A pseudo-algebra of observables for the dirac equation

Abstract

It is shown that the Dirac equation admits a natural algebra P of global pseudodifferential operators, characterized] by the property that the “Heisenberg representation” A→exp(iHt)A exp(-iHt)=A_t leaves P invariant. (For a general 4×4-matrix A of pseudodifferential operators one expects A to be a matrix of Fourier integral operators). An attempt to work with P as an algebra of observables would require certain modifications of most standard dynamical observables. An Egorov-type theorem, (theorem 3.1) can be applied to gain direct insights into particle orbits spin propagation not available with other algebras. Some inconsistencies of classical theory, like “Zitterbewegung”, non-commutativity of velocity components, Klein's paradox, would naturally disappear.