Analytic approach to confinement and monopoles in lattice SU(2)

Abstract

We extend the approach of Banks, Myerson, and Kogut for the calculation of the Wilson loop in lattice U(1) to the non-abelian SU(2) group. The original degrees of freedom of the theory are integrated out, new degrees of freedom are introduced in several steps. The centre group $Z_2$ enters automatically through the appearance of a field strength tensor $f_{\mu \nu}$, which takes on the values 0 or 1 only. It obeys a linear field equation with the loop current as source. This equation implies that $f_{\mu \nu}$ is non vanishing on a two-dimensional surface bounded by the loop, and possibly on closed surfaces. The two-dimensional surfaces have a natural interpretation as strings moving in euclidean time. In four dimensions we recover the dual Abrikosov string of a type II superconductor, i.e. an electric string encircled by a magnetic current. In contrast to other types of monopoles found in the literature, the monopoles and the associated magnetic currents are present in every configuration. With some plausible, though not generally conclusive, arguments we are directly led to the area law for large loops.