Dirac theory as a one-particle relativistic quantum mechanics in the space of unit two-component spinors

TL;DR

This paper reformulates Dirac theory as a one-particle quantum mechanics within the space of two-component spinors, providing a unified splitting method applicable to various potentials and deriving exact solutions in different limits.

Contribution

It introduces a novel splitting approach for Dirac operators into two two-component operators, applicable to vector and scalar potentials, differing from traditional methods.

Findings

Splitting of Dirac operator into two two-component operators.

Exact solutions for free two-component spinors.

Analytical expressions in nonrelativistic and ultrarelativistic limits.

Abstract

Using the example of a Dirac particle in external static fields, Dirac theory is reformulated as a one-particle quantum theory in the space of normalized two-component spinors. In this formulation, the Dirac operator ``splits'' into two two-component operators: one is bounded from below (in the nonrelativistic limit, it coincides with the Pauli operator), and the other is bounded from above. The first describes the Dirac particle, and the second can be ignored for sufficiently weak external fields. Unlike approaches based on the Foldy-Wouthuysen transformation, the ``splitting'' procedure in our approach is the same for the vector and scalar potentials. It is reduced to solving a second-order algebraic equation for the searched-for operators. A general solution to the free equation for a two-component normalized spinor is presented. Exact analytical expressions are obtained for the two-component analogs of the other Dirac operators, which are then presented in the nonrelativistic and ultrarelativistic limits.

Equations

$$i\hslash \frac{\partial \Psi (z,t)}{\partial t}={\hat{H}}_D\Psi (z,t);\text{mbox}{\hat{H}}_D=c{\alpha }_z{\hat{p}}_z+e\varphi (z)+\beta m{c}^2,$$

$${\hat{H}}_D\Psi =E\Psi .$$

$$\left[{\hat{p}}_z\frac{1}{2M(E,z)}{\hat{p}}_z+e\varphi (z)\right]\Phi =ϵ(E)\Phi ,\text{mbox}\chi =\frac{1}{2M(E,z)c}\text{mbox}{\sigma }_z{\hat{p}}_z\Phi .$$

$$\left[{\hat{p}}_z\frac{1}{2\mu (E,z)}{\hat{p}}_z+e\varphi (z)\right]\chi ={ϵ}^{\prime }\chi ,\text{mbox}\Phi =\frac{1}{2\mu (E,z)c}\text{mbox}{\sigma }_z{\hat{p}}_z\chi .$$

$$\displaystyle\hat{\Lambda}_{\vec{p}}=\frac{\vec{\Sigma}\hat{\vec{p}}}{|\vec{p}|},\mbox{\hskip 8.53581pt}\hat{P}=a\beta\hat{P}_{0};\mbox{\hskip 14.22636pt}\vec{\Sigma}=\left(\begin{array}[]{cc}\vec{\sigma}&0\\ 0&\vec{\sigma}\end{array}\right)$$

$$\displaystyle\Psi_{(+1,+1)}=\left(\begin{array}[]{cc}1\\ 0\\ 0\\ 0\end{array}\right),\mbox{\hskip 14.22636pt}\Psi_{(-1,+1)}=\left(\begin{array}[]{cc}0\\ 1\\ 0\\ 0\end{array}\right),\mbox{\hskip 14.22636pt}\Psi_{(+1,-1)}=\left(\begin{array}[]{cc}0\\ 0\\ 1\\ 0\end{array}\right),\mbox{\hskip 14.22636pt}\Psi_{(-1,-1)}=\left(\begin{array}[]{cc}0\\ 0\\ 0\\ 1\end{array}\right);$$

$$\left(-\frac{{\hslash }^2}{2M}\text{mbox}\frac{d^2}{d{z}^2}-\frac{{\hslash }^2}{2M}\text{mbox}\frac{d\ln |M|}{dz}\frac{d}{dz}+e\varphi \right){\Phi }_{\pm }=ϵ{\Phi }_{\pm },\text{mbox}\left(-\frac{{\hslash }^2}{2\mu }\text{mbox}\frac{d^2}{d{z}^2}-\frac{{\hslash }^2}{2\mu }\text{mbox}\frac{d\ln |\mu |}{dz}\frac{d}{dz}+e\varphi \right){\chi }_{\pm }={ϵ}^{\prime }{\chi }_{\pm }.$$

$$\frac{d^2{\Phi }_{\pm }}{d{z}^2}-\frac{d\ln |\tilde{M}|}{dz}\cdot \frac{d{\Phi }_{\pm }}{dz}+k_0^2\tilde{\mu }\tilde{M}{\Phi }_{\pm }=0,\text{mbox}\frac{d^2{\chi }_{\pm }}{d{z}^2}-\frac{d\ln |\tilde{\mu }|}{dz}\cdot \frac{d{\chi }_{\pm }}{dz}+k_0^2\tilde{\mu }\tilde{M}{\chi }_{\pm }=0;$$

$$\tilde{M}(z)=\frac{M(z)}{M_0}=1-\frac{e\varphi (z)}{E+m{c}^2},\text{mbox}\tilde{\mu }(z)=\frac{\mu (z)}{{\mu }_0}=1-\frac{e\varphi (z)}{E-m{c}^2};\text{mbox}{k}_0=\frac{p_0}{\hslash }=\frac{2c\sqrt{{\mu }_0{M}_0}}{\hslash };$$

