On 3D and 1D Weyl particles in a 1D box

Abstract

We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.

Appendix

1.1 A. On rotations

Let us write an ordinary spatial rotation through an angle $\theta$ around the $x^{j}$-axis, that is, $\left[ \, ct'\; x'\; y'\; z'\,\right] ^{\mathrm {T}}={\hat{\varLambda }}_{j}\left[ \, ct\; x\; y\; z\,\right] ^{\mathrm {T}}$ (i.e., $x^{\mu }\,'=(\varLambda _{j})_{\;\,\nu }^{\mu }\, x^{\nu }\Rightarrow x^{\mu }=(\varLambda _{j}^{-1})_{\;\,\nu }^{\mu }\, x^{\nu }\,'$), where

$$\begin{aligned} {\hat{\varLambda }}_{j}={\hat{\varLambda }}_{j}(\theta )=\exp \left( \mathrm {i}\theta {\hat{J}}_{j}\right) , \end{aligned}$$

(A1)

with

$$\begin{aligned} {\hat{J}}_{1}= & {} \left[ \begin{array}{cc} {\hat{0}}_{2} &{} {\hat{0}}_{2}\\ {\hat{0}}_{2} &{} {\hat{\sigma }}_{y} \end{array}\right] \,,\quad {\hat{J}}_{2}=\left[ \begin{array}{cc} {\hat{0}}_{2} &{} \frac{\mathrm {i}}{2}({\hat{1}}_{2}-{\hat{\sigma }}_{z})\\ -\frac{\mathrm {i}}{2}({\hat{1}}_{2}-{\hat{\sigma }}_{z}) &{} {\hat{0}}_{2} \end{array}\right] ,\nonumber \\ {\hat{J}}_{3}= & {} \left[ \begin{array}{cc} {\hat{0}}_{2} &{} -\frac{\mathrm {i}}{2}({\hat{\sigma }}_{x}-\mathrm {i}{\hat{\sigma }}_{y})\\ \frac{\mathrm {i}}{2}({\hat{\sigma }}_{x}+\mathrm {i}{\hat{\sigma }}_{y}) &{} {\hat{0}}_{2} \end{array}\right] . \end{aligned}$$

(A2)

Then, under this linear transformation, the Dirac wave function in (3+1) dimensions transforms as $\varPsi '(x^{k}\,',t')={\hat{S}}(\varLambda _{j})\varPsi (x^{k},t)$ ($k=1,2,3$), where ${\hat{S}}(\varLambda _{j})$, which obeys the relation $(\varLambda _{j})_{\;\,\nu }^{\mu }{\hat{\gamma }}^{\nu }={\hat{S}}^{-1}(\varLambda _{j})\,{\hat{\gamma }}^{\mu }\,{\hat{S}}(\varLambda _{j})$, is given by ${\hat{S}}(\varLambda _{j})=\exp (\mathrm {i}\theta \,\mathrm {i}{\hat{\gamma }}^{k}{\hat{\gamma }}^{l}/2)$ ($l=1,2,3$) for cyclic $\{j,k,l\}$. Because $\hat{\varvec{\varSigma }}=(\mathrm {i}{\hat{\gamma }}^{2}{\hat{\gamma }}^{3},\mathrm {i}{\hat{\gamma }}^{3}{\hat{\gamma }}^{1},\mathrm {i}{\hat{\gamma }}^{1}{\hat{\gamma }}^{2})$ ($={\hat{\gamma }}^{5}{\hat{\gamma }}^{0}\hat{\varvec{\gamma }}$), we can write the $4\times 4$ matrix ${\hat{S}}(\varLambda _{j})$ as follows:

$$\begin{aligned} {\hat{S}}(\varLambda _{j})=\exp \left( \mathrm {i}\theta \,\frac{{\hat{\varSigma }}^{j}}{2}\right) , \end{aligned}$$

