FUNCTIONAL-EQUATIONS FOR PATH-DEPENDENT PHASE FACTORS IN YANG-MILLS THEORIES

We introduce a gauge-covariant functional derivative for path-dependent quantities, and use it to analyze the behavior of the phase matrix Φy,x(Γ)=PΓexp[-∫xyA(x)·dx], under variations of the contour Γ. We give a general formula for Φy+δy,x+δx(Γ′) for a varied path in terms of Φy,x(Γ), and analyze a gauge-covariant version of the functional wave equation (or string equation") for Φ derived by Nambu and by Gervais and Neveu. We conclude that there is still a serious gap in the attempt to derive the string theory from a Yang-Mills theory. © 1979."