On Dirac Observables

Abstract

The Dirac equation admits a natural time-invariant algebra Q of unbounded operators of the Hilbert space H = L ²(ℝ³, ℂ⁴). Time-invariance means that ${{e}^{{iHt}}}Q{{e}^{ - }}^{{iHt}} = Q,$ with the unitary ${e^{iHt}}:{\mathbf{H}} \to {\mathbf{H}}.$ This is not a specific feature of the Dirac equation; any 1-st order strictly hyperbolic system (under mild additional assumptions) admits such an algebra. We propose to introduce Q as algebra of “really predictable observables” into v. Neumann’s theory of physical states and observables: Only the self-adjoint operators A ∈ Q can be really predicted. For other observables — especially for most dynamical observables — one must find an observable A⁰ ∈ Q “close” to A, perhaps only for a given physical state ψ. Energy H belongs to Q, but location (angular) momentum, spin,…, do not. This seems to explain at a glance the triumphs and difficulties Dirac’s theory encountered.