Equivalence of helicity and Euclidean self-duality for gauge fields

Abstract

In the canonical formalism for the free electromagnetic field a solution to Maxwell's equations is customarily identified with its initial gauge potential (in Coulomb gauge) and initial electric field, which together determine a point in phase space. The solutions to Maxwell's equations, all of whose plane waves in their plane wave expansions have positive helicity, thereby determine a subspace of phase space. We will show that this subspace consists of initial gauge potentials which lie in the positive spectral subspace of the operator curl together with initial electric fields conjugate to such potentials. Similarly for negative helicity. Helicity is thereby characterized by the spectral subspaces of curl in configuration space. A gauge potential on three-space has a Poisson extension to a four dimensional Euclidean half space, defined as the solution to the Maxwell-Poisson equation whose initial data is the given gauge potential. We will show that the extension is anti-self-dual if and only if the gauge potential lies in the positive spectral subspace of curl. Similarly for self-dual extension and negative spectral subspace. Helicity is thereby characterized for a normalizable electromagnetic field by the canonical formalism together with (anti-)self-duality.For a non-abelian gauge field on Minkowski space a plane wave expansion is not gauge invariant. Nor is the notion of positive spectral subspace of curl. But if one replaces the Maxwell-Poisson equation by the Yang-Mills-Poisson equation then (anti-)self-duality on the Euclidean side induces a decomposition of (now non-linear) configuration space similar to that in the electromagnetic case. The strong analogy suggests a gauge invariant definition of helicity for non-abelian gauge fields. We will establish two further properties that support this view.