$$\frac{d^{2}U}{d{z}^2}-\frac{d\ln {\mu }_{per}}{dz}\cdot \frac{dU}{dz}+k^2{\mu }_{per}{\epsilon }_{per}U=0,\text{mbox}\frac{d^{2}V}{d{z}^2}-\frac{d\ln {\epsilon }_{per}}{dz}\cdot \frac{dV}{dz}+k^2{\mu }_{per}{\epsilon }_{per}V=0.$$

$${\chi }_{\pm }=\mp \sqrt{\frac{{\mu }_0}{M_0}}\cdot \frac{i}{k_0\tilde{M}}\frac{d{\Phi }_{\pm }}{dz},\text{mbox}V=-\frac{i}{k{\mu }_{per}}\frac{dU}{dz}.$$

$$\displaystyle-\frac{\hbar^{2}}{2M}\frac{d^{2}}{dz^{2}}\left(\begin{array}[]{cc}\Phi_{+}\\ \Phi_{-}\end{array}\right)=\epsilon\left(\begin{array}[]{cc}\Phi_{+}\\ \Phi_{-}\end{array}\right),\mbox{\hskip 8.53581pt}\left(\begin{array}[]{cc}\chi_{+}\\ \chi_{-}\end{array}\right)=-\frac{i\hbar}{2Mc}\frac{d}{dz}\left(\begin{array}[]{cc}\Phi_{+}\\ -\Phi_{-}\end{array}\right).$$

$$\displaystyle\Psi_{\pm k}^{(+1)}(z,t)=\left(\begin{array}[]{cc}\sqrt{\frac{M}{M+\mu}}\\ 0\\ \pm\sqrt{\frac{\mu}{M+\mu}}\\ 0\end{array}\right)e^{\pm ikz-iE(k)t/\hbar},\mbox{\hskip 14.22636pt}\Psi_{\pm k}^{(-1)}(z,t)=\left(\begin{array}[]{cc}0\\ \sqrt{\frac{M}{M+\mu}}\\ 0\\ \mp\sqrt{\frac{\mu}{M+\mu}}\end{array}\right)e^{\pm ikz-iE(k)t/\hbar},$$

$$\displaystyle\psi_{\pm k}^{(+1)}(z,t)=\left(\begin{array}[]{cc}\mp\sqrt{\frac{\mu}{M+\mu}}\\ 0\\ \sqrt{\frac{M}{M+\mu}}\\ 0\end{array}\right)e^{\pm ikz+iE(k)t/\hbar},\mbox{\hskip 14.22636pt}\psi_{\pm k}^{(-1)}(z,t)=\left(\begin{array}[]{cc}0\\ \pm\sqrt{\frac{\mu}{M+\mu}}\\ 0\\ \sqrt{\frac{M}{M+\mu}}\end{array}\right)e^{\pm ikz+iE(k)t/\hbar}.$$

$${\Psi }_{(+)}(z,t)=\sum _{\sigma =\pm 1}{\int }_0^{\infty }\left[A_{+k}^{(\sigma )}(k){\Psi }_{+k}^{(\sigma )}(z,t)+A_{-k}^{(\sigma )}(k){\Psi }_{-k}^{(\sigma )}(z,t)\right]dk,$$

$${\psi }_{(-)}(z,t)=\sum _{\sigma =\pm 1}{\int }_0^{\infty }\left[a_{+k}^{(\sigma )}(k){\psi }_{+k}^{(\sigma )}(z,t)+a_{-k}^{(\sigma )}(k){\psi }_{-k}^{(\sigma )}(z,t)\right]dk;$$

$$\displaystyle\Psi_{\pm k}^{(\sigma)}\Rightarrow\Psi_{k}^{(\sigma)}(z,t)=u_{+}^{(\sigma)}(k)\mbox{\hskip 2.84526pt}e^{i[kz-E(k)t/\hbar]};\mbox{\hskip 8.53581pt}u_{+}^{(\sigma)}(k)=\left(\begin{array}[]{cc}\sqrt{\frac{M}{M+\mu}}\mbox{\hskip 2.84526pt}v_{(\sigma)}\\ \sqrt{\frac{\mu}{M+\mu}}\mbox{\hskip 2.84526pt}\nu_{k}\sigma_{z}v_{(\sigma)}\end{array}\right);$$

$$\displaystyle\psi_{\pm k}^{(\sigma)}\Rightarrow\psi_{k}^{(\sigma)}(z,t)=u_{-}^{(\sigma)}(k)\mbox{\hskip 2.84526pt}e^{-i[kz-E(k)t/\hbar]};\mbox{\hskip 8.53581pt}u_{-}^{(\sigma)}(k)=\left(\begin{array}[]{cc}\sqrt{\frac{\mu}{M+\mu}}\mbox{\hskip 2.84526pt}\nu_{k}\sigma_{z}v_{(\sigma)}\\ \sqrt{\frac{M}{M+\mu}}\mbox{\hskip 2.84526pt}v_{(\sigma)}\end{array}\right),$$

$${\Psi }_{(+)}(z,t)=\sum _{\sigma =\pm 1}{\int }_{-\infty }^{\infty }{A}_{st}^{(\sigma )}(k){\Psi }_k^{(\sigma )}(z,t)dk,\text{mbox}{\psi }_{(-)}(z,t)=\sum _{\sigma =\pm 1}{\int }_{-\infty }^{\infty }{a}_{st}^{(\sigma )}(k){\psi }_k^{(\sigma )}(z,t)dk;$$