(A3)

i.e., the spin operator in (3+1) dimensions $\hat{{\mathbf {S}}}=\hbar \hat{\varvec{\varSigma }}/2$ is essentially the generator of spatial rotations. In the Weyl representation, ${\hat{S}}(\varLambda _{j})$ is a block-diagonal matrix because $\hat{\varvec{\varSigma }}=\mathrm {diag}(\hat{\varvec{\sigma }},\hat{\varvec{\sigma }})$, and we obtain the following results:

$$\begin{aligned} \left[ \begin{array}{c} \varphi _{1}^{\mathrm {t}}\,'(x^{j}\,',t')\\ \varphi _{1}^{\mathrm {b}}\,'(x^{j}\,',t') \end{array}\right] =\left[ \,\cos \left( \frac{\theta }{2}\right) {\hat{1}}_{2}+\mathrm {i}\sin \left( \frac{\theta }{2}\right) {\hat{\sigma }}_{j}\,\right] \left[ \begin{array}{c} \varphi _{1}^{\mathrm {t}}(x^{j},t)\\ \varphi _{1}^{\mathrm {b}}(x^{j},t) \end{array}\right] \end{aligned}$$

(A4)

and

$$\begin{aligned} \left[ \begin{array}{c} \varphi _{2}^{\mathrm {t}}\,'(x^{j}\,',t')\\ \varphi _{2}^{\mathrm {b}}\,'(x^{j}\,',t') \end{array}\right] =\left[ \,\cos \left( \frac{\theta }{2}\right) {\hat{1}}_{2}+\mathrm {i}\sin \left( \frac{\theta }{2}\right) {\hat{\sigma }}_{j}\,\right] \left[ \begin{array}{c} \varphi _{2}^{\mathrm {t}}(x^{j},t)\\ \varphi _{2}^{\mathrm {b}}(x^{j},t) \end{array}\right] . \end{aligned}$$

(A5)

That is, the two-component wave functions (or Weyl spinors) in (3+1) dimensions $\varphi _{1}$ and $\varphi _{2}$ transform in the same way under spatial rotations. Obviously, in (1+1) dimensions, a pure (spatial) rotation is not possible.

1.2 B. On the concept of helicity

In (3+1) dimensions, the eigenstates $(\varPsi _{+})_{{\mathbf {p}}}=\left[ \,(\varphi _{1})_{{\mathbf {p}}}\;\,0\,\right] ^{\mathrm {T}}$ and $(\varPsi _{-})_{{\mathbf {p}}}=\left[ \,0\;\,(\varphi _{2})_{{\mathbf {p}}}\,\right] ^{\mathrm {T}}$ of the helicity operator ${\hat{\lambda }}\equiv \hat{\varvec{\varSigma }}\cdot \hat{{\mathbf {p}}}/\left\| {\mathbf {p}}\right\| =\hat{{\mathbf {S}}}\cdot \hat{{\mathbf {p}}}/\frac{\hbar }{2}\left\| {\mathbf {p}}\right\|$ ($=\mathrm {diag}({\hat{\lambda }}_{[2]},{\hat{\lambda }}_{[2]})$) satisfy the following relations that depend on the sign of the energy:

$$\begin{aligned} \hat{{\mathbf {S}}}\cdot \frac{\hat{{\mathbf {p}}}}{\left\| {\mathbf {p}}\right\| }\,(\varPsi _{+})_{{\mathbf {p}}}=\mathrm {sgn}(E)\,\frac{\hbar }{2}\,{\hat{1}}_{4}(\varPsi _{+})_{{\mathbf {p}}}\,,\quad \hat{{\mathbf {S}}}\cdot \frac{\hat{{\mathbf {p}}}}{\left\| {\mathbf {p}}\right\| }\,(\varPsi _{-})_{{\mathbf {p}}}=-\mathrm {sgn}(E)\,\frac{\hbar }{2}\,{\hat{1}}_{4}(\varPsi _{-})_{{\mathbf {p}}}, \end{aligned}$$

