Generalized lines of force as the gauge invariant degrees of freedom for general relativity and Yang-Mills theory

We present a set of equations that determine (with certain qualifications) what is (i) a complete canonical set of gauge-invariant variables for Yang-Mills (YM) theory that generalize Faraday’s electric lines of force and (ii) a complete canonical set of the diffeomorphism and gauge invariant observables for general relativity (GR). Both results come from solving the Hamilton-Jacobi version of the constraint equations: the Gauss law constraints of YM theory and the Gauss law and diffeomorphism constraint of GR in the Ashtekar formalism. These results are local in physical space as well as in phase space.