Complex Form of Classical and Quantum Electrodynamics

Abstract

We consider a complex covariant form of the macroscopic Maxwell equations, in a moving medium or at rest, following the original ideas of Minkowski. A compact, Lorentz invariant, derivation of the energy-momentum tensor and the corresponding differential balance equations are given. Conservation laws and quantization of the electromagnetic field will be discussed in this covariant approach elsewhere.

Physical laws should have mathematical beauty.

P.A.M. Dirac

Dedicated to Krishna Alladi on the occasion of his 60th birthday and to the memory of his late father Professor Alladi Ramakrishnan

1 Introduction

Although a systematic study of electromagnetic phenomena in media is not possible without methods of quantum mechanics, statistical physics and kinetics, in practice a standard mathematical model based on phenomenological Maxwell’s equations provides a good approximation to many important problems. As is well known, one should be able to obtain the electromagnetic laws for continuous media from those for the interaction of fields and point particles [18], [34], [42], [51], [57], [66], [91]. As a result of the hard work of several generations of researchers and engineers, the classical electrodynamics, especially in its current complex covariant form, undoubtedly satisfies Dirac’s criteria of mathematical beauty¹, being a state of the art mathematical description of nature.

In macroscopic electrodynamics, the volume (mechanical or ponderomotive) forces, acting on a medium, and the corresponding energy density and energy flux are introduced with the help of the energy-momentum tensors and differential balance relations [24], [31], [51], [72], [86], [91]. These forces occur in the equations of motion for a medium or individual charges and, in principle, they can be experimentally tested [32], [69], [74], [92] (see also the references therein). But interpretation of the results should depend on the accepted model of the interaction between the matter and radiation.

In this methodological note, we discuss a complex version of Minkowski’s phenomenological electrodynamics (at rest or in a moving medium) without assuming any particular form of material equations as far as possible. Lorentz invariance of the corresponding differential balance equations is emphasized in view of long-standing uncertainties about the electromagnetic stresses and momentum density, the so-called Abraham–Minkowski controversy (see, for example, [5], [15], [19], [22], [24], [30], [31], [32], [34], [36], [51], [62], [63], [67], [68], [69], [72], [73], [74], [78], [80], [85], [89], [92], [93], [94], [95] and the references therein).

The paper is organized as follows. In sections 2 to 4, we describe the 3D-complex version of Maxwell’s equations and derive the corresponding differential balance density laws for the electromagnetic fields. Their covariant versions are given in sections 5 to 9. The case of a uniformly moving medium is discussed in section 10 and complex Lagrangians are introduced in section 11. Some useful tools are collected in appendices A to C for the reader’s benefit.

2 Maxwell’s Equations in 3D-Complex Form

Traditionally, the macroscopic Maxwell equations in a fixed frame of reference are given by

$$\begin{aligned} {\text {curl}}\,\mathbf {E}=-\frac{1}{c}\frac{\partial \mathbf {B}}{\partial t}\ \left( \text {Faraday}\right) ,\qquad \qquad {\text {div}}\, \mathbf {B}=0\ \left( \text {no magnetic charge}\right) \end{aligned}$$

(2.1)

$$ \begin{aligned} {\text {curl}}\,\mathbf {H}=\frac{1}{c}\frac{\partial \mathbf {D}}{\partial t}+\frac{4\pi }{c}\mathbf {j}_{\text {free}}\ \left( \text {Biot} \& \text {Savart} \right) ,\quad {\text {div}}\,\mathbf {D}=4\pi \rho _{\text {free}}\ \left( \text {Coulomb}\right) . \end{aligned}$$

(2.2)

Here, $\mathbf {E}$ is the electric field,² $\mathbf {D}$ is the displacement field; $\mathbf {H}$ is the magnetic field, $\mathbf {B}$ is the induction field. These equations, which are obtained by averaging of microscopic Maxwell’s equations in the vacuum, provide a good mathematical description of electromagnetic phenomena in various media, when complemented by the corresponding material equations. In the simplest case of an isotropic medium at rest, one usually has

$$\begin{aligned} \mathbf {D}=\varepsilon \mathbf {E},\qquad \mathbf {B}=\mu \mathbf {H},\qquad \mathbf {j}=\sigma \mathbf {E}, \end{aligned}$$