$$\displaystyle f^{(+1)}_{+c}=\left(\begin{array}[]{cc}1\\ 0\\ 1\\ 0\end{array}\right),\mbox{\hskip 8.53581pt}f^{(-1)}_{+c}=\left(\begin{array}[]{cc}0\\ 1\\ 0\\ -1\end{array}\right);\mbox{\hskip 14.22636pt}f^{(+1)}_{-c}=\left(\begin{array}[]{cc}1\\ 0\\ -1\\ 0\end{array}\right),\mbox{\hskip 8.53581pt}f^{(-1)}_{-c}=\left(\begin{array}[]{cc}0\\ 1\\ 0\\ 1\end{array}\right).$$

$${\bar{p}}_z=p_z=\hslash k,\text{mbox}{\bar{H}}_D=E=\sqrt{c^2{p}_z^2+m^2{c}^4},\text{mbox}{\bar{v}}_z=\frac{p_z}{m\sqrt{1+{\left(\frac{p_z}{mc}\right)}^2}}.$$

$${\bar{v}}_z=\frac{p_z}{m_{rel}};\text{mbox}{m}_{rel}=m\sqrt{1+{\left(\frac{p_z}{mc}\right)}^2}=\frac{m}{\sqrt{1-{\left(\frac{{\bar{v}}_z}{c}\right)}^2}}.$$

Full text

1 Introduction

According to the “official” point of view Dir ; Fok ; Bere ; Ber ; Ici , the Dirac theory is a field theory that needs to be quantized; while a consistent Dirac theory describing the quantum dynamics of a single electron is, in principle, impossible. But, in our opinion, it is worth returning to this conclusion, especially since in fact there is still no consensus on the nature of the Dirac equation. This is evidenced, for example, by the recent controversy Bia ; Sil0 around the Dirac theory in the standard and Foldy-Wouthuysen Fol representations.

Note that a successful description of the fine structure of the spectrum of the hydrogen atom was given precisely within the framework of the one-particle Dirac model; and only the Lamb shift requires a multiparticle (field) approach for its explanation. The excellent agreement of the Dirac model with experimental data means that on the scales large enough compared to the Compton wavelength, up to the atomic scales, this one-particle model is quite justified; and the fact that relativistic corrections are small in comparison with the corresponding energy eigenvalues calculated in the framework of the nonrelativistic model suggests that in this problem, the Dirac theory is consistent, in the nonrelativistic limit, with the Pauli theory.

The problem is that it is far from easy to see this agreement at the level of their formalisms, since these two theories have different symmetries. The key role here is played by the fact that in the nonrelativistic theory, the operation of spatial inversion does not lead to any new physical results, while in the relativistic theory the inclusion of inversion in the symmetry group requires the simultaneous consideration of a pair of spinors (see Ber , p. 94). In other words, in the Pauli theory the state of a particle is given by a two-component spinor, while in the Dirac theory this is done with the help of two two-component spinors – the large and small components of the Dirac bispinor in the standard representation, which, unlike the Pauli spinor, obeys the system of four first-order differential equations (DE), rather than two DE of the second order.

The consequences of doubling the number of wave functions and equations turned out to be so unusual that all known attempts to give them a consistent physical interpretation within the framework of a one-particle quantum theory did not lead to success. Among these consequences, we single out the following three: (1) the four-component wave function is “numbered”, in addition to the spin, by the (relative) intrinsic parity; but this doubling of quantum numbers creates an unusual situation: on the one hand, since this function describes the states of an electron, both of its quantum “numbers” should be considered as intrinsic degrees of freedom of this particle (see, for example, Fok ; Lip ); on the other hand, this would contradict the Pauli theory that predicts the existence of only one intrinsic degree of freedom for an electron; in addition, one must take into account the fact that the number of intrinsic degrees of freedom of a particle does not depend on the velocity with which it moves; (2) the transition from a two-component wave function to a four-component one leads to the fact that in the Dirac theory, instead of $(2\times 2)$-matrix operators of the Pauli theory, there appear $(4\times 4)$-matrix operators; among them there are new operators that have no analogs in the Pauli theory, and there are some that are known (for example, the velocity operator), but they acquire so unusual properties in the space of four-component wave functions, that the question of their physical interpretation remains open to this day; (3) the new equation has solutions not only with positive, but also with negative energies; this calls into question the possibility of applying the well-known quantum mechanical interpretation of the energy eigenvalues in the Dirac theory.

Our goal is to consider all these issues using the example of an electron in an external electric field which is inhomogeneous along the direction of a particle motion. We follow the approach Chu , according to which, at a fixed energy, the quantum dynamics of an electron in the four-dimensional space-time is equivalent to the quantum dynamics of the ’heavy’ and the ’light’ quasiparticles in the usual three-dimensional space, which are described, respectively, by the large and small components of the Dirac bispinor with the positive and negative intrinsic parities. This quantity is not the intrinsic degree of freedom of the electron (Section 2). In the case considered, the division into the subensembles of heavy and light quasiparticles reflects the fact that, at a fixed energy and $z$-component of spin, the quantum dynamics of the electron in four-dimensional space-time is analogous to the propagation of a linearly polarized plane monochromatic electromagnetic wave in a nonuniform medium with dispersion; the subensembles of heavy and light quasiparticles are, respectively, analogs of the electric and magnetic components of this wave (Section 3). In Section 4 it is shown that the general solution of the Dirac equation for a free electron contains only solutions with positive energies: states with negative energies are in principle unavailable for a free electron with a positive energy, since spontaneous transitions of such an electron into the region of negative energies contradict the energy conservation law (Section 5). An explanation of the unusual properties of the velocity operator is given in Section 6.

