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

Abstract

We introduce a family of four Dirac operators in 1+1 dimensions: $\hat{h}_{A}=-i\hbar c\,\hat{\mathcal{\mathit{\mathrm{\Gamma}}}}_{A}\,\partial/\partial x$ ($A=1,2,3,4$) for $x\in\Omega=[a,b]$. Here, $\{\hat{\mathit{\mathrm{\Gamma}}}_{A}\}$ is a complete set of $2\times2$ matrices: $\hat{\mathit{\mathrm{\Gamma}}}_{1}=\hat{1}$, $\hat{\mathit{\mathrm{\Gamma}}}_{2}=\hat{\alpha}$, $\hat{\mathit{\mathrm{\Gamma}}}_{3}=\hat{\beta}$, and $\hat{\mathit{\mathrm{\Gamma}}}_{4}=i\hat{\beta}\hat{\alpha}$, where $\hat{\alpha}$ and $\hat{\beta}$ are the usual Dirac matrices. We show that the hermiticity of each of the operators $\hat{h}_{A}$ implies that $C_{A}(x=b)=C_{A}(x=a)$, where the real-valued quantities $C_{A}=c\psi^{\dagger}\hat{\mathit{\mathrm{\Gamma}}}_{A}\psi$, the bilinear densities, are precisely the components of a Clifford number $\hat{C}$ in the basis of the matrices $\hat{\mathcal{\mathit{\mathrm{\Gamma}}}}_{A}$; moreover, $\hat{C}/2c\varrho$ is a density matrix ($\varrho$ is the probability density). Because we know the most general family of self-adjoint boundary conditions for $\hat{h}_{2}$ in the Weyl representation (and also for $\hat{h}_{1}$), we can obtain similar families for $\hat{h}_{3}$ and $\hat{h}_{4}$ in the Weyl representation using only the aforementioned family for $\hat{h}_{2}$ 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.