Yang-Mills equations and parallel propagation on closed paths

A new variable for Yang-Mills theory is introduced. This variable, denoted by H, is the differential holonomy operator, i.e., the variation of the holonomy operator associated with a variation of a specific set of closed paths in Minkowski space. The main purpose of this paper is to show how the vacuum Yang-Mills equations can be restated as relatively simple equations for H. We will present two separate approaches to this problem. The first is a global approach involving Stokes’s theorem and global regularity whereas the second uses purely local arguments. The self-dual non-Abelian case and Maxwell case are considered as particular examples.