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

S. De Vincenzo

Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, Apartado Postal 47145, Caracas 1041-A, Venezuela. e-mail: salvatore.devincenzo@ucv.ve

Received 30 May 2014; ]]> Accepted 19 August 2014.

Abstract

We introduce a family of four Dirac operators in 1+1 dimensions: ĥ_A = -iħcˆΓ_A ∂/∂x (A = 1, 2, 3,4) for x ∉ Ω = [α, b]. Here, {ˆΓ_A} is a complete set of 2 x 2 matrices: ˆΓ₁ = ˆ1, ˆΓ₂ = ˆα, ˆΓ₃ = ˆβ, and ˆΓ₄ = iˆβˆα, where ˆα and ˆβ are the usual Dirac matrices. We show that the hermiticity of each of the operators ĥ_A implies that C_A (x = b) = C_A (x = α), where the real-valued quantities C_A = cψ†ˆΓ_Aψ, the bilinear densities, are precisely the components of a Clifford number Ĉ in the basis of the matrices ˆΓ_A; 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

PACS: 03.65.-w, 03.65.Ca, 03.65.Pm

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