1 INTRODUCTION

For fermions with mass \(m\) and charge \(e\), moving in an external electromagnetic field, the Dirac equation can be written in the following form

$$\begin{gathered} \left( {{{p}_{0}} - e{{A}_{0}}\left( {{\mathbf{r}},t} \right)} \right)\psi \left( {{\mathbf{r}},t} \right) \\ = \left( {{\mathbf{\alpha }}\left( {{\mathbf{p}} - e{\mathbf{A}}\left( {{\mathbf{r}},t} \right)} \right) + \beta m} \right)\psi \left( {{\mathbf{r}},t} \right). \\ \end{gathered} $$

Hereinafter, we use system of units \(\hbar = c = 1\) and the Minkowski space signature

$${{g}_{{\alpha \beta }}} = diag\left[ {1, - 1, - 1, - 1} \right].$$

\(\psi \left( {{\mathbf{r}},t} \right)\) is fermion bispinor wave function; \({{A}_{0}}\left( {{\mathbf{r}},t} \right),{\mathbf{A}}\left( {{\mathbf{r}},t} \right)\) are potentials of the electromagnetic field; \({{\alpha }^{k}},\beta \) are the four-dimensional Dirac matrices, \(k = 1,2,3\); \({{p}_{0}} = i\frac{\partial }{{\partial t}},{\mathbf{\rho }} = - i\vec {\nabla }\).

Dirac also obtained the following second-order equation

$$\left[ {{{{\left( {{{p}_{0}} - e{{A}_{0}}} \right)}}^{2}} - {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} - {{m}^{2}} + e{\mathbf{\Sigma H}} - ie{\mathbf{\alpha {\rm E}}}} \right]\psi = 0,$$

where \({\mathbf{\Sigma }} = \left( \begin{gathered} {\mathbf{\sigma }}\,\,\,0 \hfill \\ 0\,\,\,{\mathbf{\sigma }} \hfill \\ \end{gathered} \right)\), \({{\sigma }^{k}}\) are the two-dimensional Pauli matrices, \({\mathbf{H}} = {\text{curl}}{\mathbf{A}},\,\,\,{\mathbf{{\rm E}}} = - \frac{{\partial {\mathbf{A}}}}{{\partial t}} - \nabla {{A}_{0}}\) are magnetic and electric fields. Below, we shall consider stationary states, when \({{p}_{0}}\psi = E\psi \), where \(E\) is a fermion energy \(\left( {\psi \left( {{\mathbf{r}},t} \right) = {{e}^{{ - iEt}}}\psi \left( {\mathbf{r}} \right)} \right)\).

2 SELF-ADJOINT SECOND-ORDER EQUATION WITH SPINOR WAVE FUNCTION

Let the bispinor wave function be as following

$$\psi \left( {{\mathbf{r}},t} \right) = \left( \begin{gathered} u\left( {\mathbf{r}} \right) \hfill \\ v\left( {\mathbf{r}} \right) \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}}.$$

Then

$$\begin{gathered} \left( {E - e{{A}_{0}} - m} \right)u = {\mathbf{\sigma }}\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)v, \hfill \\ \left( {E - e{{A}_{0}} + m} \right)v = {\mathbf{\sigma }}\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)u, \hfill \\ \end{gathered} $$

and we can obtain equations either for spinor \(u\), or for spinor \(v\).

For spinor \(u\left( {\mathbf{r}} \right)\) the equation is written as follows:

$$\begin{gathered} \left[ {{{{\left( {E - e{{A}_{0}}} \right)}}^{2}} - {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} - {{m}^{2}} + e{\mathbf{\sigma H}}} \right. \\ \left. { - \frac{1}{{E - e{{A}_{0}} + m}}ie{\mathbf{\sigma {\rm E}\sigma }}\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)} \right]u\left( {\mathbf{r}} \right) = 0. \\ \end{gathered} $$

In the case of stationary states, electromagnetic potentials \({{A}_{0}}\left( {\mathbf{r}} \right),{{A}^{k}}\left( {\mathbf{r}} \right)\) do not depend on time.

