Remarks about the invariant algebra of the Dirac equation

Abstract:
There exists an algebra P of pseudodifferential operators which is left invariant by conjugation with the propagator exp(-iHt) of the Dirac Hamiltonian. Essentially this is the algebra of operators propagating "smoothly" under the Heisenberg transform. It is decoupled by the Foldy-Wouthuysen transform. It seems that Dirac and his contemporaries were not aware of the existence of P.

The author interprets this fact by arguing that "only observables belonging to P are precisely predictable, under J.v.Neumann's rules of quantum theory. For any other observable - especially for most of the dynamical observables - one must find an approximation within P , (which then may be precisely predicted) and the closeness of the approximation reflects the precision one can expect from any measurement." For more details we refer to our book [Chapter 10 of  The Technique of Pseudodifferential Operators; London Math. Soc. Lecture Notes, Vol 202].