Characterizing Klein–Fock–Gordon–Majorana Particles in (1 + 1) Dimensions

Abstract

Theoretically, in (1 + 1) dimensions, one can have Klein–Fock–Gordon–Majorana (KFGM) particles. More precisely, these are one-dimensional (1D) Klein–Fock–Gordon (KFG) and Majorana particles at the same time. In principle, the wave equations considered to describe such first-quantized particles are the standard 1D KFG equation and/or the 1D Feshbach–Villars (FV) equation, each with a real Lorentz scalar potential and some kind of Majorana condition. The aim of this paper is to analyze the latter assumption fully and systematically; additionally, we introduce specific equations and boundary conditions to characterize these particles when they lie within an interval (or on a line with a tiny hole at a point). In fact, we write first-order equations in the time derivative that do not have a Hamiltonian form. We may refer to these equations as first-order 1D Majorana equations for 1D KFGM particles. Moreover, each of them leads to a second-order equation in time that becomes the standard 1D KFG equation when the scalar potential is independent of time. Additionally, we examine the nonrelativistic limit of one of the first-order 1D Majorana equations.

Appendices

Appendix A

As we have seen, we can write $2+2=4$ first-order (non-Hamiltonian) 1D Majorana equations in time for the 1D KFGM particles, namely, Eqs. (44) and (48) (by using the standard Majorana condition), and (51) and (55) (by using the nonstandard Majorana condition). Likewise, we can also write four second-order 1D Majorana equations in time for these particles, i.e., four second-order equations for the components $\varphi$ and $\chi$ of $\Psi$. In effect, applying the operator $\hat{\textrm{E}}$ to both sides of Eq. (44), and using the relation $\hat{\textrm{E}}^{*}=-\hat{\textrm{E}}$, gives the following equation:

$$\begin{aligned} \left[ \,\hat{\textrm{E}}^{2}-(c\,\hat{\textrm{p}})^{2} -(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right] \varphi =(\hat{\textrm{E}}\, S)(\varphi +\varphi ^{*}). \end{aligned}$$

(A1)

Similarly, from Eq. (48), the following equation is obtained:

$$\begin{aligned} \left[ \,\hat{\textrm{E}}^{2}-(c\,\hat{\textrm{p}})^{2}-(\textrm{m}c^{2})^{2}-2\, \textrm{m}c^{2}S\,\right] \chi =-(\hat{\textrm{E}}\, S)(\chi +\chi ^{*}). \end{aligned}$$

(A2)

These two equations correspond to the Majorana condition $\Psi =\Psi _{c}$, that is, $\psi =\psi ^{*}$. If we add Eqs. (A1) and (A2) and use the relations given in Eqs. (3) and (4) (the latter with $V=0$), it is confirmed that $\psi =\varphi +\chi$ satisfies the 1D KFG equation (i.e., Eq. 39), namely,

$$\begin{aligned} \left[ \,\hat{\textrm{E}}^{2}-(c\,\hat{\textrm{p}})^{2} -(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right] \psi =0, \end{aligned}$$

(A3)

as expected. Note that only when the scalar potential does not explicitly depend on time, the complex components $\varphi$ and $\chi$ of $\Psi$ also satisfy this equation. If this is not the case, only the functions $\textrm{Re}(\varphi )$ and $\textrm{Re}(\chi )$ can satisfy the 1D KFG equation (see the discussion following Eq. (45) through Eq. (50)).

Similarly, applying the operator $\hat{\textrm{E}}$ to both sides of Eq. (51), and using the relation $\hat{\textrm{E}}^{*}=-\hat{\textrm{E}}$, gives the equation

$$\begin{aligned} \left[ \,\hat{\textrm{E}}^{2}-(c\,\hat{\textrm{p}})^{2} -(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right] \varphi =(\hat{\textrm{E}}\, S)(\varphi -\varphi ^{*}). \end{aligned}$$

(A4)

In the same manner, applying $\hat{\textrm{E}}$ to Eq. (55) gives the following equation:

$$\begin{aligned} \left[ \,\hat{\textrm{E}}^{2}-(c\,\hat{\textrm{p}})^{2} -(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right] \chi =-(\hat{\textrm{E}}\, S)(\chi -\chi ^{*}). \end{aligned}$$

(A5)

The latter two equations correspond to the Majorana condition $\Psi =-\Psi _{c}$, that is, $\psi =-\psi ^{*}$. If we add Eqs. (A4) and (A5) and use the relations given in Eqs. (3) and (4) (the latter with $V=0$), it is again found that $\psi =\varphi +\chi$ satisfies the 1D KFG equation (i.e., Eq. A3), as expected. Clearly, if $(\hat{\textrm{E}}\, S)=0$, then $\varphi$ and $\chi$ also satisfy the 1D KFG equation. If $(\hat{\textrm{E}}\, S)\ne 0$, then only the functions $\textrm{Im}(\varphi )$ and $\textrm{Im}(\chi )$ can satisfy this equation (see the discussion following Eq. (53) through Eq. (57)).

