Feynman Path Integral Formulation

8.5 Wilson Lines and Static Potential 283be many that have this property for the topology of a four-torus), with length T(see Fig. 8.2). One then enumerates all the geodesics that lie at a fixed distance Rfrom the original one, and computes the associated correlation between the Wilsonlines. After averaging the Wilson line correlation over many metric configurations,one extracts the potential from the R dependence of the correlation of Eq. (8.46).In general since two geodesics will not be at a fixed geodesic distance from eachother in the presence of curvature, one needs to introduce some notion of averagedistance, which then gives the spatial separations of the sources R.On the lattice one can construct the analog of the Wilson line for one heavyparticle,L(x,y,z) =exp { }−M∑l i , (8.47)iwhere edges are summed in the “t” direction, and the path is closed by the periodicityof the lattice in the t direction. One can envision the simplicial lattice as dividedup in hypercubes, in which case the points x,y,z can be taken as the remaining labelsfor the Wilson line.Fig. 8.2 Correlations betweenWilson lines closed by thelattice periodicity.TRFor a single line one expects< L(x,y,z) > = < exp { }−M∑l i > ∼ e− ˜MT , (8.48)iwhere T is the linear size of lattice in the t direction and ˜M the renormalized mass.The correlation between Wilson lines at average “distance” R is then given by− 1 [T log < L(x,y,0) L(x,y,R) >]< L(x,y,0) >< L(x,y,R) >∼ V (R) . (8.49)T ≫ RNumerical studies suggest that the correct qualitative features of the potentialemerge close to the critical point. In particular it was found that the potential isattractive close to the critical point, and for two equal mass particles of mass μscales, as expected, like the mass squared. As for any correlation in gravity, the accuratedetermination of the potential as a function of distance R is a more difficult