(2.3)

where $\varepsilon$ is the dielectric constant, $\mu$ is the magnetic permeability, and $\sigma$ describes the conductivity of the medium (see, for example, [1], [6], [7], [15], [16], [18], [21], [23], [28], [34], [37], [51], [57], [70], [72], [82], [86], [88], [90], [91] for fundamentals of classical electrodynamics).

Introduction of two complex fields

$$\begin{aligned} \mathbf {F}=\mathbf {E}+i\mathbf {H},\qquad \mathbf {G}=\mathbf {D}+i\mathbf {B} \end{aligned}$$

(2.4)

allows one to rewrite the phenomenological Maxwell equations in the following compact form

$$\begin{aligned} \frac{i}{c}\left( \frac{\partial \mathbf {G}}{\partial t}+4\pi \mathbf {j} \right) ={\text {curl}}\,\mathbf {F},\qquad \mathbf {j}=\mathbf {j}^{*}, \end{aligned}$$

(2.5)

$$\begin{aligned} {\text {div}}\,\mathbf {G}=4\pi \rho ,\qquad \rho =\rho ^{*}, \end{aligned}$$

(2.6)

where the asterisk stands for complex conjugation (see also [6], [47] and [79]). As we shall demonstrate, different complex forms of Maxwell’s equations are particularly convenient for study of the corresponding “energy-momentum” balance equations for the electromagnetic fields in the presence of the “free” charges and currents in a medium.

3 Hertz Symmetric Stress Tensor

We begin from a complex 3D-interpretation of the traditional symmetric energy-momentum tensor [72]. By definition,

$$\begin{aligned}&T_{pq}=\frac{1}{16\pi }\left[ F_{p}G_{q}^{*}+F_{p}^{*}G_{q} +F_{q}G_{p}^{*}+F_{q}^{*}G_{p}\right. \\&\qquad \quad -\left. \delta _{pq}\left( \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G}\right) \right] =T_{qp}\qquad \left( p,q=1,2,3\right) \nonumber \end{aligned}$$

(3.1)

and the corresponding “momentum” balance equation,

$$\begin{aligned}&\left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j}\times \mathbf {B}\right) _{p}+\frac{\partial }{\partial t}\left[ \frac{1}{4\pi c}\left( \mathbf {D} \times \mathbf {B}\right) \right] _{p}\\&\ =\frac{\partial T_{pq}}{\partial x_{q}}+\frac{1}{16\pi }\left[ {\text {curl}}\,\left( \mathbf {F\times G}^{*}+\mathbf {F}^{*} \times \mathbf {G}\right) \right] _{p}\nonumber \\&\quad +\frac{1}{16\pi }\left( F_{q}\frac{\partial G_{q}^{*}}{\partial x_{p}}-G_{q}\frac{\partial F_{q}^{*}}{\partial x_{p}}+F_{q}^{*} \frac{\partial G_{q}}{\partial x_{p}}-G_{q}^{*}\frac{\partial F_{q} }{\partial x_{p}}\right) ,\nonumber \end{aligned}$$

(3.2)

can be obtained from Maxwell’s equations (2.5)–(2.6) as a result of elementary but rather tedious vector calculus calculations usually omitted in textbooks. (We use Einstein summation convention over any two repeated indices unless otherwise stated. In this paper, Greek indices run from 0 to 3, while Latin indices may have values from 1 to 3 inclusive.)

Proof.