Appendix B

Let us examine the nonrelativistic approximation of one of the first-order 1D Majorana equations in time, for example, Eq. (44). As we know, the latter equation for $\varphi \in {\mathbb {C}}$ is completely equivalent to Eq. (46), namely,

$$\begin{aligned} \left[ \,-\hbar ^{2}\frac{\partial ^{2}}{\partial t^{2}}+\hbar ^{2}c^{2} \frac{\partial ^{2}}{\partial x^{2}}-(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right] \textrm{Re}(\varphi )=0, \end{aligned}$$

(B1)

plus the relation given in Eq. (47), namely,

$$\begin{aligned} \textrm{Im}(\varphi )=\frac{\hbar }{\textrm{m}c^{2}}\,\frac{\partial }{\partial t}\,\textrm{Re}(\varphi ). \end{aligned}$$

(B2)

Let us note that Eq. (B1) can also be written as follows:

$$\begin{aligned} \textrm{Re}\left[ \,\left( \,-\hbar ^{2}\frac{\partial ^{2}}{\partial t^{2}}+\hbar ^{2}c^{2}\frac{\partial ^{2}}{\partial x^{2}}-(\textrm{m}c^{2})^{2}-2\,\textrm{m}c^{2}S\,\right) \varphi \,\right] =0. \end{aligned}$$

(B3)

Now, we choose the typical ansatz that connects $\varphi$ to its nonrelativistic approximation $\varphi _{\textrm{NR}}$, namely,

$$\begin{aligned} \varphi =\varphi _{\textrm{NR}}\,\textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t}, \end{aligned}$$

(B4)

and therefore,

$$\begin{aligned} \varphi _{t}=\left[ (\varphi _{\textrm{NR}})_{t}-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }\, \varphi _{\textrm{NR}}\right] \textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t} \end{aligned}$$

(B5)

and

$$\begin{aligned} \varphi _{tt}=\left[ (\varphi _{\textrm{NR}})_{tt}-\textrm{i} \frac{2\textrm{m}c^{2}}{\hbar }\,(\varphi _{\textrm{NR}})_{t}-\frac{(\textrm{m}c^{2})^{2}}{\hbar ^{2}}\, \varphi _{\textrm{NR}}\right] \textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t}. \end{aligned}$$

(B6)

In the nonrelativistic approximation, we have that

$$\begin{aligned} \left| \,\textrm{i}\hbar \,(\varphi _{\textrm{NR}})_{t}\,\right| \ll \textrm{m}c^{2}\left| \, \varphi _{\textrm{NR}}\,\right| \;\Rightarrow \; \left| \,(\varphi _{\textrm{NR}})_{t}\,\right| \ll \frac{\textrm{m}c^{2}}{\hbar }\left| \, \varphi _{\textrm{NR}}\,\right| \end{aligned}$$

(B7)

and

$$\begin{aligned} \left| \,\textrm{i}\hbar \,(\varphi _{\textrm{NR}})_{tt}\,\right| \ll \textrm{m}c^{2}\left| \, (\varphi _{\textrm{NR}})_{t}\,\right| \;\Rightarrow \;\left| \,(\varphi _{\textrm{NR}})_{tt}\,\right| \ll \frac{\textrm{m}c^{2}}{\hbar } \left| \,(\varphi _{\textrm{NR}})_{t}\,\right| . \end{aligned}$$

(B8)

Consequently, in this regime, the relations given in Eqs. (B5) and (B6) can be written as follows:

$$\begin{aligned} \varphi _{t}=-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }\,\varphi _{\textrm{NR}}\, \textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t} \end{aligned}$$

(B9)

and

$$\begin{aligned} \varphi _{tt}=\left[ -\textrm{i}\frac{2\textrm{m}c^{2}}{\hbar }\,(\varphi _{\textrm{NR}})_{t} -\frac{(\textrm{m}c^{2})^{2}}{\hbar ^{2}}\,\varphi _{\textrm{NR}}\right] \textrm{e}^{-\textrm{i} \frac{\textrm{m}c^{2}}{\hbar }t}. \end{aligned}$$

(B10)

Substituting the latter expression into Eq. (B3), we obtain the following result:

$$\begin{aligned} \textrm{Re}\left[ \,\textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t} \left( \,-\hat{\textrm{E}}+\frac{\hat{\textrm{p}}^{2}}{2\textrm{m}}+S\,\right) \varphi _{\textrm{NR}}\,\right] =0. \end{aligned}$$

(B11)

Clearly, this is not the Schrödinger equation with the scalar interaction, i.e., $\varphi _{\textrm{NR}}$ in Eq. (B11) does not necessarily obey this equation.

Similarly, the relation that gives the imaginary part of $\varphi$ (Eq. B2) can also be written as follows:

$$\begin{aligned} \textrm{Im}(\varphi )=\textrm{Re}\left( \frac{\hbar }{\textrm{m}c^{2}}\,\varphi _{t}\right) . \end{aligned}$$

(B12)

Substituting Eqs. (B4) and (B9) into Eq. (B12), we obtain the result

$$\begin{aligned} \textrm{Im}\left( \varphi _{\textrm{NR}}\,\textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t}\right) =\textrm{Re}\left( -\textrm{i}\,\varphi _{\textrm{NR}}\,\textrm{e}^{-\textrm{i}\frac{\textrm{m}c^{2}}{\hbar }t}\right) , \end{aligned}$$

(B13)

which is always true because $\textrm{Im}(z)=\textrm{Re}(-\textrm{i}z)$, for all $z\in {\mathbb {C}}$. Thus, nothing new is obtained from Eq. (B2) and the nonrelativistic limit of Eq. (44) reduces to Eq. (B11). Finally, $\varphi _{\textrm{NR}}$ is obtained from Eq. (B11), $(\varphi$ is given in Eq. (B4) and $\chi =\varphi ^{*})$.