The Maxwell, Yang-Mills and Einstein equations and closed path parallel propagation

Abstract

Two linear partial differential equations (and several non-linear generalizations) for a function H, defined on the unit sphere, will be described and their relationship to gauge fields discussed. The two equations are ∂H + ∂ A = 0 and ∂H + [H, A] + ∂ A = 0 , where the differential operators ∂ and ∂ are given by either ∂ A= −sinθ ∂ ∂θ + i sinθ ∂ ∂φ ( A/sinθ) , ∂ H= −sinθ ∂ ∂θ + i sinθ ∂ ∂φ ( H/sinθ) , or (using complex stereographic coordinates (ζ, ζ ) given by ζ = e iφ cot θ/2) ∂ A = 2P 2 ∂ ∂ζ ( A/P) , ∂ H = 2P 2 ∂ ∂ζ ( H/P) with P = 1 2 (1 + ζ ζ) . The function A is to be considered as (essentially) an arbitrary function of three variables u, θ, φ (or u, ζ, ζ ) where u is a particular given function of (θ, φ) and four parameters x a , ( a = 0, 1, 2, 3), i.e. A has the form A ( u(θ, φ, x a ), θ, φ). In eq. (1), H and A are scalar valued while in eq. (2) they are matrix (or Lie algebra) valued. The claim now is that eq. (1) is equivalent to the vacuum Maxwell equations and eq. (2) is equivalent to the anti-self-dual Yang-Mills equations. The x a , which only appear as parameters in the equation, represent the Minkowski space-time points. In both equations the function H( x a , θ, φ) has a simple geometric interpretation as an infinitesimal holonomy operator (or parallel propagator) associated with a particular family of paths, while the function A ( u, φ, θ) is the free characteristic data for the respective fields. In both cases the fields F ab and vector potential γ a are easily determined from knowledge of H. Eqs. (1) and (2) have been generalized to the following cases; 1. a) the full vacuum Yang-Mills equations on Minkowski space; 2. b) the Maxwell and Yang-Mills equations on a given asymptotically flat space-time; 3. c) the asymptotically flat vacuum solutions of the Einstein equations, with the self or anti-self-dual fields as a simple special case. Only in the maxwell and self-dual Yang-Mills case are the equations linear. In each of the succeeding cases the interactions between H and its complex conjugate H gets more complicated. Moreover the equations become integro-differential equations which is a manifestation of the non-Huygens nature of the propagation.