Indeed, in a 3D-complex form,

$$\begin{aligned}&\frac{\partial }{\partial x_{q}}\left( F_{p}G_{q}^{*}+F_{q}G_{p}^{*}-\delta _{pq}\mathbf {F}\cdot \mathbf {G}^{*}\right) \\&\quad =\frac{\partial F_{p}}{\partial x_{q}}G_{q}^{*}+F_{p}\frac{\partial G_{q}^{*}}{\partial x_{q}}+\frac{\partial F_{q}}{\partial x_{q}}G_{p} ^{*}+F_{q}\frac{\partial G_{p}^{*}}{\partial x_{q}}-\frac{\partial }{\partial x_{p}}\left( F_{q}G_{q}^{*}\right) \nonumber \\&\quad =F_{q}\left( \frac{\partial G_{p}^{*}}{\partial x_{q}} -\frac{\partial G_{q}^{*}}{\partial x_{p}}\right) +\left( \frac{\partial F_{p}}{\partial x_{q}}-\frac{\partial F_{q}}{\partial x_{p}}\right) G_{q}^{*}\nonumber \\&\quad \quad +F_{p}{\text {div}}\,\mathbf {G}^{*}+G_{p}^{*}{\text {div}}\,\mathbf {F}\nonumber \\&\quad =F_{p}{\text {div}}\,\mathbf {G}^{*}-\left( \mathbf {F} \times {\text {curl}}\,\mathbf {G}^{*}\right) _{p}+G_{p}^{*}{\text {div}}\,\mathbf {F-}\left( \mathbf {G}^{*}\times {\text {curl}}\,\mathbf {F}\right) _{p}\nonumber \end{aligned}$$

(3.3)

due to an identity [86]:

$$\begin{aligned} \left( \mathbf {A}\times {\text {curl}}\,\mathbf {B}\right) _{p} =A_{q}\left( \frac{\partial B_{q}}{\partial x_{p}}-\frac{\partial B_{p} }{\partial x_{q}}\right) . \end{aligned}$$

(3.4)

Taking into account the complex conjugate, we derive

$$\begin{aligned} \frac{1}{2}&\frac{\partial }{\partial x_{q}}\left[ F_{p}G_{q}^{*} +F_{p}^{*}G_{q}+F_{q}G_{p}^{*}+F_{q}^{*}G_{p}-\delta _{pq}\left( \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G}\right) \right] \\&\quad =\frac{1}{2}\left( \mathbf {F}\,{\text {div}}\,\mathbf {G}^{*}-\mathbf {G}^{*}\times {\text {curl}}\,\mathbf {F}+\mathbf {F}^{*}{\text {div}}\,\mathbf {G}-\mathbf {G}\times {\text {curl}}\, \mathbf {F}^{*}\right) _{p}\nonumber \\&\qquad +\frac{1}{2}\left( \mathbf {G}\,{\text {div}}\,\mathbf {F}^{*}-\mathbf {F}^{*}\times {\text {curl}}\,\mathbf {G}+\mathbf {G}^{*}{\text {div}}\,\mathbf {F}-\mathbf {F}\times {\text {curl}}\, \mathbf {G}^{*}\right) _{p}\nonumber \end{aligned}$$

(3.5)

as our first important fact.

On the other hand, in view of Maxwell’s equations (2.5)–(2.6), one gets

$$\begin{aligned}&\mathbf {F}\,{\text {div}}\,\mathbf {G}^{*}-\mathbf {G}^{*} \times {\text {curl}}\,\mathbf {F}\\&\quad =4\pi \rho \mathbf {F}+\frac{i}{c}\left( \frac{\partial \mathbf {G} }{\partial t}\times \mathbf {G}^{*}+4\pi \mathbf {j\times G}^{*}\right) \nonumber \end{aligned}$$

(3.6)

and, with the help of its complex conjugate,

$$\begin{aligned}&\mathbf {F}\,{\text {div}}\,\mathbf {G}^{*}-\mathbf {G}^{*} \times {\text {curl}}\,\mathbf {F}+\mathbf {F}^{*}{\text {div}}\, \mathbf {G}-\mathbf {G}\times {\text {curl}}\,\mathbf {F}^{*}\\&\quad =4\pi \rho \left( \mathbf {F}+\mathbf {F}^{*}\right) +\frac{i}{c}\frac{\partial }{\partial t}\left( \mathbf {G}\times \mathbf {G}^{*}\right) +\frac{4\pi i}{c}\mathbf {j\times }\left( \mathbf {G}^{*}-\mathbf {G}\right) ,\nonumber \end{aligned}$$