(B1)

and therefore,

$$\begin{aligned} \hat{{\mathbf {S}}}_{[2]}\cdot \frac{\hat{{\mathbf {p}}}_{[2]}}{\left\| {\mathbf {p}}_{[2]}\right\| }\,(\varphi _{1})_{{\mathbf {p}}}=\mathrm {sgn}(E)\,\frac{\hbar }{2}\,{\hat{1}}_{2}(\varphi _{1})_{{\mathbf {p}}}\,,\quad \hat{{\mathbf {S}}}_{[2]}\cdot \frac{\hat{{\mathbf {p}}}_{[2]}}{\left\| {\mathbf {p}}_{[2]}\right\| }\,(\varphi _{2})_{{\mathbf {p}}}=-\mathrm {sgn}(E)\,\frac{\hbar }{2}\,{\hat{1}}_{2}(\varphi _{2})_{{\mathbf {p}}},\nonumber \\ \end{aligned}$$

(B2)

where ${\hat{\lambda }}_{[2]}\equiv \hat{\varvec{\sigma }}\cdot \hat{{\mathbf {p}}}_{[2]}/\left\| {\mathbf {p}}_{[2]}\right\| =\hat{{\mathbf {S}}}_{[2]}\cdot \hat{{\mathbf {p}}}_{[2]}/\frac{\hbar }{2}\left\| {\mathbf {p}}_{[2]}\right\|$ and $\left\| {\mathbf {p}}_{[2]}\right\| =\left\| {\mathbf {p}}\right\|$. Thus, the eigenvalues of the operators ${\hat{\lambda }}$ and ${\hat{\lambda }}_{[2]}$ only indicate whether the direction of the spin of the particle in question is parallel or antiparallel to its respective momentum; however, all of these eigenvalues are also dependent on the sign of the energy.

Let us now introduce the so-called (Hermitian) classical velocity operator $\hat{\mathrm {\mathrm {\mathbf {v}}}}_{\mathrm {cl}}\equiv c^{2}\hat{\mathrm {\mathbf {p}}}\hat{\mathrm {E}}^{-1}$ (which corresponds to the formula of classical relativistic mechanics that provides the velocity as a function of momentum and energy), where $\hat{\mathrm {E}}$ is the Dirac Hamiltonian operator [27]. Clearly, if $\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}}$ acts on the Dirac plane-wave solution $\varPsi _{{\mathbf {p}}}$, one obtains the eigenvalue $\mathrm {{\mathbf {v}}}_{\mathrm {cl}}=c^{2}{\mathbf {p}}/E$, i.e., $\mathrm {{\mathbf {v}}}_{\mathrm {cl}}=\mathrm {sgn}(E)\, c\,{\mathbf {p}}/\left\| {\mathbf {p}}\right\|$ ($\Rightarrow \left\| \mathrm {{\mathbf {v}}}_{\mathrm {cl}}\right\| =c$, as expected). Then, we can use these results to write the relations in (B1) and (B2) such that they do not depend on the sign of the energy, that is,

$$\begin{aligned} \hat{{\mathbf {S}}}\cdot \frac{\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}}}{c}\,(\varPsi _{+})_{{\mathbf {p}}}=\frac{\hbar }{2}\,{\hat{1}}_{4}(\varPsi _{+})_{{\mathbf {p}}}\,,\quad \hat{{\mathbf {S}}}\cdot \frac{\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}}}{c}\,(\varPsi _{-})_{{\mathbf {p}}}=-\frac{\hbar }{2}\,{\hat{1}}_{4}(\varPsi _{-})_{{\mathbf {p}}}, \end{aligned}$$

(B3)

and

$$\begin{aligned} \hat{{\mathbf {S}}}_{[2]}\cdot \frac{(\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]}}{c}\,(\varphi _{1})_{{\mathbf {p}}}=\frac{\hbar }{2}\,{\hat{1}}_{2}(\varphi _{1})_{{\mathbf {p}}}\,,\quad \hat{{\mathbf {S}}}_{[2]}\cdot \frac{(\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]}}{c}\,(\varphi _{2})_{{\mathbf {p}}}=-\frac{\hbar }{2}\,{\hat{1}}_{2}(\varphi _{2})_{{\mathbf {p}}}, \end{aligned}$$