2 The concepts of ’heavy’ and ’light’ Schrödinger quasiparticles in the Dirac theory

Let us consider the case when an electron moves along the $OZ$-axis in an external scalar field $\phi(z)$. In the standard representation, the Dirac equation for this quasi-$(1+1)$-dimensional system has the form

[TABLE]

where \Phi=\left(\begin{array}[]{cc}\Phi_{+}\\ \Phi_{-}\end{array}\right) and \chi=\left(\begin{array}[]{cc}\chi_{+}\\ \chi_{-}\end{array}\right) are, respectively, the large and small components of the bispinor \Psi=\left(\begin{array}[]{cc}\Phi\\ \chi\end{array}\right); \alpha_{z}=\left(\begin{array}[]{cc}0&\sigma_{z}\\ \sigma_{z}&0\end{array}\right), \beta=\left(\begin{array}[]{cc}\textbf{1}&0\\ 0&-\textbf{1}\end{array}\right), $\sigma_{z}$ is the Pauli matrix, $m$ is the electron mass, and $\hat{p}_{z}=-i\hbar\mbox{\hskip 2.84526pt}d/dz$. Since the external field does not depend on time, the search for a general solution to this equation is reduced to solving the stationary Dirac equation

[TABLE]

As was shown in Chu , this equation can be written in the form of the “Pauli equation” for the large component $\Phi$ to be a two-component wave function describing the state of a “heavy Schrödinger quasiparticle” with the effective mass $M(E,z)=[E+mc^{2}-e\phi(z)]/(2c^{2})$ and energy $\epsilon(E)=E-mc^{2}$, as well as the equality connecting the small component with the large –

[TABLE]

In addition, it can be written in the form of the “Pauli equation” for the small component $\chi$ to be a two-component wave function describing the state of the “light quasiparticle” with the effective mass $\mu(E,z)=[E-mc^{2}-e\phi(z)]/(2c^{2})$ and energy $\epsilon^{\prime}(E)=E+mc^{2}$, as well as the equality connecting the large component with the small –

[TABLE]

Both these quasiparticles have the same electrical charge.

Thus, in the four-dimensional space-time the quantum dynamics of an electron with a given energy is equivalent to two interconnected quantum dynamics in the ordinary three-dimensional space – the quantum dynamics of the heavy quasiparticle and the quantum dynamics of the light quasiparticle. It is important to emphasize that the Dirac equation (2) is not equivalent to the system of the “Pauli equations” (3) and (4) if they are considered without the equalities connecting the large and small components. A theory, where these components are independent of each other, a priori violates the symmetry of the Dirac equation.

The well-known “Dirac theory” in the Foldy-Wouthuysen representation is just such a theory. The transformation underlying this representation differs from the identity transform only when it acts on functions that do not obey the Dirac equation. Otherwise, the bispinor $\Psi_{+}(\mathbf{x})$ in Fol (see Exp. (15)) is identically equal to $\Psi(\mathbf{x})$, and the bispinor $\Psi_{-}(\mathbf{x})$ is identically equal to 0. Thus, either the Foldy-Wouthuysen representation is identical to the Dirac one, but then the large and small components are obviously related to each other; or this representation cannot be considered as one of unitary representations of the Dirac theory (see also Bia ).

As is known Bere , the components of the Dirac bispinor are “numbered” by the quantum numbers $\sigma$ and $\eta_{P}$, which define the spin component along the preferred direction, as well as the relative internal parity; $\sigma=\pm 1$ and $\eta_{P}=\pm 1$. This follows from the transformation properties of the bispinor with respect to the action of the helicity $\hat{\Lambda}_{\vec{p}}$ and parity (spatial inversion) $\hat{P}$ operators

[TABLE]

when they are considered in the rest frame of the particle; the operator $\hat{\Lambda}_{\vec{p}}$ describes the projection of the spin $\frac{\hbar}{2}\hat{\vec{\Sigma}}$ onto the direction $\vec{n}=\hat{\vec{p}}/|\vec{p}|$; $a^{2}=\pm 1$ (about two possible definitions of the operator $\hat{P}$ see Ber ); $\hat{P}_{0}$ is the orbital parity operator: $\hat{P}_{0}f(\vec{r},t)=f(-\vec{r},t)$, where $f(\vec{r},t)$ is an arbitrary function.