2.1. Let \({{A}_{0}}\left( {\mathbf{r}} \right) = 0,\,\,\,{{A}^{k}}\left( {\mathbf{r}} \right) \ne 0\). Then the equation for spinor \(u\left( {\mathbf{r}} \right)\) is self-adjoint and

$$Eu\left( {\mathbf{r}} \right) = \left( { \pm \sqrt {{{m}^{2}} + {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} + e{\mathbf{\sigma H}}} } \right)u\left( {\mathbf{r}} \right).$$

2.2. Now let us consider the case of \({{A}^{k}}\left( {\mathbf{r}} \right) = 0,{{A}_{0}}\left( {\mathbf{r}} \right) \ne 0\). In this case, the last term in the equation for spinor \(u\left( {\mathbf{r}} \right)\) is a non-self-adjoint operator. Let us perform a nonunitary similarity transformation of this equation and spinor \(u\left( {\mathbf{r}} \right)\)

$$\Phi \left( {\mathbf{r}} \right) = gu\left( {\mathbf{r}} \right),$$

where

$$g = {{\left( {E - e{{A}_{0}} + m} \right)}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}.$$

As a result, the equation for spinor \(u\left( {\mathbf{r}} \right)\) is reduced to form

$$\begin{gathered} g\left[ {{{{\left( {E - e{{A}_{0}}} \right)}}^{2}} - {{{\mathbf{p}}}^{2}} - {{m}^{2}} - \frac{1}{{\left( {E - e{{A}_{0}} + m} \right)}}i{\mathbf{\sigma {\rm E}\sigma p}}} \right] \\ \times \,\,{{g}^{{ - 1}}}\Phi \left( {\mathbf{r}} \right) = 0 \\ \end{gathered} $$

and finally, to

$$\begin{gathered} \left[ {{{{\left( {E - e{{A}_{0}}} \right)}}^{2}} - {{{\mathbf{p}}}^{2}} - {{m}^{2}} - \frac{3}{4}\frac{1}{{{{{\left( {E - e{{A}_{0}} + m} \right)}}^{2}}}}{{{\mathbf{{\rm E}}}}^{2}}} \right. \\ \left. { + \,\,\frac{1}{2}\frac{1}{{E - e{{A}_{0}} + m}}{\text{div}}{\mathbf{{\rm E}}} + \frac{1}{{E - e{{A}_{0}} + m}}{\mathbf{\sigma }}\left( {{\mathbf{{\rm E}}} \times {\mathbf{p}}} \right)} \right] \\ \times \,\,\Phi \left( {\mathbf{r}} \right) = 0. \\ \end{gathered} $$

For a central symmetric Coulomb potential, variable separation is allowed in spherical coordinates \(\left( {r,\theta ,\varphi } \right)\)

$$\Phi \left( {r,\theta ,\varphi } \right) = F\left( r \right)\chi _{\kappa }^{{{{m}_{\varphi }}}}\left( \theta \right){{e}^{{i{{m}_{\varphi }}\varphi }}},$$

\(\chi _{\kappa }^{{{{m}_{\varphi }}}}\) are spherical spinors; \({{m}_{\varphi }} = - j, - j + 1,...j,\) is azimuthal component of angular momentum \(j\); \(\kappa \) is the quantum number of the Dirac equation

$$\kappa = \mp 1, \mp 2,... = \left\{ \begin{gathered} - \left( {l + 1} \right),\,\,\,j = l + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2} \hfill \\ \,\,\,\,\,\,\,\,\,l,\,\,\,\,\,\,\,\,\,j = l - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2} \hfill \\ \end{gathered} \right.,$$

\(j,l\) are quantum numbers of the fermion angular and orbital momenta.

Taking into consideration the known relations

$$({\text{a}})\;{{{\mathbf{p}}}^{2}} = {\mathbf{p}}_{r}^{2} + {\mathbf{p}}_{{\theta ,\varphi }}^{2},\;{\text{where}}\;{\mathbf{p}}_{r}^{2} = - \frac{1}{{{{r}^{2}}}}\frac{\partial }{{\partial r}}\left( {{{r}^{2}}\frac{\partial }{{\partial r}}} \right),$$
$$\begin{gathered} {\mathbf{p}}_{{\theta ,\varphi }}^{2}\chi _{\kappa }^{{{{m}_{\varphi }}}} = - \frac{1}{{{{r}^{2}}}}\left( {\frac{{{{\partial }^{2}}}}{{\partial {{\theta }^{2}}}} + {\text{cotan}}\theta \frac{\partial }{{\partial \theta }} + \frac{1}{{{{{\sin }}^{2}}\theta }}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}} \right) \\ \times \,\,\chi _{\kappa }^{{{{m}_{\varphi }}}} = \frac{{\kappa \left( {\kappa + 1} \right)}}{{{{r}^{2}}}}\chi _{\kappa }^{{{{m}_{\varphi }}}}. \\ \end{gathered} $$

(b) \(e{\mathbf{\sigma }}\left( {{\mathbf{{\rm E}}} \times {\mathbf{p}}} \right) = - e\frac{{dV}}{{dr}}\frac{1}{r}{\mathbf{\sigma L}}\), where \({\mathbf{L}} = {\mathbf{r}} \times {\mathbf{p}}\) is the orbital momentum operator.

