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Revista mexicana de física
Print version ISSN 0035-001X
Abstract
DE VINCENZO, S.. Changes of representation and general boundary conditions for Dirac operators in 1+1 dimensions. Rev. mex. fis. [online]. 2014, vol.60, n.5, pp.401-408. ISSN 0035-001X.
We introduce a family of four Dirac operators in 1+1 dimensions: ĥA = -iħc^ΓA ∂/∂x (A = 1, 2, 3,4) for x ∉ Ω = [α, b]. Here, {^ΓA} is a complete set of 2 x 2 matrices: ^Γ1 = ^1, ^Γ2 = ^α, ^Γ3 = ^β, and ^Γ4 = i^β^α, where ^α and ^β are the usual Dirac matrices. We show that the hermiticity of each of the operators ĥA implies that CA (x = b) = CA (x = α), where the real-valued quantities CA = cψ†^ΓAψ, the bilinear densities, are precisely the components of a Clifford number Ĉ in the basis of the matrices ^ΓA; moreover, Ĉ/2cρ is a density matrix (ρ is the probability density). Because we know the most general family of self-adjoint boundary conditions for ĥ2 in the Weyl representation (and also for ĥ1), we can obtain similar families for ĥ3 and ĥ4 in the Weyl representation using only the aforementioned family for ĥ2 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.
Keywords : Dirac operators; bilinear densities; changes of representation; boundary conditions; Foldy-Wouthuysen representation.