Equivalence of Helicity and Euclidean Self-Duality for Gauge Fields
MATHEMATICAL PICTURE LANGUAGE
Leonard Gross - Cornell University
via Zoom Video Conferencing: https://harvard.zoom.us/j/779283357
Circularly polarized light (i.e. helicity) is a concept defined in terms of
plane wave expansions of solutions to Maxwell's equations. We wish to find an analogous concept for classical and quantized Yang-Mills fields. Since the classical (hyperbolic) Yang-Mills equation is a non-linear equation, a gauge invariant plane wave expansion does not exist. We will first
show, in electromagnetism, an equivalence between the usual plane wave characterization of helicity and a characterization in terms of (anti-)self duality of a gauge potential on a half space of Euclidean R^4. The transition from Minkowski space to Euclidean space is implemented by the
Maxwell-Poisson equation. We will then replace the Maxwell- Poisson equation by the Yang-Mills-Poisson equation to find a decomposition of the Yang-Mills configuration space into submanifolds arguably corresponding to positive and negative helicity. This is a report on the paper .