(3.7)

or

$$\begin{aligned}&\frac{1}{2}\left( \mathbf {F}\,{\text {div}}\,\mathbf {G}^{*} -\mathbf {G}^{*}\times {\text {curl}}\,\mathbf {F}+\mathbf {F}^{*}{\text {div}}\,\mathbf {G}-\mathbf {G}\times {\text {curl}}\, \mathbf {F}^{*}\right) \\&\quad =4\pi \left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j}\times \mathbf {B} \right) +\frac{1}{c}\frac{\partial }{\partial t}\left( \mathbf {D} \times \mathbf {B}\right) ,\nonumber \end{aligned}$$

(3.8)

providing the second important fact. (Up to the constant, the first term on the right-hand side represents the density of Lorentz’s force acting on the “free” charges and currents in the medium under consideration [85], [86].)

In view of (3.8) and (3.5), we can write

$$\begin{aligned}&4\pi \left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j}\times \mathbf {B}\right) _{p}+\frac{1}{c}\frac{\partial }{\partial t}\left( \mathbf {D}\times \mathbf {B}\right) _{p}\\&\quad =\frac{1}{2}\frac{\partial }{\partial x_{q}}\left[ F_{p}G_{q}^{*}+F_{p}^{*}G_{q}+F_{q}G_{p}^{*}+F_{q}^{*}G_{p}-\delta _{pq}\left( \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G}\right) \right] \nonumber \\&\quad \quad -\frac{1}{2}\left( \mathbf {G}\,{\text {div}}\,\mathbf {F}^{*}-\mathbf {F}^{*}\times {\text {curl}}\,\mathbf {G}+\mathbf {G}^{*}{\text {div}}\,\mathbf {F}-\mathbf {F}\times {\text {curl}}\, \mathbf {G}^{*}\right) _{p}\nonumber \\&\quad =\frac{1}{4}\frac{\partial }{\partial x_{q}}\left[ F_{p}G_{q}^{*}+F_{p}^{*}G_{q}+F_{q}G_{p}^{*}+F_{q}^{*}G_{p}-\delta _{pq}\left( \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G}\right) \right] \nonumber \\&\quad \quad -\frac{1}{4}\left( \mathbf {G}\,{\text {div}}\,\mathbf {F}^{*}-\mathbf {F}^{*}\times {\text {curl}}\,\mathbf {G}+\mathbf {G}^{*}{\text {div}}\,\mathbf {F}-\mathbf {F}\times {\text {curl}}\, \mathbf {G}^{*}\right) _{p}\nonumber \\&\quad \quad +\frac{1}{4}\left( \mathbf {F}\,{\text {div}}\,\mathbf {G}^{*}-\mathbf {G}^{*}\times {\text {curl}}\,\mathbf {F}+\mathbf {F}^{*}{\text {div}}\,\mathbf {G}-\mathbf {G}\times {\text {curl}}\, \mathbf {F}^{*}\right) _{p}\nonumber \\&\quad =4\pi \frac{\partial T_{pq}}{\partial x_{q}}+\frac{1}{4}\left( \mathbf {F}\,{\text {div}}\,\mathbf {G}^{*}-\mathbf {G}^{*} {\text {div}}\,\mathbf {F}+\mathbf {F}^{*}{\text {div}}\, \mathbf {G}-\mathbf {G}\,{\text {div}}\,\mathbf {F}^{*}\right) _{p}\nonumber \\&\quad \quad +\frac{1}{4}\left( \mathbf {F}\times {\text {curl}}\, \mathbf {G}^{*}-\mathbf {G}^{*}\times {\text {curl}}\,\mathbf {F} +\mathbf {F}^{*}\times {\text {curl}}\,\mathbf {G}-\mathbf {G} \times {\text {curl}}\,\mathbf {F}^{*}\right) _{p}.\nonumber \end{aligned}$$

