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Through the use of a lattice U(1) Ward-Takahashi identity, one can find aprecise definition of flux and electric four-current that does not rely on thecontinuum limit. The magnetic four-current defined for example by theDeGrand-Toussaint construction introduces order a^2 errors in the fielddistributions. We advocate using a single definition of flux in order to beconsistent with both the electric and magnetic Maxwell's equations at anylattice spacing. In a U(1) theory the monopoles are slightly smeared by thischoice, i.e. are no longer associated with a single lattice cube. In Abelianprojected SU(2) the consistent definition suggests further modifications. Forsimulations in the scaling window, we do not foresee large changes in thestandard analysis of the dual Abrikosov vortex in the maximal Abelian gaugebecause the order a^2 corrections have small fluctuations and tend to cancelout. However in other gauges, the consequences of our definitions could lead tolarge effects which may help in understanding the choice of gauge. We alsoexamine the effect of truncating all monopoles except for the dominant clusteron the profile of the dual Abrikosov vortex.Comment: 12 pages, 4 eps figures, Confinement 2003 contributed tal

CONSISTENT DEFINITIONS OF FLUX AND ELECTRIC AND MAGNETIC CURRENT IN ABELIAN PROJECTED SU(2) LATTICE GAUGE THEORY