In the rest frame, $\hat{\Lambda}_{\vec{p}}\Rightarrow\hat{\Lambda}_{0}=\vec{\Sigma}\vec{n}_{0}$ is the operator of the spin component along the direction $\vec{n}_{0}$ (which can be chosen arbitrarily), and $\hat{P}\Rightarrow\hat{P}_{int}=a\beta$ is an intrinsic parity operator (according to Ber , one should speak of relative intrinsic parity, but we will omit the word ’relative’). And since the operators $\hat{\Lambda}_{0}$ and $\hat{P}_{int}$ commute with each other, they have a common set of eigenvectors $\Psi_{(\sigma,\eta_{P})}$. If the vector $\vec{n}_{0}$ is directed along the $OZ$ axis, then this set is

[TABLE]

bispinors $\Psi_{(+1,+1)}$ and $\Psi_{(+1,-1)}$ correspond to the eigenvalue $+\hbar/2$ of the operator $\hat{\Lambda}_{0}$, while the bispinors $\Psi_{(-1,+1)}$ and $\Psi_{(-1,-1)}$ correspond to the eigenvalue $-\hbar/2$. At the same time, the eigenvalue $+a$ of the operator $\hat{P}_{int}$ is associated with the bispinors $\Psi_{(+1,+1)}$ and $\Psi_{(-1,+1)}$, while the eigenvalue $-a$ is associated with the bispinors $\Psi_{(+1,-1)}$ and $\Psi_{(-1,-1)}$.

Thus, the heavy quasiparticle has the positive intrinsic parity $+a$, and the light quasiparticle has the negative intrinsic parity $-a$; the heavy and light spin-up quasiparticles are described, respectively, by the odd components $\Phi_{+}$ and $\chi_{+}$ of the Dirac bispinor, while the heavy and light spin-down quasiparticles are described by its even components $\Phi_{-}$ and $\chi_{-}$, respectively. Note that the intrinsic parity is not the second, in addition to the spin, intrinsic degree of freedom of the electron, because there is a relationship between the large and small components – the subensembles of the heavy and light quasiparticles form a single whole which can not be treated as a statistical mixture; the physical meaning of this ’single whole’ will be explained in the next section.

3 Optical-mechanical analogy

The role of the subensembles of the heavy and light quasiparticles in the description of the quantum dynamics of a relativistic electron is most clearly demonstrated by the optical-mechanical analogy, which is presented below (see also Ici ). Let us write Eqs. (3) and (4) for the components $\Phi_{+}$, $\chi_{+}$, $\Phi_{-}$ and $\chi_{-}$ in the form

[TABLE]

If to multiply the left and right sides of the first and second equations by $2M$ and $2\mu$, respectively, and to combine the right-hand sides of the equations with expressions on the left-hand sides containing the potential $\phi(z)$, then, with taking into account the expressions for the effective masses, we obtain

[TABLE]

where

[TABLE]

$M_{0}=(E+mc^{2})/(2c^{2})$, $\mu_{0}=(E-mc^{2})/(2c^{2})$.

These equations coincide in form with Eqs. (15) and (16) in Born (see p. 79), which describe the dynamics of the electrical component $E_{x}=U(z)\exp(-i\omega t)$ and the magnetic component $H_{y}=V(z)\exp(-i\omega t)$ of the electromagnetic TE-wave ($\omega=ck$), which moves along the $OZ$-axis and scatters on a dielectric layer inhomogeneous in this direction::

[TABLE]

That is, the equations for the $\Phi_{+}$ and $\Phi_{-}$ components which describe the heavy quasiparticles with $\sigma=+1$ and $\sigma=-1$, coincide in form with the equations for the electric component of the electromagnetic waves with the right and left linear polarization, respectively; while the equation for the $\chi_{+}$ and $\chi_{-}$ components which describe the light quasiparticles with $\sigma=+1$ and $\sigma=-1$, coincide in form with the equations for the magnetic component of the electromagnetic waves with the right and left linear polarization, respectively. Here, the relative effective mass $\tilde{\mu}$ is analogous to the relative permittivity $\varepsilon_{per}$, and the relative effective mass $\tilde{M}$ is analogous to the relative magnetic permeability $\mu_{per}$.

The link between spinors $\Phi_{\pm}$ and $\chi_{\pm}$ (see (3)) is similar to the link between the electric and magnetic components of the electromagnetic TE-wave (see Eq. (13b) in Born ):

[TABLE]

The presence of the factor $\sqrt{\mu_{0}/M_{0}}$ (its values lie in the interval $[0,1)$) in the first relation reflects the fact that there is no complete analogy between the quantum ensemble of the Dirac electron and the electromagnetic wave, since the speed of an electron never reaches the speed of light. If $E=mc^{2}$, then the $\chi_{+}$ and $\chi_{-}$ components are equal to zero (like the magnetic field of a resting electron). The contributions of the subensembles of the heavy and light quasiparticles are equal only in the limit $|p_{z}|\to\infty$.

From this analogy it follows that the Dirac electron can propagate in the external field if $\tilde{M}\tilde{\mu}>0$; that is, just in the same way, as an electromagnetic wave can propagate in a dispersive medium if $\varepsilon_{per}\mu_{per}>0$. In particular, Klein tunneling in the region where both effective masses are negative (see Chu ) is analogous to the propagation of an electromagnetic wave in a medium with the negative permittivity and negative permeability. In both cases, the speed of energy transfer and the phase speed of the wave have opposite directions.

