersymmetry, in supergravity, in Kaluza-Klein theory and in other attempts at dimensional reduction, there is obvious interest in understanding the gauge theoretic options in this context. Quantization of parameters for consistency of the quantum theory is familiar from the Dirac monopole example. The 3-dimensional massive gauge theory and the 4-dimensional chiral SU(2) theory are novel instances of this phenomenon. One would like to settle the question whether the former is indeed relevant in a high temperature reduction, as we have conjectured, and one wants to ascertain whether there are any physically interesting applications of the latter. [The usual constraint that axial vector anomalies be absent already seems to eliminate 4-dimensional theories which are globally inconsistent.] Yet another interesting thread connects the present work with the general question of massive, yet gauge invariant vector fields. Since the advent of unification schemes based on Yang-Mills models, this question has become phenomenologically important. Most frequently it is addressed in the context of the Higgs mechanism, which however is frequently viewed to be aesthetically and phenomenologically unattractive. Attempts to replace the scalar Higgs fields by fermion, anti-fermion bound states have not produced thus far convincing and acceptable models.³² It is therefore interesting to observe that in the two examples of gauge invariant, massive vector fields without Higgs scalars -- viz. the original Schwinger model of massless, 2-dimensional spinor electrodynamics and the present 3-dimensional, topologically massive gauge fields - the vector meson mass arises from a topological mechanism and not directly from fermion condensates.³³ Is there a topological reason, as yet undiscovered, for massive, gauge invariant vector particles in 4-dimensions?