(B4)

respectively (where $(\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]}=\mathrm {sgn}(E)\, c\,\hat{{\mathbf {p}}}_{[2]}/\left\| {\mathbf {p}}_{[2]}\right\|$ and $\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}}=\mathrm {diag}((\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]},(\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]})$). In this way, the eigenvalues of the operators $\hat{{\mathbf {S}}}\cdot \hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}}/c$ and $\hat{{\mathbf {S}}}_{[2]}\cdot (\hat{\mathrm {{\mathbf {v}}}}_{\mathrm {cl}})_{[2]}/c$ indicate whether the direction of the spin of the particle in question is parallel or antiparallel to the movement of the particle. For example, the spin of the 3D Weyl particle described by $(\varphi _{1})_{{\mathbf {p}}}$ is always parallel to its direction of motion, but the spin of the 3D Weyl particle described by $(\varphi _{2})_{{\mathbf {p}}}$ is always antiparallel to its direction of motion.

As we have seen, the eigenstates of the operator $\hat{\mathrm {p}}/\left| \mathrm {p}\right|$ in (1+1) dimensions, $(\varPsi _{+})_{\mathrm {p}}=\left[ \,(\varphi _{1})_{\mathrm {p}}\;\,0\,\right] ^{\mathrm {T}}$ and $(\varPsi _{-})_{\mathrm {p}}=\left[ \,0\;\,(\varphi _{2})_{\mathrm {p}}\,\right] ^{\mathrm {T}}$, comply with relations that depend on the sign of the energy, namely

$$\begin{aligned} \frac{\hat{\mathrm {p}}}{\left| \mathrm {p}\right| }(\varPsi _{+})_{\mathrm {p}}=\mathrm {sgn}(E){\hat{1}}_{2}(\varPsi _{+})_{\mathrm {p}}\,,\quad \frac{\hat{\mathrm {p}}}{\left| \mathrm {p}\right| }(\varPsi _{-})_{\mathrm {p}}=-\mathrm {sgn}(E){\hat{1}}_{2}(\varPsi _{-})_{\mathrm {p}}, \end{aligned}$$

(B5)

from which similar relations for $(\varphi _{1})_{\mathrm {p}}$ and $(\varphi _{2})_{\mathrm {p}}$ are immediately obtained, namely

$$\begin{aligned} \frac{\hat{\mathrm {p}}_{[1]}}{\left| \mathrm {p}_{[1]}\right| }(\varphi _{1})_{\mathrm {p}}=\mathrm {sgn}(E)(\varphi _{1})_{\mathrm {p}}\,,\quad \frac{\hat{\mathrm {p}}_{[1]}}{\left| \mathrm {p}_{[1]}\right| }(\varphi _{2})_{\mathrm {p}}=-\mathrm {sgn}(E)(\varphi _{2})_{\mathrm {p}}, \end{aligned}$$

(B6)

where $\hat{\mathrm {p}}/\left| \mathrm {p}\right| =\hat{\mathrm {p}}_{[1]}{\hat{1}}_{2}/\left| \mathrm {p}_{[1]}\right|$ and $\left| \mathrm {p}_{[1]}\right| =\left| \mathrm {p}\right|$. Clearly, the operators $\hat{\mathrm {p}}/\left| \mathrm {p}\right|$ and $\hat{{\mathbf {S}}}\cdot \hat{{\mathbf {p}}}/\left\| {\mathbf {p}}\right\|$, as well as $\hat{\mathrm {p}}_{[1]}/\left| \mathrm {p}_{[1]}\right|$ and $\hat{{\mathbf {S}}}_{[2]}\cdot \hat{{\mathbf {p}}}_{[2]}/\left\| {\mathbf {p}}_{[2]}\right\|$, have a certain similarity (when acting on their respective chiral plane-wave eigenstates). The Dirac plane-wave $\varPsi _{\mathrm {p}}$ is also an eigensolution of the (Hermitian) classical velocity operator $\hat{\mathrm {v}}_{\mathrm {cl}}\equiv c^{2}\hat{\mathrm {p}}\hat{\mathrm {E}}^{-1}$ and has eigenvalue $\mathrm {v}_{\mathrm {cl}}=c^{2}\mathrm {p}/E=\mathrm {sgn}(E)\, c\,\mathrm {p}/\left| \mathrm {p}\right|$ ($\hat{\mathrm {E}}\,(=\hat{\mathrm {h}})$ is the Dirac Hamiltonian operator in (1+1) dimensions). This fact allows us to write the relations in (B5) and (B6) in a form independent of the energy sign, namely

