Reply to "Comment on 'Electronic Maxwell's equations'"

The unification of 'electronic waves and electromagnetic waves', as described in [1], is a unified approach to describing these two waves in the form of expression, where the electronic wave can be described as a three-elements vector wave rather than a four-elements Dirac state. Although the electronic Maxwell's equations (EMEs) in [1] are constructed in the form of Maxwell's equations, they are identical to the Dirac equation. Within the theory of EMEs, electrons and positrons are described by φ and χ which are referred to as 'electronic waves' and are the counterparts of E and B of electromagnetic waves, respectively. All these four quantities are vector waves in three-dimensional space. This is why we define the 'unification' of electronic waves and electromagnetic waves in terms of vector waves in [1].

Because EMEs are identical to the Dirac equation, we solve the Klein problem using the proposed EMEs method in the paper. In this connection, the solutions of the Dirac equation are interpreted correctly in [1] as we have considered the case of the creation of electron–positron pairs on page 7 of [1] where 'if a is increased further, the reflection coefficient falls below −1, producing electron–positron pairs' is claimed for the reflection coefficient

$$\begin{equation*}R=\frac{a(m-\varepsilon {c}^{-2})-{p}_{0}p}{m{V}_{0}}-1.\end{equation*}$$

In our opinion, there are two advantages of the 'similarity between electronic waves and electromagnetic waves'. First, there is similarity of the wave properties of electronic waves and electromagnetic waves. Second, there are computational benefits in practice. The explanation is elaborated using the equations below. Although the dyadic Dirac state

$$\begin{equation}\left[\begin{matrix}\hfill \varphi \hfill \\ \hfill \chi \hfill \end{matrix}\right]=\left[\begin{matrix}\hfill f\boldsymbol{I}-\mathrm{i}\boldsymbol{\sigma }\cdot \boldsymbol{\varphi }\hfill \\ \hfill \mathrm{i}g\boldsymbol{I}-\mathrm{i}\boldsymbol{\sigma }\cdot \mathrm{i}\boldsymbol{\chi }\hfill \end{matrix}\right]\end{equation} \tag{ 1 }$$

has eight components, it is confined by the conditions

$$\begin{equation}{f}^{2}+\boldsymbol{\varphi }\cdot \boldsymbol{\varphi }=0\end{equation} \tag{ 2 }$$

$$\begin{equation}{g}^{2}+\boldsymbol{\chi }\cdot \boldsymbol{\chi }=0\end{equation} \tag{ 3 }$$

$$\begin{equation}fg+\boldsymbol{\varphi }\cdot \boldsymbol{\chi }=0\end{equation} \tag{ 4 }$$

$$\begin{equation}g\boldsymbol{\varphi }-f\boldsymbol{\chi }=\boldsymbol{\varphi }\times \boldsymbol{\chi },\end{equation} \tag{ 5 }$$

where (4) can be deduced from (2), (3) and (5), reducing the number of degrees of freedom to four. In practice, we can calculate φ or χ instead of all eight components like we do in solving problems of electromagnetism. Additionally, the claim that in the new formulation there is no 'need to deal with quantum operators' is true, because there are no quaternions or Pauli matrices in the EMEs

$$\begin{equation*}\begin{cases}\bar{\nabla }\cdot \boldsymbol{\varphi }+\left(-\frac{1}{c}\frac{\partial }{\partial t}-k{\Phi}+\mathrm{i}\mu \right)g=0\quad \\ \bar{\nabla }\cdot \boldsymbol{\chi }+\left(\frac{1}{c}\frac{\partial }{\partial t}+k{\Phi}+\mathrm{i}\mu \right)f=0\quad \\ \bar{\nabla }\times \boldsymbol{\varphi }+\bar{\nabla }f=\left(-\frac{1}{c}\frac{\partial }{\partial t}-k{\Phi}+\mathrm{i}\mu \right)\boldsymbol{\chi }\quad \\ \bar{\nabla }\times \boldsymbol{\chi }+\bar{\nabla }g=\left(\frac{1}{c}\frac{\partial }{\partial t}+k{\Phi}+\mathrm{i}\mu \right)\boldsymbol{\varphi }\quad \end{cases}.\end{equation*}$$

Therefore, we no longer deal with quaternions once we obtain the EMEs.

The proof of Lorentz invariance here is not a simple copy from that of the Dirac equation. It is seen that, because the rest mass of an electron is not zero, to retain the Lorentz invariance of the EMEs, we have to write the dyadic Dirac state as (1) instead of

$$\begin{equation*}\left[\begin{matrix}\hfill \varphi \hfill \\ \hfill \chi \hfill \end{matrix}\right]=\left[\begin{matrix}\hfill -\mathrm{i}\boldsymbol{\sigma }\cdot \boldsymbol{\varphi }\hfill \\ \hfill -\mathrm{i}\boldsymbol{\sigma }\cdot \mathrm{i}\boldsymbol{\chi }\hfill \end{matrix}\right]\end{equation*}$$

which is the intuitive counterpart of electromagnetic waves. As a result, we make the following suggestion that f and g (i.e. the longitudinal fields) are attributes of the rest mass of the electron.

As for the reference, we overlooked the original paper [2], although we did know this famous experiment.

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