University of California,

Berkeley, USA

E-mail: hocordes@email.msn.com

**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].