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

We introduce a family of four Dirac operators in 1+1 dimensions: ĥA = -iħcˆΓA ∂/∂x (A = 1, 2, 3,4) for x ∉ Ω = [α, b]. Here, {ˆΓA} 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 ĥA implies that C A (x = b) = C A (x = α), where the real-valued quantities C A = cψ†ˆΓAψ, the bilinear densities, are precisely the components of a Clifford number Ĉ in the basis of the matrices ˆΓA; 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.