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

• Autores: S. De Vincenzo
• Localización: Revista Mexicana de Física, ISSN-e 0035-001X, Vol. 60, Nº. 5, 2014, págs. 401-408
• Idioma: inglés
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• Resumen

  • We introduce a family of four Dirac operators in 1+1 dimensions: [no format] (A=1,2,3,4) for x Exist "Omega" = [a,b]. Here (T_(A)) is a complete set of 2 X 2 matrices: [no format], where "alfa" and "Beta" are the usual Dirac matrices. We show that the hermiticity of each of the operators h_(A) implies that C_(A)(x=b) = C_(A)(x=a), where the real-valued quantities [no format], the bilinear densities, are precisely the components of a Clifford number C in the basis of the matrices T_(A); moreover, [no format] is a density matrix ("g" is the probability density). Because we know the most general family of self-adjoint boundary conditions for h2 in the Weyl representation (and also for h1), we can obtain similar families for h3 and h4 in the Weyl representation using only the aforementioned family for 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 (self-adjoint) Dirac Hamiltonian in the standard representation and boundary conditions for the free Dirac Hamiltonian in the Foldy-Wouthuysen representation.