Thus, the doubling of the number of wave functions, in the transition from the Pauli theory to the Dirac one, is due to the fact that, in the four-dimensional space-time, the quantum dynamics of an electron with a given energy and $z$-component of spin is analogous to propagation in a dispersive medium of a plane monochromatic electromagnetic wave with a given linear polarization (see also section 5). This analogy reveals the physical meaning of the link between the large and small components of the Dirac bispinor and, therefore, clarifies the question of why the doubling of the number of wave functions in the Dirac theory does not mean the appearance of a second intrinsic degree of freedom for the electron. From the point of view of this analogy, the idea of separating the large and small components of the Dirac bispinor, as is done in the Foldy-Wouthuysen theory, has no physical sense.

4 Plane-wave solutions of the Dirac equation for a free electron

Let us consider Eq. (2) (see also Eq. (3)) for $\phi=0$:

[TABLE]

Eqs. (34) for the components of $\Phi$ have two independent particular solutions $e^{ikz}$ and $e^{-ikz}$, where $k=\sqrt{E^{2}-m^{2}c^{4}}/c\hbar$; thus, the eigenvalue $E$ is four-fold degenerate.

Note that the expression for $k$ makes sense not only for positive, but also for negative energies. Therefore, leaving for the time being the question of the physical meaning of states with negative energy, we write down four independent solutions for this case as well. If the parameter $k$ is considered independent ($k>0$) and $E(k)=\sqrt{(\hbar kc)^{2}+m^{2}c^{4}}$, then for the positive energy $E(k)$ we have

[TABLE]

where $M=[E(k)+mc^{2}]/(2c^{2})$ and $\mu=[E(k)-mc^{2}]/(2c^{2})$; for the negative energy $-E(k)$

[TABLE]

All the eight solutions (43) and (52) at $t=0$ are orthogonal to each other in the state space with the scalar product $\langle\psi|\phi\rangle=\int_{-\infty}^{\infty}\psi^{\dagger}(z)\phi(z)dz$. Here the solutions with $e^{ikz}$ and $e^{-ikz}$ ($k>0$) are orthogonal, since they are eigenfunctions of the momentum operator, which correspond to the different eigenvalues $+\hbar k$ and $-\hbar k$. Based on them, we construct a pair of solutions orthogonal to each other

[TABLE]

[TABLE]

we assume that the functions $A^{(\sigma)}_{\pm k}(k)$ and $a^{(\sigma)}_{\pm k}(k)$, which are determined by the initial state at $t=0$, belong to the space $C^{\infty}_{0}(0,\infty)$, consisting of infinitely differentiable functions identically equal to zero in the interval $(-\infty,0]$; for any integer $n>0$ all such functions, when $k\to 0$ and $k\to\infty$, tend to zero faster than the functions $k^{n}$ and $1/k^{n}$, respectively. The solution (53), constructed with making use of the basis functions (43), will be considered as a general solution of the Dirac equation in this problem. As for the solution (54), it should be discarded as having no physical meaning (see Section 5).

Now let’s compare our approach with the standard one (see Sections 1.1.3 and 1.1.4 in Bere and paragraph 23 in Ber ). To do this, let us rewrite the solutions (43) and (52), assuming that the parameter $k$ can now take not only positive, but also negative values. For the positive energy we obtain

[TABLE]

for the negative energy –

[TABLE]

where v_{(+1)}=\left(\begin{array}[]{cc}1\\ 0\end{array}\right), v_{(-1)}=\left(\begin{array}[]{cc}0\\ 1\end{array}\right); $\nu_{k}=k/|k|$. These solutions coincide with those of the standard approach Bere (see Sections 1.1.3 and 1.1.4) if to consider them for $\vec{p}=(0,0,p_{z})$. The difference arises in the construction of localized states from these stationary states.

In our approach, for each value of $\sigma$, two wave packets are formed (one consists of waves that move to the right along the $OZ$-axis, another consists of waves that move in the opposite direction). However, in the standard approach, only one packet is generated for each $\sigma$:

[TABLE]

the functions $A^{(\sigma)}_{st}(k)$ and $a^{(\sigma)}_{st}(k)$ are assumed to belong to the Schwartz space. (According to our approach, these functions form a narrower class of functions than the Schwartz space: $A^{(\sigma)}_{st}(k)=A^{(\sigma)}_{+k}(k)$ $a^{(\sigma)}_{st}(k)=a^{(\sigma)}_{+k}(k)$ $k>0$; $A^{(\sigma)}_{st}(k)=A^{(\sigma)}_{-k}(-k)$ $a^{(\sigma)}_{st}(k)=a^{(\sigma)}_{-k}(-k)$ $k<0$; $A^{(\sigma)}_{st}(0)=a^{(\sigma)}_{st}(0)=0$; however, this difference in the definition of the weight functions $A^{(\sigma)}_{st}(k)$ and $a^{(\sigma)}_{st}(k)$ is not related to the main topic of our paper and is not considered in more detail here.) The general solution, according to Bere ; Ber , is given by the superposition $\Psi_{(+)}(z,t)+\psi_{(-)}(z,t)$, where the function $\psi_{(-)}(z,t)$ that describes the nonphysical electron states with negative energies is expressed (with making use of the charge conjugation transformation) through the positron states with positive energies.

