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CRC Press eBooks

Dirac Algebra and Foldy-Wouthuysen Transform

Heinz Otto Cordes

Published Apr 24, 2019• 1 citation
Publisher

Abstract

We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.

DOI
10.1201/9780429187810-28

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We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.
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 eiHtQe−iHt=Q,{{e}^{{iHt}}}Q{{e}^{ - }}^{{iHt}} = Q, eiHtQe−iHt=Q, with the unitary eiHt:H→H.{e^{iHt}}:{\mathbf{H}} \to {\mathbf{H}}. eiHt:H→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.
On etudie les representation equivalents de l'equation de Dirac a l'aide de generalisations de la transformation de Foldy-Wouthuysen. On decrit et on compare les differentes methodes de diagonalisation de l'Hamiltonien de Dirac libre ou avec interaction. Des applications montrent l'interet des transformations exprimees sous forme fermee. On discute la definition et l'interpretation des observables, conduisant a distinguer le centre de charge du centre de masse. On resout les difficultees liees aux interactions dependant du temps. On traite les transformations generales conduisant a des representations equivalentes de l'equation de Dirac. On decrit les transformations unitaires soit comme des rotations dans un espace a 2 ou 3 dimensions, soit sous forme parametrique ; Une methode permet de passer d'un Hamiltonien initial donne a un Hamiltonien de forme differente, moyennant certaines contraintes. On met l'accent sur les formes diagonisables. On compare les tranformations de Foldy-Wouthuysen generalisees et celles de Lorentz. Il n'y a pas d'equivalence possible mais toutes sont des cas particuliers de transformations plus generales liees au groupe de Sitter