(3.9)

Finally, in the last two lines, one can utilize the following differential vector calculus identity,

$$\begin{aligned}&\left[ \mathbf {A}\,{\text {div}}\,\mathbf {B}-\mathbf {B}\,{\text {div}}\, \mathbf {A}+\mathbf {A}\times {\text {curl}}\,\mathbf {B}-\mathbf {B} \times {\text {curl}}\,\mathbf {A}-{\text {curl}}\,\left( \mathbf {A} \times \mathbf {B}\right) \right] _{p}\\&\quad =A_{q}\frac{\partial B_{q}}{\partial x_{p}}-B_{q}\frac{\partial A_{q} }{\partial x_{p}},\nonumber \end{aligned}$$

(3.10)

see (A.5), with $\mathbf {A}=\mathbf {F},$ $\mathbf {B}=\mathbf {G}^{*}$ and its complex conjugates, in order to obtain (3.2) and/or (3.16), which completes the proof. (An independent proof will be given in section 7.)

Derivation of the corresponding differential “energy” balance equation is much simpler. By (2.5),

$$\begin{aligned} \mathbf {F}\cdot \frac{\partial \mathbf {G}^{*}}{\partial t}+\mathbf {F}^{*}\cdot \frac{\partial \mathbf {G}}{\partial t}+4\pi \mathbf {j}\cdot \left( \mathbf {F}+\mathbf {F}^{*}\right) =\frac{c}{i}{\text {div}}\,\left( \mathbf {F}\times \mathbf {F}^{*}\right) \end{aligned}$$

(3.11)

due to a familiar vector calculus identity (A.1):

$$\begin{aligned} {\text {div}}\,\left( \mathbf {A}\times \mathbf {B}\right) =\mathbf {B} \cdot {\text {curl}}\,\mathbf {A}-\mathbf {A}\cdot {\text {curl}}\, \mathbf {B}. \end{aligned}$$

(3.12)

In a traditional form,

$$\begin{aligned} \frac{1}{4\pi }\left( \mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial t}+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial t}\right) +\mathbf {j} \cdot \mathbf {E}+{\text {div}}\,\left( \frac{c}{4\pi }\mathbf {E} \times \mathbf {H}\right) =0 \end{aligned}$$

(3.13)

(see, for example, [18], [86]), where one can substitute

$$\begin{aligned}&\mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial t}+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial t}=\frac{1}{2}\frac{\partial }{\partial t}\left( \mathbf {E}\cdot \mathbf {D}+\mathbf {H}\cdot \mathbf {B}\right) \\&\quad +\frac{1}{2}\left( \mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial t}-\mathbf {D}\cdot \frac{\partial \mathbf {E}}{\partial t}+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial t}-\mathbf {B}\cdot \frac{\partial \mathbf {H} }{\partial t}\right) .\nonumber \end{aligned}$$

(3.14)

As a result, 3D-differential “energy-momentum” balance equations are given by

$$\begin{aligned}&\frac{\partial }{\partial t}\left( \frac{\mathbf {E}\cdot \mathbf {D} +\mathbf {H}\cdot \mathbf {B}}{8\pi }\right) +{\text {div}}\,\left( \frac{c}{4\pi }\mathbf {E}\times \mathbf {H}\right) +\mathbf {j}\cdot \mathbf {E} \\&\ +\frac{1}{8\pi }\left( \mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial t}-\mathbf {D}\cdot \frac{\partial \mathbf {E}}{\partial t}+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial t}-\mathbf {B}\cdot \frac{\partial \mathbf {H} }{\partial t}\right) =0\nonumber \end{aligned}$$