$$\begin{aligned} \frac{\hat{\mathrm {v}}_{\mathrm {cl}}}{c}(\varPsi _{+})_{\mathrm {p}}={\hat{1}}_{2}(\varPsi _{+})_{\mathrm {p}}\,,\quad \frac{\hat{\mathrm {v}}_{\mathrm {cl}}}{c}(\varPsi _{-})_{\mathrm {p}}=-{\hat{1}}_{2}(\varPsi _{-})_{\mathrm {p}}, \end{aligned}$$

(B7)

and

$$\begin{aligned} \frac{(\hat{\mathrm {v}}_{\mathrm {cl}})_{[1]}}{c}(\varphi _{1})_{\mathrm {p}}=(\varphi _{1})_{\mathrm {p}}\,,\quad \frac{(\hat{\mathrm {v}}_{\mathrm {cl}})_{[1]}}{c}(\varphi _{2})_{\mathrm {p}}=-(\varphi _{2})_{\mathrm {p}}, \end{aligned}$$

(B8)

respectively (where $(\hat{\mathrm {v}}_{\mathrm {cl}})_{[1]}=\mathrm {sgn}(E)\, c\,\hat{\mathrm {p}}_{[1]}/\left| \mathrm {p}_{[1]}\right|$ and $\hat{\mathrm {v}}_{\mathrm {cl}}=(\hat{\mathrm {v}}_{\mathrm {cl}})_{[1]}{\hat{1}}_{2}$). Clearly, the eigenvalues of the operators $\hat{\mathrm {v}}_{\mathrm {cl}}/c$ and $(\hat{\mathrm {v}}_{\mathrm {cl}})_{[1]}/c$ indicate whether the particle in question, whether it is a 1D Dirac particle or a 1D Weyl particle, actually moves to the right or to the left. For example, the 1D Weyl particle described by $(\varphi _{1})_{\mathrm {p}}$ always moves to the right (left), but the 1D Weyl particle described by $(\varphi _{2})_{\mathrm {p}}$ moves to the left (right).

1.3 C. On the boundary conditions for the Weyl equations

We have obtained the most general families of boundary conditions for the (time-dependent) Weyl equations given in Eq. (14) (i.e., in (3+1) dimensions), where the (self-adjoint) Weyl Hamiltonian operators present are precisely the operators $\hat{\mathrm {H}}_{a,j}$ given in Eq. (17). Each of the three families of boundary conditions [labeled by $j=1,2,3$ and given in Eqs. (31), (33) and (35)] is parametrized by a unitary $2\times 2$ matrix, that is, by $2^{2}=4$ real parameters. A feasible parameterization for these unitary matrices, for example, for the matrix ${\hat{A}}_{1}$ in Eq. (31), is given by

$$\begin{aligned} {\hat{A}}_{1}=\exp (\mathrm {i}\mu )\left[ \begin{array}{cc} m_{0}-\mathrm {i}m_{3} &{} -m_{2}-\mathrm {i}m_{1}\\ m_{2}-\mathrm {i}m_{1} &{} m_{0}+\mathrm {i}m_{3} \end{array}\right] , \end{aligned}$$

(C1)

where $\mu \in [0,\pi )$, and real quantities $m_{0}$, $m_{1}$, $m_{2}$ and $m_{3}$, satisfy $(m_{0})^{2}+(m_{1})^{2}+(m_{2})^{2}+(m_{3})^{2}=1$ (but also $\det ({\hat{A}}_{1})=\exp (\mathrm {i}2\mu )$) [28]. For other interesting examples of Hamiltonians operators whose self-adjoint extensions (or sets of general boundary conditions) are characterized in terms of unitary matrices, see Refs. [29, 30].

