VORTEX OPERATORS IN GAUGE FIELD THEORIES

We study several related aspects of the t Hooft vortex operator. The first chapter reviews the current picture of the vacuum of quantum chromodynamics, the idea of dual field theories, and the idea of the vortex operator.;In the third chapter we discuss the dependence of the Green's functions of the Wilson and t Hooft operators on the nature of the vacuum. We emphasize the cluster properties of the Green's functions rather than the vacuum expectation values. We explain t Hooft's result relating the expectation values of the Wilson and t Hooft operators. We then show that the vortex operator in a massive Abelian theory always has surface-like clustering, and we see how this appears in a graphical expansion. We emphasize that the form of Green's functions in terms of Feynman graphs is the same in Higgs and symmetric phases, and that the difference appears in the sum over all tadpole trees.;In the fourth chapter we consider systems which have fields in the fundamental representation, so that there are no vortex operators. When these fields enter only weakly into the dynamics, as is the case in QCD and in real superconductors, we would expect to be able to define a vortex-like operator. We show that any such operator can no longer be "local looplike", but must have commutators at long range. We can still find an operator with useful properties, its cluster property, though more complicated than that of the usual vortex operator, still appears to distinguish Higgs, confining and perturbative phases. To test this, we consider a U(1) lattice gauge theory with two matter fields, one singly charged (fundamental) and one doubly charged (adjoint). When the fundamental field is weakly coupled, we find the expected phase transitions. When it is strongly coupled, our operator still appears to be a good order parameter, a discontinuous change in its behavior leads us to find a new phase transition. We give some discussion of how operators can be good or bad order parameters.;The second chapter deals with the Abelian vortex operator written in terms of elementary fields and with the calculation of its Green's functions. The Dirac veto problem appears in a new guise. We present a two dimensional "solvable model" of a Dirac string. This leads us to a new solution of the veto problem; we discuss its extension to four dimensions. We then show how the Green's functions can be expressed more neatly in terms of Wu and Yang's geometrical idea of "sections". The renormalization of the Green's functions of two kinds of Abelian looplike operators, the Wilson loop and the vortex operator, is studied. In each case the possible divergences are easily determined with the aid of the operator product expansion, and for both operators only an overall multiplicative renormalization is needed. In the case of the vortex this involves a surprising cancellation.