(3.15)

and

$$\begin{aligned}&-\frac{\partial }{\partial t}\left[ \frac{1}{4\pi c}\left( \mathbf {D} \times \mathbf {B}\right) \right] _{p}+\frac{\partial T_{pq}}{\partial x_{q} }-\left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j}\times \mathbf {B}\right) _{p}\\&\ +\frac{1}{8\pi }\left[ {\text {curl}}\,\left( \mathbf {E\times D}+\mathbf {H}\times \mathbf {B}\right) \right] _{p}\nonumber \\&\quad +\frac{1}{8\pi }\left( \mathbf {E}\cdot \frac{\partial \mathbf {D} }{\partial x_{p}}-\mathbf {D}\cdot \frac{\partial \mathbf {E}}{\partial x_{p} }+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial x_{p}}-\mathbf {B} \cdot \frac{\partial \mathbf {H}}{\partial x_{p}}\right) =0,\nonumber \end{aligned}$$

(3.16)

respectively (see also [32], [62]). The real form of the symmetric stress tensor (3.1), namely,

$$\begin{aligned}&T_{pq}=\frac{1}{8\pi }\left[ E_{p}D_{q}+E_{q}D_{p}+H_{p}B_{q}+H_{q} B_{p}\right. \\&\qquad \qquad \left. -\delta _{pq}\left( \mathbf {E}\cdot \mathbf {D} +\mathbf {H}\cdot \mathbf {B}\right) \right] \qquad \left( p,q=1,2,3\right) ,\nonumber \end{aligned}$$

(3.17)

is due to Hertz [72].

Equations (3.15)–(3.16) are related to a fundamental concept of conservation of mechanical and electromagnetic energy and momentum. Here, these balance conditions are presented in differential forms in terms of the corresponding local field densities. They can be integrated over a given volume in $\left. \mathbb {R} \right. ^{3}$ in order to obtain, in a traditional way, the corresponding conservation laws of the electromagnetic fields (see, for example, [50], [51], [88], [90], [91]). These laws made it necessary to ascribe a definite linear momentum and energy to the field of an electromagnetic wave, which can be observed, for example, as light pressure.

Note. At this point, the Lorentz invariance of these differential balance equations is not obvious in our 3D-analysis. But one can introduce the four-vector $x^{\mu }=\left( ct,\mathbf {r}\right)$ and try to match (3.15)–(3.16) with the expression,

$$\begin{aligned} \frac{\partial }{\partial x^{\nu }}T_{\mu }^{\ \nu }=\frac{\partial T_{\mu }^{\ 0} }{\partial x_{0}}+\frac{\partial T_{\mu }^{\ q}}{\partial x_{q}}\qquad \left( \mu ,\nu =0,1,2,3;\quad p,q=1,2,3\right) , \end{aligned}$$

(3.18)

as an initial step, in order to guess the corresponding four-tensor form. An independent covariant derivation will be given in section 7.

Note. In an isotropic non-homogeneous variable medium (without dispersion and/or compression), when $\mathbf {D}=\varepsilon \left( \mathbf {r},t\right) \mathbf {E}$ and $\mathbf {B}=\mu \left( \mathbf {r} ,t\right) \mathbf {H},$ the “ponderomotive forces” in (3.15) and (3.16) take the form [86]:

$$\begin{aligned}&\mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial x^{\nu }}-\mathbf {D} \cdot \frac{\partial \mathbf {E}}{\partial x^{\nu }}+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial x^{\nu }}-\mathbf {B}\cdot \frac{\partial \mathbf {H}}{\partial x^{\nu }}\\&\quad =\dfrac{\partial \varepsilon }{\partial x^{\nu }}\mathbf {E}^{2} +\dfrac{\partial \mu }{\partial x^{\nu }}\mathbf {H}^{2}=\left( \begin{array} [c]{c} \dfrac{1}{c}\left( \dfrac{\partial \varepsilon }{\partial t}\mathbf {E} ^{2}+\dfrac{\partial \mu }{\partial t}\mathbf {H}^{2}\right) \\ \mathbf {E}^{2}\nabla \varepsilon +\mathbf {H}^{2}\nabla \mu \end{array} \right) ,\nonumber \end{aligned}$$

(3.19)

which may be interpreted as a four-vector “energy-force” acting from an inhomogeneous and time-variable medium. Its covariance is analyzed in section 7.