On the other hand, all boundary conditions that are part of each of these three families of self-adjoint boundary conditions cancel the boundary term in Eq. (18), which implies that

$$\begin{aligned} \left. c\left[ \,\varphi _{a,j}^{\dagger }{\hat{\sigma }}_{j}\varphi _{a,j}\,\right] \right| _{0}^{\ell }\equiv \left. \left[ \, J_{a,j}\,\right] \right| _{0}^{\ell }=0\quad \Rightarrow \quad J_{a,j}(x^{j}=\ell ,t)=J_{a,j}(x^{j}=0,t), \end{aligned}$$

(C2)

where $J_{a,j}=J_{a,j}(x^{j},t)$ is the probability current density [11]. Thus, all of the self-adjoint boundary conditions lead to the equality of $J_{a,j}$ at the ends of the box. Within each general family of boundary conditions, there are boundary conditions that simply cancel the probability current density at these extremes; they are called confining boundary conditions. For example, the following confining boundary conditions for the Weyl Hamiltonian $\hat{\mathrm {H}}_{a,1}$ are contained in Eq. (31): $\varphi _{a}^{\mathrm {t}}(x=\ell ,t)=\varphi _{a}^{\mathrm {t}}(x=0,t)=0$ (${\hat{A}}_{1}=-{\hat{1}}_{2}$), i.e., the upper component of the wave function $\varphi _{a,1}\equiv \varphi _{a}$ can satisfy the Dirichlet boundary condition; $\varphi _{a}^{\mathrm {b}}(x=\ell ,t)=\varphi _{a}^{\mathrm {b}}(x=0,t)=0$ (${\hat{A}}_{1}=+{\hat{1}}_{2}$), i.e., the lower component of the wave function $\varphi _{a,1}\equiv \varphi _{a}$ can also satisfy the Dirichlet boundary condition. However, the entire two-component Weyl wave function $\varphi _{a,1}\equiv \varphi _{a}$ does not support this boundary condition at the walls of the box, i.e., the latter is not contained in Eq. (31). This result is also fulfilled by the Dirac wave function [31]. Likewise, there are also boundary conditions that do not cancel $J_{a,j}$ at the ends of the box; they are called non-confining boundary conditions. For example, the following non-confining boundary conditions for the Weyl Hamiltonian $\hat{\mathrm {H}}_{a,1}$ are also contained in Eq. (31): $\varphi _{a}^{\mathrm {t}}(x=\ell ,t)=\varphi _{a}^{\mathrm {t}}(x=0,t)$ and $\varphi _{a}^{\mathrm {b}}(x=\ell ,t)=\varphi _{a}^{\mathrm {b}}(x=0,t)$ (${\hat{A}}_{1}=+{\hat{\sigma }}_{x}$), i.e., the wave function $\varphi _{a,1}\equiv \varphi _{a}$ can satisfy the periodic boundary condition; $\varphi _{a}^{\mathrm {t}}(x=\ell ,t)=-\varphi _{a}^{\mathrm {t}}(x=0,t)$ and $\varphi _{a}^{\mathrm {b}}(x=\ell ,t)=-\varphi _{a}^{\mathrm {b}}(x=0,t)$ (${\hat{A}}_{1}=-{\hat{\sigma }}_{x}$), i.e., the wave function $\varphi _{a,1}\equiv \varphi _{a}$ can also satisfy the antiperiodic boundary condition.

As was noted in Sect. 3, the (time-dependent) Weyl equations with the (self-adjoint) Hamiltonian operators $\hat{\mathrm {H}}_{a,1}$ and $\hat{\mathrm {H}}_{a,3}$ can provide purely real-valued solutions. Thus, if we impose on the respective wave functions the reality condition, these wave functions and their respective complex conjugates must satisfy the same boundary conditions, in which case the unitary matrices ${\hat{A}}_{1}$ and ${\hat{A}}_{3}$ must each be orthogonal. For example, in this case, the unitary matrix ${\hat{A}}_{1}$ in Eq. (C1) takes the form

$$\begin{aligned} {\hat{A}}_{1}=\left[ \begin{array}{cc} m_{0} &{} -m_{2}\\ m_{2} &{} m_{0} \end{array}\right] , \end{aligned}$$

