Novel Principles and the Charge-Symmetric Design of Dirac’s Quantum Mechanics: I. Enhanced Eriksen’s Theorem and the Universal Charge-Index Formalism for Dirac’s Equation in (Strong) External Static Fields

Abstract

The presented enhanced version of Eriksen’s theorem defines an universal transform of the Foldy–Wouthuysen type and in any external static electromagnetic field (ESEMF) reveals a discrete symmetry of Dirac’s equation (DE), responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism (CIF) obeying the charge-index conservation law (CICL). Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac’s quantum mechanics (DQM), which resolves all the riddles of the standard DE theory (SDET). Besides the abstract CIF algebra, the paper discusses: (1) the novel accurate charge-symmetric definition of the electric-current density; (2) DE in the true-particle representation, where electrons and positrons coexist on equal footing; (3) flawless “natural” scheme of second quantization; and (4) new physical grounds for the Fermi–Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein’s tunneling (KT) in Klein’s zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier.

Appendix: The Charge-Conjugation Transform and True Positron States

Let us consider the function $\widetilde{\psi }$ which satisfies DE with the reversed sign of e:

$$\begin{aligned} \mathrm{i}\hslash \,\frac{\partial \widetilde{\psi }}{\partial t} =\widetilde{\mathcal {H}}\widetilde{\psi }, \quad \widetilde{\mathcal {H}}=\hat{\mathcal {H}}_{e\rightarrow -e}. \end{aligned}$$

(117)

Taking into account the relations²⁷

$$\begin{aligned} \widetilde{\mathcal {H}}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {H}}{^*}\hat{\gamma }^{\phantom {\dag }}_2,\;\;\hat{\mathcal {H}}=\hat{\gamma }^{\phantom {\dag }}_2\widetilde{\mathcal {H}}{^*}\hat{\gamma }^{\phantom {\dag }}_2,\;\; \hat{\gamma }^{\phantom {\dag }}_2=\left( \!\begin{array}{cl} 0 &{} \hat{\sigma }_\mathrm{y}\\ -\hat{\sigma }_\mathrm{y}&{} 0 \end{array}\!\right) ,\;\; \hat{\sigma }_\mathrm{y}=\left( \begin{array}{cr} 0 &{} -\mathrm{i}\\ \mathrm{i}&{} 0 \end{array}\right) , \end{aligned}$$

(118)

we easily get the well known charge-conjugation transform $\hat{C}$ [2, 11] connecting $\widetilde{\psi }$ with a solution $\psi$ of the initial DE:²⁸

$$\begin{aligned} \widetilde{\psi }=\hat{C}\psi \equiv \hat{\gamma }^{\phantom {\dag }}_2\psi ^*,\quad \psi =\hat{C}\widetilde{\psi }\equiv \hat{\gamma }^{\phantom {\dag }}_2\widetilde{\psi }{^*}. \end{aligned}$$

(119)

By analogy with Sect. 5, consider the collection of operators arising in the block-diagonalization of Hamiltonian $\widetilde{\mathcal {H}}$:

$$\begin{aligned} \widetilde{\mathcal {A}}=\hat{\mathcal {A}}_{e\rightarrow -e},\;\;\widetilde{\mathcal {B}}=\hat{\mathcal {B}}_{e\rightarrow -e},\;\; \widetilde{K}^\mathrm{a}=\hat{K}^\mathrm{a}_{e\rightarrow -e},\;\;\widetilde{K}^\mathrm{b}=\hat{K}^\mathrm{b}_{e\rightarrow -e}. \end{aligned}$$

(120)

Superscripts “$\mathrm{a}$” and “$\mathrm{b}$” here, as before, refer to physical and nonphysical states, respectively, of Hamiltonian $\widetilde{\mathcal {H}}$.  It seems evident that the charge-conjugation transform (119), applied to nonphysical states of Hamiltonian $\hat{\mathcal {H}}$,  generates the physical states of Hamiltonian $\widetilde{\mathcal {H}}$.  Similarly, the nonphysical states of $\widetilde{\mathcal {H}}$  are the charge-conjugate physical states of $\hat{\mathcal {H}}$.

