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Revista Mexicana de Física

Changes of representation and general boundary conditions for Dirac operators in 1+1 dimensions

Salvatore De Vincenzo

Published Jan 1, 2014• 4 citations
OpenAlex

Abstract

h2 and changes of representation among the Dirac matrices. Using these results, we also determine families of general boundary conditions for all these operators in the standard representation. We also find and discuss connections between boundary conditions for the free (selfadjoint) Dirac Hamiltonian in the standard representation and boundary conditions for the free Dirac Hamiltonian in the Foldy-Wouthuysen representation.

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The Necessary and Sufficient Conditions for Transformation from Dirac Representation to Foldy-Wouthuysen Representation

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arXiv:0804.0333•Apr 2, 2008•3 citations

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h2 and changes of representation among the Dirac matrices. Using these results, we also determine families of general boundary conditions for all these operators in the standard representation. We also find and discuss connections between boundary conditions for the free (selfadjoint) Dirac Hamiltonian in the standard representation and boundary conditions for the free Dirac Hamiltonian in the Foldy-Wouthuysen representation.
The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.
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.
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