(C3)

where $(m_{0})^{2}+(m_{2})^{2}=1$, and therefore, $\det ({\hat{A}}_{1})=+1$ (because $m_{1}=m_{3}=0$ and $\mu =0$). Likewise, ${\hat{A}}_{1}$ in Eq. (C1) can also take the form

$$\begin{aligned} {\hat{A}}_{1}=\left[ \begin{array}{cc} m_{3} &{} m_{1}\\ m_{1} &{} -m_{3} \end{array}\right] , \end{aligned}$$

(C4)

where $(m_{1})^{2}+(m_{3})^{2}=1$, and therefore, $\det ({\hat{A}}_{1})=-1$ (because $m_{0}=m_{2}=0$ and $\mu =\pi /2$) [28]. Contrarily, the (time-dependent) Weyl equation with the (self-adjoint) Hamiltonian operator $\hat{\mathrm {H}}_{a,2}$ cannot provide real-valued solutions; thus, the corresponding wave functions support any boundary condition included in Eq. (33).

It is a known fact that families of boundary conditions for relativistic (and non-relativistic) Hamiltonian operators that describe a quantum particle inside a finite interval, and on the entire real line except the finite interval, are similar (in the latter case, if the interval is very small, we have the real line with a single point excluded — a hole —, and a particle in this kind of system can be modeled through proper boundary conditions only, and with potentials only — with singular potentials and with smooth potentials —). For example, in Ref. [32], 1D Dirac point interactions were recently modeled with a two-parameter potential that has a scalar part and an electrostatic part, and each is essentially a Dirac delta function. Likewise, in Ref. [33], some self-adjoint extensions of the 1D Dirac Hamiltonian operator for massive “1D-fermions” confined to the interval $\varOmega =[-L/2,+L/2]$ were recently studied (the latter results can be conveniently extended to the region ${\mathbb {R}}-\varOmega$, which the authors call the dual geometry of $\varOmega$). Essentially, the results in Ref. [33] were applied to the calculation of the so-called Casimir energy of a massive Dirac field confined in a 1D finite filament (of length L). Precisely, the most general family of (self-adjoint) boundary conditions for the Dirac Hamiltonian operator [in Eq. (3) of Ref. [33]] is given by

$$\begin{aligned} \left[ \begin{array}{c} \psi _{1}(x=-L/2)\\ \psi _{2}(x=+L/2) \end{array}\right] =\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}\left[ \begin{array}{c} \psi _{1}(x=+L/2)\\ \psi _{2}(x=-L/2) \end{array}\right] \end{aligned}$$

(C5)

(see Eq. (23) in Ref. [33]), where $\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}$ is a unitary matrix (this is because $\hat{{\mathbb {U}}}$ and ${\hat{\gamma }}_{0}$ are also unitary matrices). The latter family of boundary conditions is similar to the most general family of boundary conditions for the self-adjoint Weyl operator $\hat{\mathrm {H}}_{a,3}$. In fact, the set of boundary conditions for $\hat{\mathrm {H}}_{a,3}$ is given in Eq. (35), but it can also be written as the set in Eq. (C5), namely

$$\begin{aligned} \left[ \begin{array}{c} \varphi _{a}^{\mathrm {t}}(z=0)\\ \varphi _{a}^{\mathrm {b}}(z=\ell ) \end{array}\right] =({\hat{A}}_{3}{\hat{\sigma }}_{x})^{-1}\left[ \begin{array}{c} \varphi _{a}^{\mathrm {t}}(z=\ell )\\ \varphi _{a}^{\mathrm {b}}(z=0) \end{array}\right] , \end{aligned}$$

(C6)

where $({\hat{A}}_{3}{\hat{\sigma }}_{x})^{-1}$ is a unitary matrix (this is because ${\hat{A}}_{3}$ and ${\hat{\sigma }}_{x}$ are also unitary matrices). For simplicity, the variable t was dropped from the functions $\varphi _{a}^{\mathrm {t}}$ and $\varphi _{a}^{\mathrm {b}}$ in Eq. (C6). Again, all the boundary conditions for the (massive) 1D Dirac particle in the box are also valid for the (massless) 3D Weyl particle in the box.