(c) \( - {\mathbf{\sigma L}}\chi _{\kappa }^{{{{m}_{\varphi }}}} = \left( {\kappa + 1} \right)\chi _{\kappa }^{{{{m}_{\varphi }}}}\), we obtain the Schrödinger type of equation with effective potential for radial function \(F\left( r \right)\)

$$\begin{gathered} \frac{1}{{{{r}^{2}}}}\frac{d}{{dr}}\left( {{{r}^{2}}\frac{d}{{dr}}} \right)F\left( r \right) + \left( {{{{\left( {E - V} \right)}}^{2}} - {{m}^{2}} - \frac{{\kappa \left( {\kappa + 1} \right)}}{{{{r}^{2}}}} - \frac{3}{4}\frac{{{{{\left( {\frac{{dV}}{{dr}}} \right)}}^{2}}}}{{{{{\left( {E - V + m} \right)}}^{2}}}}} \right. \\ \left. { - \,\,\frac{1}{2}\frac{{\frac{{{{d}^{2}}V}}{{d{{r}^{2}}}}}}{{\left( {E - V + m} \right)}} + \frac{1}{{\left( {E - V + m} \right)}}\frac{1}{r}\frac{{dV}}{{dr}}\kappa } \right)F\left( r \right) = 0. \\ \end{gathered} $$

Here \(V\left( {\mathbf{r}} \right) = e{{A}_{0}}\left( {\mathbf{r}} \right).\)

This equation is convenient to use for the analysis of fermion motion in Coulomb fields of different intensity.

3 FOLDY–WOUTHUYSEN REPRESENTATION AND CHIRAL SYMMETRY

For the case of \({{A}^{k}}\left( {\mathbf{r}} \right) \ne 0,{{A}_{0}}\left( {\mathbf{r}} \right) = 0\), the equation for spinor \(u\left( {\mathbf{r}} \right)\) can be written in the Hamiltonian form

$$Eu\left( {\mathbf{r}} \right) = Hu\left( {\mathbf{r}} \right),$$

where \(H = \pm \sqrt {{{m}^{2}} + {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} + e{\mathbf{\sigma H}}} .\)

This equation can be easily compared to the Dirac equation in the Foldy–Wouthuysen representation (FW). In this representation, the bispinor wave function can be written as

$$\begin{gathered} {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} u\left( {\mathbf{r}} \right) \hfill \\ \,\,\,\,0 \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}},\,\,\,\,E > 0, \hfill \\ {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} \,\,\,\,0 \hfill \\ v\left( {\mathbf{r}} \right) \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}},\,\,\,\,E < 0, \hfill \\ \end{gathered} $$

and the Hamiltonian

$${{H}_{{{\text{FW}}}}} = \beta \sqrt {{{m}^{2}} + {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} + e\Sigma {\mathbf{H}}} .$$

In the case when \({{A}^{k}}\left( {\mathbf{r}} \right) = 0,{{A}_{0}}\left( {\mathbf{r}} \right) \ne 0\), the equation for spinor \(u\left( {\mathbf{r}} \right)\) cannot be written in the closed Hamiltonian form

$$\begin{gathered} \left[ {{{{\left( {E - e{{A}_{0}}} \right)}}^{2}} - {{{\mathbf{p}}}^{2}} - {{m}^{2}} + e{\mathbf{\sigma }}{{{\mathbf{H}}}^{{^{{^{{^{{}}}}}}}}}} \right. \\ \left. { - \,\,\frac{1}{{E - e{{A}_{0}} + m}}ie{\mathbf{\sigma {\rm E}\sigma p}}} \right]u\left( {\mathbf{r}} \right) = 0. \\ \end{gathered} $$

This can be done only using the successive approximation method. First, we substitute the value \(E = {{E}_{0}}\) into the denominator of this equation’s last term with \({{A}^{k}}\left( {\mathbf{r}} \right) = 0\). Then, we perform a transformation with \({{g}_{0}}\), which provides self-adjointness to the equation. Here we obtain operator \({{H}_{1}}\)\(\left( {{{E}_{1}}\Phi \left( {\mathbf{r}} \right) = {{H}_{1}}\Phi \left( {\mathbf{r}} \right)} \right)\). Then, this process repeats with value \(E = {{H}_{1}}\) in the denominator of the last term of the equation for spinor \(u\left( {\mathbf{r}} \right)\) up to the level of accuracy required for the considered physical problem.

