Treffer: A precise pseudodifferential Foldy-Wouthuysen transform for the Dirac equation

The author discusses the existence of a unitary pseudodifferential operator \(U\) in the algebra of strictly classical pseudodifferential operators on \({\mathbb R}^3\) such that \(U\) precisely decouples the electronic and positronic part of the Dirac equation \(\partial\psi/\partial t+H\psi=0\) with the Dirac Hamiltonian \(H\) acting on \(x=(x_1,x_2,x_3)\). This situation holds for rather general potentials and without supersymmetry. A spectral split is introduced which can amount to a clean split into an electron and positron part of the Hamiltonian. Accordingly, a new algebra \(Q\) of precisely predictable observables amounts to a restriction of observables to those which are also reduced by this split. This interpretation makes the 1-particle Dirac theory consistent and thus eliminates a compelling need for Fermi-Dirac statistics since one no longer has to worry about negative energy states.