Inserting $\hat{\mathcal {H}}=\hat{G}^\dag \hat{\mathcal {H}}^\mathrm{o}\hat{G}$ into $\widetilde{\mathcal {H}}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {H}}^*\hat{\gamma }^{\phantom {\dag }}_2$, one can easily obtain

$$\begin{aligned} \widetilde{\mathcal {H}}=\widetilde{G}^\dag \widetilde{\mathcal {H}}^\mathrm{o}\widetilde{G},\quad \widetilde{\mathcal {H}}^\mathrm{o}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {H}}^{\mathrm{o}*}\hat{\gamma }^{\phantom {\dag }}_2=\left( \begin{array}{cc} -\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{b}*}\hat{\sigma }_\mathrm{y}&{} 0 \\ 0 &{} -\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{a}*}\hat{\sigma }_\mathrm{y}\end{array}\right) , \end{aligned}$$

(121)

where the block-diagonalizing unitary matrix $\widetilde{G}$ for Hamiltonian $\widetilde{\mathcal {H}}$ is defined via $\hat{G}$ (44) as

$$\begin{aligned} \widetilde{G}=-\hat{\gamma }^{\phantom {\dag }}_2\hat{G}^*\hat{\gamma }^{\phantom {\dag }}_2,\quad \widetilde{G}^\dag =-\hat{\gamma }^{\phantom {\dag }}_2\hat{G}^\mathrm{T}\hat{\gamma }^{\phantom {\dag }}_2. \end{aligned}$$

(122)

As usual, the superscript $\mathrm{T}$ designates transposed matrices.

In accordance with (44), the operators $\widetilde{\mathcal {A}}$ and $\widetilde{\mathcal {B}}$, implementing the block-diagonalization of $\widetilde{\mathcal {H}}$, can be found immediately from the relation

$$\begin{aligned} \widetilde{G}^\dag =\bigg (\begin{array}{cc}\widetilde{\mathcal {A}}_1 &{} \widetilde{\mathcal {B}}_1 \\ \widetilde{\mathcal {A}}_2 &{} \widetilde{\mathcal {B}}_2 \end{array}\bigg ). \end{aligned}$$

(123)

As a result, we get:

$$\begin{aligned} \widetilde{\mathcal {A}}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {B}}^*\hat{\sigma }_\mathrm{y}, \;\; \widetilde{\mathcal {B}}=-\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {A}}^*\hat{\sigma }_\mathrm{y}, \;\; \widetilde{K}^\mathrm{a}=-\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{b}*}\hat{\sigma }_\mathrm{y}, \;\; \widetilde{K}^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{a}*}\hat{\sigma }_\mathrm{y}. \end{aligned}$$

(124)

It is obvious that the respective $\phi$-functions

$$\begin{aligned} \widetilde{\phi }^\mathrm{a}=\hat{\sigma }_\mathrm{y}\phi ^{\mathrm{b}*},\quad \widetilde{\phi }^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\phi ^{\mathrm{a}*}\end{aligned}$$

(125)

satisfy the equations

$$\begin{aligned} \widetilde{K}^\mathrm{a}\widetilde{\phi }^\mathrm{a}=\widetilde{\mathcal {E}}^\mathrm{a}\widetilde{\phi }^\mathrm{a},\quad \widetilde{K}^\mathrm{b}\widetilde{\phi }^\mathrm{b}=\widetilde{\mathcal {E}}^\mathrm{b}\widetilde{\phi }^\mathrm{b}, \end{aligned}$$

(126)

with the eigenvalues $\widetilde{\mathcal {E}}^\mathrm{a}$, $\widetilde{\mathcal {E}}^\mathrm{b}$ being equal to the respective energies, with opposite sign, of states $\phi ^\mathrm{b}$ and $\phi ^\mathrm{a}$, whose complex conjugate counterparts stand in (125):

$$\begin{aligned} \widetilde{\mathcal {E}}^\mathrm{a}=-\mathcal {E}^\mathrm{b}:\;\hat{K}^\mathrm{b}\phi ^\mathrm{b}=\mathcal {E}^\mathrm{b}\phi ^\mathrm{b};\qquad \widetilde{\mathcal {E}}^\mathrm{b}=-\mathcal {E}^\mathrm{a}:\;\hat{K}^\mathrm{a}\phi ^\mathrm{a}=\mathcal {E}^\mathrm{a}\phi ^\mathrm{a}. \end{aligned}$$

(127)

Summarizing the above considerations, we come to the exact charge-conjugation symmetry relations, connecting physical states of Hamiltonians $\widetilde{\mathcal {H}}$ and $\hat{\mathcal {H}}$ with respective nonphysical ones of $\hat{\mathcal {H}}$ and $\widetilde{\mathcal {H}}$,  and vice versa:

$$\begin{aligned} \begin{array}{lll} \widetilde{\psi }^\mathrm{a}=\widetilde{\mathcal {A}}\widetilde{\phi }^\mathrm{a}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {B}}^*\phi ^{\mathrm{b}*}=\hat{\gamma }^{\phantom {\dag }}_2\psi ^{\mathrm{b}*},\;\; &{}\widetilde{K}^\mathrm{a}=-\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{b}*}\hat{\sigma }_\mathrm{y},\;\;&{}\widetilde{\phi }^\mathrm{a}=\phantom {-}\hat{\sigma }_\mathrm{y}\phi ^{\mathrm{b}*}; \\ \psi ^\mathrm{a}=\hat{\mathcal {A}}\phi ^\mathrm{a}=\hat{\gamma }^{\phantom {\dag }}_2\widetilde{\mathcal {B}}^*\widetilde{\phi }^{\mathrm{b}*}=\hat{\gamma }^{\phantom {\dag }}_2\widetilde{\psi }^{\mathrm{b}*},\;\; &{}\hat{K}^\mathrm{a}=-\hat{\sigma }_\mathrm{y}\widetilde{K}^{\mathrm{b}*}\hat{\sigma }_\mathrm{y},\;\;&{}\phi ^\mathrm{a}=\phantom {-}\hat{\sigma }_\mathrm{y}\widetilde{\phi }^{\mathrm{b}*}; \\ \widetilde{\psi }^\mathrm{b}=\widetilde{\mathcal {B}}\widetilde{\phi }^\mathrm{b}=\hat{\gamma }^{\phantom {\dag }}_2\hat{\mathcal {A}}^*\phi ^{\mathrm{a}*}=\hat{\gamma }^{\phantom {\dag }}_2\psi ^{\mathrm{a}*},\;\; &{}\widetilde{K}^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{a}*}\hat{\sigma }_\mathrm{y},\;\;&{}\widetilde{\phi }^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\phi ^{\mathrm{a}*}; \\ \psi ^\mathrm{b}=\hat{\mathcal {B}}\phi ^\mathrm{b}=\hat{\gamma }^{\phantom {\dag }}_2\widetilde{\mathcal {A}}^*\widetilde{\phi }^{\mathrm{a}*}=\hat{\gamma }^{\phantom {\dag }}_2\widetilde{\psi }^{\mathrm{a}*},\;\; &{}\hat{K}^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\widetilde{K}^{\mathrm{a}*}\hat{\sigma }_\mathrm{y},\;\;&{}\phi ^\mathrm{b}=-\hat{\sigma }_\mathrm{y}\widetilde{\phi }^{\mathrm{a}*}. \end{array} \end{aligned}$$

(128)

As a general result, we can see from (120), (128) an exact relation between paulionic ($\hat{K}^\mathrm{a}$) and pseudo-antipaulionic ($\hat{K}^\mathrm{b}$) Hamiltonians, first published in [1]:

$$\begin{aligned} \hat{K}^\mathrm{b}=-\big (\hat{\sigma }_\mathrm{y}\hat{K}^{\mathrm{a}*}\hat{\sigma }_\mathrm{y}\big )_{e\rightarrow -e}. \end{aligned}$$

(129)

The functions $\widetilde{\psi }^\mathrm{a}$ and $\widetilde{\phi }^\mathrm{a}$ in the above must be regarded as physical wave functions of positrons and antipaulions, respectively, whose energy spectra coincide and tend to $+\infty$ with increasing momentum. Simultaneously, $\widetilde{\psi }^\mathrm{b}$ and $\widetilde{\phi }^\mathrm{b}$ must be treated as nonphysical (unobservable) states of pseudo-electrons and pseudo-paulions, respectively, because their coincident energy spectra extend to $-\infty$. The general relations (128) are, naturally, in full agreement with definitions of electronic and positronic $\psi$-functions in the SDET [2, 11].

In the free space one easily notices that, for a given momentum, an electronic state, as a physical solution of the primary DE, is described by exactly the same $\psi$-function as the positronic state with the mentioned momentum. This fact reflects the general specific feature of the SDET, that physical states with different electric charges here are not orthogonal. Just this circumstance makes the SDET operate with the orthonormal system of eigenstates of the primary DE, a half of which are not realized in nature in their original form. The complete wreck of the STED inevitably happens in strong fields, where the theory is unable to correctly define observable and unobservable states of Dirac’s particle.

The advanced theory exploits the mathematical apparatus (see Sect. 8), in which the physically observable electronic and positronic states are described by orthonormalized wave functions. In the bispinor true-particle representation (see equation (79)) one can see that the subsystem of positronic functions is connected with its standard representation via a simple unitary transform:

$$\begin{aligned} \psi ^\mathrm{b}_\mathrm{u}=\hat{\Gamma }\widetilde{\psi }^\mathrm{a},\quad \widetilde{\psi }^\mathrm{a}=\hat{\Gamma }^\dag \psi ^\mathrm{b}_\mathrm{u}, \quad \hat{\Gamma }=\hat{\mathcal {B}}\widetilde{\mathcal {A}}^\dag , \end{aligned}$$

(130)

which confirms equal rights of electrons and positrons in the true-particle representation.