5 The role of states with negative energies in the Dirac theory

Since these two different general solutions imply fundamentally different physical interpretations of the Dirac equation, we consider in detail those arguments that are used in the standard approach against the exclusion of states with negative energies from the general solution of this problem, as well as against the one-particle formulation of the Dirac theory. They are as follows (see p. 13 in Bere ):

• (A)

The existence of solutions of the Dirac equation of two types - with positive and negative frequencies - is of fundamental importance. It leads to the conclusion that in relativistic quantum mechanics it is impossible to preserve the usual interpretation of nonrelativistic quantum mechanics, according to which the eigenvalues of the Hamiltonian have the meaning of the values of the particle energy. Indeed, the frequencies $\omega$ are the eigenvalues of the Hamiltonian of a free electron. Therefore, if the usual interpretation of the eigenvalues of the Hamiltonian were valid, then this would mean the existence of states with negative energy for a free electron and the absence of the lowest energy state. In turn, it would follow from this that when interacting with other particles, the electron could give up its energy indefinitely, passing to ever lower energy states, which is physically absurd.

• (B)

Therefore, we can say that there is a certain unified particle that can be in two charge states that differ in the sign of the charge. This unified particle should be described by the Dirac equation, and it is natural to assume that positive and negative frequencies correspond to its two charge states. In other words, we will associate electronic states with solutions with a positive frequency, and positron states with solutions with a negative frequency.

The arguments and inferences highlighted in bold in the point (A) regarding the role of solutions with negative energy were first expressed by Dirac himself and led him to the concept that is now commonly referred to as the“Dirac Sea” Dir . However, as will be seen from what follows, the direct transfer of the usual physical interpretation of the eigenvalues of operators onto negative energy-eigenvalues is baseless and leads to an incorrect understanding of the role of negative-energy states in the Dirac theory.

Indeed, if the usual interpretation of the eigenvalues of the Hamiltonian is applicable in this problem, then for a free electron this does not at all mean the existence of physical states with negative energy and the absence of a state with the lowest energy. The fact is that spontaneous transitions of a free electron from states with positive energy to states with negative energy are, in principle, impossible – during such transitions an electron at rest with an energy equal to $mc^{2}$ should emit photons with an energy exceeding this value, but this would contradict the law of conservation of energy. Thus, the states with negative energy are in principle inaccessible to a free electron – they are alien to it, and there is no need to include them in the general solution of the problem for a free particle.

The role of states with negative energy in the Dirac theory, as well as the nature of spontaneous transitions in the region $E<0$, can be clarified with making use of the charge conjugation transformation, which transforms the states $\Psi_{e}$ and $\Psi_{p}$ of an electron and positron with a given positive energy into the states $\psi_{p}$ and $\psi_{e}$, respectively, of “positron” and “electron” with negative energy, and vice versa. This transformation can be understood as the operation of “specular reflection” of the physical states of both particles relative to the level $E=0$ with a simultaneous change in the sign of the electric charge (as well as the signs of $\sigma$ and $\eta_{P}$): in this case the (nonphysical) electron and positron states, $\psi_{e}$ and $\psi_{p}$ with negative energy are, respectively, “mirror images” of (physical) electron and positron states $\Psi_{e}$ and $\Psi_{p}$ with positive energy. Thus, the concepts of ’upper’ and ’lower’ energy levels in the region $E>0$ acquire the opposite meaning in the region $E<0$. In particular, the “mirror images” of the (physical) spontaneous transitions of a positron in the region $E>mc^{2}$ from the upper levels to the lower ones are nothing but the transitions of an “electron” from lower to upper levels in the region $E<-mc^{2}$. That is, contrary to the arguments in point (A), the usual interpretation of nonrelativistic quantum mechanics does not lead to a physically absurd scenario of spontaneous transitions in the region $E<-mc^{2}$.

The highlighted inference in point (B) should also be recognized as erroneous, since the (single) Dirac equation, in principle, cannot describe a “unified particle that can be in two charge states”. This contradicts the analogy between the systems of Eqs. (23) and (24) described in Section 3.

The fact is that this analogy can be considered not only as an optical-mechanical one, but also as an analogy between the equations (24), describing the electric and magnetic components of the electromagnetic wave field, and the equations (23), describing the “electric” and “magnetic” components of the Dirac field. From this analogy it follows that the “electric” and “magnetic” components of the Dirac field form a single whole. That is, the Dirac equation does not describe the field of a ’unified particle with two charge states’. It describes the field of the very electron – a particle that can be in only one charge state. Moreover, this field is a superposition of waves with only positive (physical) frequencies.

In other words, describing a ’unified particle with two charge states’ requires not one the Dirac equation, but two. In this regard, of interest is the modified standard (’field’) formulation of the Dirac theory, developed in Git . In this approach, this ’unified particle’ is described by the Schrödinger equation (5.32) for an eight-component spinor, which is equivalent to two Dirac equations – one for the Dirac field with a positive electric charge, and the other for the Dirac field with a negative charge. An arbitrary state of a ’unified particle’ is a superposition of the Hamiltonian eigenvectors corresponding only to positive energy-eigenvalues.

6 Velocity operator in the Dirac theory

So, in the proposed approach, all four components of the Dirac bispinor, as well as the Dirac equation itself, describe the state of a particle that can be in only one charge state (this is either the electron, or the positron, or any other fermion), and the general solution (53), the Dirac equation for a free particle contains states only with positive energy. As a consequence, within the framework of this one-particle approach it is possible to explain (see Chu ) the well-known Klein paradox; as regards the so-called Zitterbewegung phenomenon (see, for example, pp. 952 in Mess ) – a rapidly oscillating change in time of the mean value of the coordinate operator $\vec{r}$ of a free particle, in this approach it does not arise in principle due to the absence of contributions of non-physical states with negative energies in the general solution (53).