Finally, as might be expected, within the family given in Eq. (C5), and in Eq. (C6), we also have the boundary condition commonly used in the so-called MIT bag model for hadronic structure (certainly, in its one-dimensional version) [34]. This confining boundary condition can be obtained from Eq. (C5), where the matrix $\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}$ satisfies the relation $\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}\,{\hat{\alpha }}+{\hat{\alpha }}\,\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}={\hat{0}}_{2}$ (see Eq. (27) in Ref. [33]), by setting $\theta =-\pi /2$ and $\eta =0$, and therefore, $\hat{{\mathbb {U}}}{\hat{\gamma }}_{0}=(-1)^{r}(-\mathrm {i}){\hat{\sigma }}_{x}$. Thus, one obtains

$$\begin{aligned} \psi _{1}(-L/2)=(-1)^{r}(-\mathrm {i})\,\psi _{2}(-L/2),\quad \psi _{1}(+L/2)=(-1)^{r}(+\mathrm {i})\,\psi _{2}(+L/2). \end{aligned}$$

(C7)

Interestingly, the latter boundary condition can also be written explicitly in a form independent of the particular choice of the gamma matrices (and, as is known, also in a Lorentz-invariant way), namely

$$\begin{aligned} \mathrm {i}\, n_{\mu }{\hat{\gamma }}^{\mu }\psi =\psi \;\mathrm {at}\; x=-L/2\;\mathrm {and}\; x=+L/2, \end{aligned}$$

(C8)

where $n^{\mu }=(0,-1)$ at $x=-L/2$, and $n^{\mu }=(0,+1)$ at $x=+L/2$ (i.e., the unit two-vector normal to the surface of the box is pointing outward from the wall). In addition, we have that ${\hat{\gamma }}^{0}=(-1)^{r}{\hat{\sigma }}_{x}$ and ${\hat{\gamma }}^{1}={\hat{\gamma }}^{0}{\hat{\alpha }}={\hat{\gamma }}^{0}{\hat{\sigma }}_{z}=(-1)^{r}(-\mathrm {i}){\hat{\sigma }}_{y}$ (see Eq. (26) in Ref. [33]). Similarly, the usual MIT bag boundary condition can be obtained from the family in Eq. (C6) by setting $({\hat{A}}_{3}{\hat{\sigma }}_{x})^{-1}=(-\mathrm {i}){\hat{\sigma }}_{x}$. Thus, one obtains $\varphi _{a}^{\mathrm {t}}(z=0)=(-\mathrm {i})\varphi _{a}^{\mathrm {b}}(z=0)$, $\varphi _{a}^{\mathrm {t}}(z=\ell )=(+\mathrm {i})\varphi _{a}^{\mathrm {b}}(z=\ell )$.

On the other hand, in (1+1) dimensions, the most general family of self-adjoint boundary conditions for each of the (time-dependent) Weyl equations given in Eq. (36) is characterized by a phase, i.e., by a single real parameter. All boundary conditions present in these two families of boundary conditions cancel the boundary term in Eq. (39), which implies that

$$\begin{aligned} \left. \left[ \,\varphi _{a}^{*}\varphi _{a}\,\right] \right| _{0}^{\ell }\equiv \left. \left[ \,\varrho _{a}\,\right] \right| _{0}^{\ell }=0\quad \Rightarrow \quad \varrho _{a}(x=\ell ,t)=\varrho _{a}(x=0,t), \end{aligned}$$

(C9)

where $\varrho _{a}=\varrho _{a}(x,t)$ is the probability density. In this case, each Weyl equation leads to an atypical continuity equation, in which the probability density is precisely proportional to the probability current density, namely $\partial (\varphi _{a}^{*}\varphi _{a})/\partial t+(-1)^{a-1}\partial (c\,\varphi _{a}^{*}\varphi _{a})/\partial x=0$. With that said, it is clear that all boundary conditions within the two one-parametric families of boundary conditions are non-confining boundary conditions, i.e., none of them can cancel the probability current density at the ends of the box.