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 Φ_{y, x}( Γ)= P_Γexp[-∫ _x^yA( x)·d x], under variations of the contour Γ. We give a general formula for Φ_{y+ δy, x+ δx( Γ‧)} for a varied path in terms of Φ_{y, 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.