Remarks about the invariant algebra of the Dirac equation
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