Let us consider the case of \({{E}_{0}} = m\) as an example. Then

$$E\Phi \left( {\mathbf{r}} \right) = {{H}_{1}}\Phi \left( {\mathbf{r}} \right) = \left( {e{{A}_{0}} \pm \sqrt {{{m}^{2}} + {{{\mathbf{p}}}^{2}} + \frac{3}{{16{{m}^{2}}}}{{{\mathbf{{\rm E}}}}^{2}} - \frac{1}{{4m}}{\text{div}}{\mathbf{{\rm E}}} - \frac{1}{{2m}}{\mathbf{\sigma }}\left( {{\mathbf{{\rm E}}} \times {\mathbf{p}}} \right)} } \right)\Phi \left( {\mathbf{r}} \right).$$

Limiting by \(\sim{1 \mathord{\left/ {\vphantom {1 {{{m}^{2}}}}} \right. \kern-0em} {{{m}^{2}}}}\) the expansion in powers of \(m\) of the expression under the square root of this equation, we obtain an expression coinciding with the known nonrelativistic Foldy–Wouthuysen expansion for the \(\left( + \right)\) sign in front of the square root.

3.1 Chiral Foldy–Wouthuysen representation

In this case, matrices \({{\alpha }^{k}},\beta \) in the Weyl representation commonly applied in the Standard Model are used: \(\beta = {{\gamma }_{0}} = {{\rho }_{1}}\); \({{\alpha }^{k}} = \beta {{\gamma }^{k}} = {{\rho }_{3}}{{\sigma }^{k}}\); \({{\gamma }^{k}} = \beta {{\alpha }^{k}} = - i{{\rho }_{2}}{{\sigma }^{k}}\); \({{\gamma }^{5}} = i{{\gamma }^{0}}{{\gamma }^{1}}{{\gamma }^{2}}{{\gamma }^{3}} = {{\rho }_{3}}\); \({{\Sigma }^{i}} = {{E}_{{4 \times 4}}}{{\sigma }^{i}}\); \({{E}_{{4 \times 4}}}\) is the unit matrix.

If we represent bispinor \(\psi \) in the following form

$$\psi \left( {{\mathbf{r}},t} \right) = \left( \begin{gathered} {{\psi }_{L}}\left( {\mathbf{r}} \right) \hfill \\ {{\psi }_{R}}\left( {\mathbf{r}} \right) \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}},$$

then the term \(\beta m\) in the Dirac equation mixes spinors \({{\psi }_{L}}\left( {\mathbf{r}} \right),{{\psi }_{R}}\left( {\mathbf{r}} \right)\), so that the Dirac equation with a nonzero fermion mass does not possess chiral symmetry.

On the contrary, there are no summands mixing spinors \({{\psi }_{L}},{{\psi }_{R}}\) in the second-order equation, so it can be written in the following form

$$\begin{gathered} \left[ {{{{\left( {{{p}_{0}} - e{{A}_{0}}} \right)}}^{2}} - {{{\left( {{\mathbf{p}} - e{{{\mathbf{A}}}_{0}}} \right)}}^{2}} - {{m}^{2}} + {\mathbf{\sigma H}} - ie{\mathbf{\sigma {\rm E}}}} \right]{{\psi }_{L}} = 0, \hfill \\ \left[ {{{{\left( {{{p}_{0}} - e{{A}_{0}}} \right)}}^{2}} - {{{\left( {{\mathbf{p}} - e{{{\mathbf{A}}}_{0}}} \right)}}^{2}} - {{m}^{2}} + {\mathbf{\sigma H}} + ie{\mathbf{\sigma {\rm E}}}} \right]{{\psi }_{R}} = 0. \hfill \\ \end{gathered} $$

Earlier, similar equations were considered by Feynman and Gell–Mann.

When using the equations stated above, we have all the effects of quantum electrodynamics reproduced with the additional assumption that electrons and positrons are created or annihilated in pairs.

Fig. 1.
figure 1

(a) Attractive Coulomb field \(\left( {V = - {{Z\alpha } \mathord{\left/ {\vphantom {{Z\alpha } \rho }} \right. \kern-0em} \rho }} \right)\); effective potentials of the second-order equation: (b) at \(Z < {{Z}_{{{\text{cr}}}}}\), (c) at \(Z > {{Z}_{{{\text{cr}}}}}\).

Full size image

These equations display chiral symmetry. Presence or absence of fermion mass does not influence the chiral symmetry of the equations.

It is remarkable that these equations allow a closed expression for the FW Hamiltonian in the chiral representation.

