On the boundary conditions for the 1D Weyl-Majorana particle in a box

In (1+1) space-time dimensions, we can have two particles that are Weyl and Majorana particles at the same time---1D Weyl-Majorana particles. That is, the right-chiral and left-chiral parts of the two-component Dirac wave function that satisfies the Majorana condition, in the Weyl representation, describe these particles, and each satisfies their own Majorana condition. Naturally, the nonzero component of each of these two two-component wave functions satisfies a Weyl equation. We investigate and discuss this issue and demonstrate that for a 1D Weyl-Majorana particle in a box, the nonzero components, and therefore the chiral wave functions, only admit the periodic and antiperiodic boundary conditions. From the latter two boundary conditions, we can only construct four boundary conditions for the entire Dirac wave function. Then, we demonstrate that these four boundary conditions are also included within the most general set of self-adjoint boundary conditions for a 1D Majorana particle in a box.