Bessel function expressions for Dirac fields

Abstract

Penrose’s null initial data techniques are used to obtain a solution to the electromagnetism-free Dirac equation, involving Bessel functions. Each Dirac field is split up into an infinite number of elementary contributions whose densities satisfy Dirac-like distributional field density equations on the interior of the future cone of an origin of real Minkowski space. The initial data hypersurface for all the elementary fields is conveniently chosen to be the future null cone of the origin. It is observed that the field density equations are invariant under proper Lorentz transformations. The entire fields are then expressed as explicit scaling invariant convolutions taken over the initial data cone.