A Microscopic Semiclassical Connning Eld Equation for U(1) Lattice Gauge Theory in 2+1 Dimensions I Introduction

We present a semiclassical nonlinear eld equation for the connning eld in 2+1{dimensional U (1) lattice gauge theory (compact QED). The equation is derived directly from the underlying microscopic quantum Hamiltonian by means of truncation. Its nonlinearities express the dynamic creation of magnetic monopole currents leading to the connnement of the electric eld between two static electric charges. We solve the equation numerically and show that it can be interpreted as a London relation in a dual superconductor. In the last few years, several studies of connnement in lattice gauge theories have suc-ceded in observing not only the connnement potential 1, 2], but also the formation of ux tubes between two static charges 3, 4]. The most promising mechanism for this eeect is the dual superconductor hypothesis 5, 6, 7] which assumes that the electric eld is connned by magnetic monopole currents forming the walls of the ux tube, just like superconducting electric currents connne the magnetic eld in an ordinary superconductor. The monopoles are presumed to originate dynamically from the pe-riodicity of the Hamiltonian as tunneling eeects between neighboring minima 8, 9]. Abelian monopoles occur naturally in U(1) lattice gauge theory (compact QED). As U(1) is a subgroup of SU(2) and SU(3), so-called \Abelian" monopoles (monopoles in the maximally Abelian projection) 10] may also play a role in the QCD mechanism