However, in the standard representation there is one more property of the operator $\vec{r}$, which is treated as a serious flaw of this theory (see, for example, p. 343 in Dir , p. 305 in Fok , and also Sil0 The fact is that the velocity operator $\vec{v}=d\vec{r}/dt$ is not equal to the ratio of the momentum operator and the relativistic mass, as is the case in the special theory of relativity (STR); instead, it is represented by the matrix operator $c\vec{\alpha}$ whose different components do not commute with each other and have only two eigenvalues $+c$ and $-c$, which also contradicts SRT.

It should be admitted that the problem of eigenvalues and eigenfunctions of the velocity operator in the Dirac theory loses the physical meaning prescribed by nonrelativistic quantum mechanics. Now the eigenvalues of the velocity operator acquire a different physical meaning, and this is easy to establish. To do this, note that the velocity operator $\hat{v}_{z}=c\alpha_{z}$ has the following eigenvectors corresponding to the eigenvalues $+c$ and $-c$:

[TABLE]

It is easy to check that the bispinors $f^{(+1)}_{+c}$ and $f^{(-1)}_{+c}$ coincide, up to phase, with the bispinors $\Psi_{\infty}^{(+1)}(z,t)$ and $\Psi_{\infty}^{(-1)}(z,t)$ (see (43)) describing the states with the momentum $p_{z}=\infty$; the bispinors $f^{(+1)}_{-c}$ and $f^{(-1)}_{-c}$ coincide, up to phase, with the wave functions $\Psi_{-\infty}^{(+1)}(z,t)$ and $\Psi_{-\infty}^{(-1)}(z,t)$, describing the states with the momentum $p_{z}=-\infty$. As we see, the eigenvectors of the velocity operator describe the states of a particle with the limiting momentum values $p_{z}=\pm\infty$, and its eigenvalues are the limiting values of the velocity $\pm c$. Now it becomes clear why all three components of the velocity operator do not commute with each other: if they were commuting, then the absolute values of all three velocity components of a particle with infinite momentum would be equal to $c$, which would contradict STR.

As for the link between the velocity, momentum and energy, characteristic of STR, in the Dirac theory it is realized between the mean values of the corresponding operators. Consider, for example, the stationary state $\Psi_{+k}^{(+1)}(z,t)$. The average values $\bar{p}_{z}=\langle\Psi_{+k}^{(+1)}|\hat{p}_{z}|\Psi_{+k}^{(+1)}\rangle$ and $\bar{H}_{D}=\langle\Psi_{+k}^{(+1)}|\hat{H}_{D}|\Psi_{+k}^{(+1)}\rangle$ are obvious, and $\bar{v}_{z}=\langle\Psi_{+k}^{(+1)}|\hat{v}_{z}|\Psi_{+k}^{(+1)}\rangle$ can be easily calculated:

[TABLE]

The last magnitude can also be written as

[TABLE]

Thus, for a free particle with a precisely specified momentum and energy, the Dirac theory, as a quantum theory of a relativistic particle, exactly reproduces the link between the velocity and momentum of a particle, which is characteristic of STR (and what is important – there is no need to go over to the semiclassical limit). It should be also stressed that the large and small components of the Dirac bispinor are equally important in calculating the average values of physical quantities in this theory.

7 Conclusion

For an external static field, which slowly varies on the scales comparable with the Compton wavelength, a one-particle formulation of the Dirac theory in the standard representation is developed. According to this formulation the quantum dynamics of an electron with a given energy $E$ in the four-dimensional space-time is equivalent to two quantum dynamics in the ordinary three-dimensional space: one of them, described by the large component of the Dirac bispinor, is the quantum dynamics of the heavy quasiparticle with the effective mass $M(E,z)=m+\mu(E,z)$, energy $\epsilon(E)=E-mc^{2}$ and positive (relative) intrinsic parity; the other, described by the small component of the Dirac bispinor, is the quantum dynamics of the light quasiparticle with the effective mass $\mu(E,z)$, energy $\epsilon(E)+2mc^{2}$, and negative (relative) intrinsic parity.

Relative intrinsic parity, distinguishing the large and small components, is not the second intrinsic degree of freedom of an electron. It is shown that, given energy and spin-component, the quantum dynamics of the Dirac electron, under an external scalar electric field, nonuniform along the direction of a particle motion, is analogous to the propagation of a plane monochromatic electromagnetic wave with a given linear polarization in a dispersive medium, nonuniform in the direction of a wave motion. The quantum ensemble of such an electron has a structure to be similar to that of this wave: the subensemble of heavy quasiparticles is analog of the electric component of the wave, and the subensemble of light quasiparticles is analog of the magnetic component of the wave. From this it follows that the subensembles of heavy and light quasiparticles, in principle, cannot be separated from each other – they form a quantum ensemble which is not a statistical mixture.

It should also be emphasized that the properties of the velocity operator in the Dirac theory, which look as paradoxical in existing approaches, receive a simple physical explanation within the framework of the presented one-particle formulation; where states with negative energy have no physical meaning, and the large and small components of the Dirac bispinor are equally essential in the description of the quantum dynamics of an electron in the four-dimensional space-time.