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.
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Cordes, H.O. (2000). On Dirac Observables. In: Balakrishnan, A.V. (eds) Semigroups of Operators: Theory and Applications. Progress in Nonlinear Differential Equations and Their Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8417-4_6
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DOI: https://doi.org/10.1007/978-3-0348-8417-4_6
Publisher Name: Birkhäuser, Basel
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