Indeed, the following relation can be derived from them

$${{p}_{0}}{{\psi }_{{{\text{FW}}}}}\left( {{\mathbf{r}},t} \right) = {{H}_{{{\text{FW}}}}}{{\psi }_{{{\text{FW}}}}}\left( {{\mathbf{r}},t} \right),$$

where, for stationary states

$$\begin{gathered} {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} {{\psi }_{L}} \hfill \\ \,0 \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}},\,\,\,\,E > 0, \hfill \\ {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} \,\,0 \hfill \\ \,{{\psi }_{R}} \hfill \\ \end{gathered} \right){{e}^{{iEt}}},\,\,\,\,E < 0. \hfill \\ \end{gathered} $$
$${{H}_{{{\text{FW}}}}} = e{{A}_{0}} + {{\gamma }_{5}}\sqrt {{{m}^{2}} + {{{\left( {{\mathbf{p}} - e{\mathbf{A}}} \right)}}^{2}} + e\Sigma {\mathbf{H}} - ie{\mathbf{\alpha {\rm E}}}} .$$

For the Hamiltonian, another representation of wave functions is also possible

$$\begin{gathered} {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} \,\,0 \hfill \\ \,{{\psi }_{R}} \hfill \\ \end{gathered} \right){{e}^{{ - iEt}}},\,\,\,\,E > 0, \hfill \\ {{\psi }_{{{\text{FW}}}}} = \left( \begin{gathered} {{\psi }_{L}} \hfill \\ \,0 \hfill \\ \end{gathered} \right){{e}^{{iEt}}},\,\,\,\,E < 0. \hfill \\ \end{gathered} $$

4 CONCLUSIONS

(4.1) Self-adjoint second-order equations with spinor wave functions describing fermion quantum-mechanical motion are obtained in this paper. Connection of these equations with the equation of Schrödinger type with effective potentials and with the Dirac equation in the Foldy–Wouthuysen representation is shown.

(4.2) The cause of the absence of a closed FW transformation at nonzero scalar potential of electromagnetic field \({{A}_{0}}\left( {{\mathbf{r}},t} \right)\) is found using four-dimensional matrices \({{\gamma }^{0}},{{\gamma }^{k}},{{\gamma }^{5}}\) in the Dirac–Pauli representation. On the contrary, a closed expression for the FW Hamiltonian in the case of the general expression for electromagnetic field A0(r, t), Ak(r, t) is discovered using matrices \({{\gamma }^{0}},{{\gamma }^{k}},{{\gamma }^{5}}\) in the chiral representation. Then, the second-order equations with spinor wave functions display chiral symmetry irrespective of the presence or absence of fermion mass.

(4.3) Nonunitary similarity transformations required for the derivation of self-adjoint equations of the Schrödinger type with effective potentials for real radial wave functions can induce new physical consequences.

Let us consider two of them:

(4.3.1) Let us analyze the Schrödinger type equation with an effective Coulomb potential by considering three domains depending on \(Z\) in the initial Coulomb field \(V\left( r \right) = - {{Z{{e}^{2}}} \mathord{\left/ {\vphantom {{Z{{e}^{2}}} r}} \right. \kern-0em} r}\):

At ground state \(1{{S}_{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em} \!\lower0.7ex\hbox{$2$}}}}}\) in the first domain \(1 \leqslant Z < \frac{{Z137\sqrt 3 }}{2} \approx 118.7\) at \(r \to 0\), there is a positive barrier \(\sim{1 \mathord{\left/ {\vphantom {1 {{{r}^{2}}}}} \right. \kern-0em} {{{r}^{2}}}}\) with the subsequent potential well. In the second domain for \(Z\,\,\,\,\left( {119 \leqslant Z \leqslant 137} \right)\), the potential well \(\sim{K \mathord{\left/ {\vphantom {K {{{r}^{2}}}}} \right. \kern-0em} {{{r}^{2}}}}\) remains, where coefficient \(K \leqslant {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0em} 8}\), which allows fermion stationary bound states. In the third domain with \(Z > 137\), there is a potential well with coefficient \(K > {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0em} 8}\), which indicates the mode of particle “fall” onto the center.

(4.3.2) Real radial wave functions of the Dirac equation in the external gravitational Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman fields are square-un-integrable near the event horizons. With the transition to the Schrödinger-type equation with effective potentials, the radial wave functions become square-integrable in all allowed domains of definition, and wave functions on the event horizons are zero.

Hence, the use of the self-adjoint second-order equations with spinor wave functions expands the capabilities of the quantum mechanics of fermion motion in the exterior electromagnetic and gravitational fields.