4 “Angular Momentum” Balance

The 3D-“linear momentum” differential balance equation (3.16), can be rewritten in a more compact form,

$$\begin{aligned} \frac{\partial T_{pq}}{\partial x_{q}}=\mathscr {F}_{p}+\frac{\partial \mathscr {G}_{p}}{\partial t},\qquad \overrightarrow{\mathscr {G}}=\frac{1}{4\pi c}\left( \mathbf {D}\times \mathbf {B}\right) , \end{aligned}$$

(4.1)

with the help of the Hertz symmetric stress tensor $T_{pq}=T_{qp}$ defined by (3.17). A “net force” is given by

$$\begin{aligned} \mathscr {F}_{p}&=\left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j} \times \mathbf {B}\right) _{p}-\frac{1}{8\pi }\left[ {\text {curl}}\,\left( \mathbf {E\times D}+\mathbf {H}\times \mathbf {B}\right) \right] _{p} \\&-\frac{1}{8\pi }\left( \mathbf {E}\cdot \frac{\partial \mathbf {D}}{\partial x_{p}}-\mathbf {D}\cdot \frac{\partial \mathbf {E}}{\partial x_{p}}+\mathbf {H} \cdot \frac{\partial \mathbf {B}}{\partial x_{p}}-\mathbf {B}\cdot \frac{\partial \mathbf {H}}{\partial x_{p}}\right) .\nonumber \end{aligned}$$

(4.2)

In this notation, we state the 3D-“angular momentum” differential balance equation as follows

$$\begin{aligned} \frac{\partial M_{pq}}{\partial x_{q}}=\mathscr {T}_{p}+\frac{\partial \mathscr {L}_{p}}{\partial t},\qquad \overrightarrow{\mathscr {L}} =\mathbf {r\times }\overrightarrow{\mathscr {G}},\quad \overrightarrow{\mathscr {T} }=\mathbf {r\times }\overrightarrow{\mathscr {F}}, \end{aligned}$$

(4.3)

where the “field angular momentum density” is defined by

$$\begin{aligned} \overrightarrow{\mathscr {L}}=\frac{1}{4\pi c}\mathbf {r\times }\left( \mathbf {D}\times \mathbf {B}\right) \end{aligned}$$

(4.4)

and the “flux of angular momentum” is described by the following tensor [37]:

$$\begin{aligned} M_{pq}=e_{prs}x_{r}T_{sq}. \end{aligned}$$

(4.5)

(Here, $e_{pqr}$ is the totally anti-symmetric Levi-Civita symbol with $e_{123}=+1$). An elementary example of conservation of the total angular momentum is discussed in [86].

Proof.

Indeed, in view of (4.1), one can write

$$\begin{aligned} \frac{\partial M_{pq}}{\partial x_{q}}&=e_{prs}T_{sr}+e_{prs}x_{r} \frac{\partial T_{sq}}{\partial x_{q}}\\&=e_{pqr}x_{q}\mathscr {F}_{r}+\frac{\partial }{\partial t}\left( e_{pqr}x_{q}\mathscr {G}_{r}\right) ,\nonumber \end{aligned}$$

(4.6)

which completes the proof.

Note. Once again, in 3D-form, the Lorentz invariance of this differential balance equation for the local densities is not obvious. An independent covariant derivation will be given in section 8.

5 Complex Covariant Form of Macroscopic Maxwell’s Equations

With the help of complex fields $\mathbf {F}=\mathbf {E}+i\mathbf {H}$ and $\mathbf {G}=\mathbf {D}+i\mathbf {B},$ we introduce the following anti-symmetric four-tensor,

$$\begin{aligned} Q^{\mu \nu }=-Q^{\nu \mu }=\left( \begin{array} [c]{cccc} 0 &{} -G_{1} &{} -G_{2} &{} -G_{3}\\ G_{1} &{} 0 &{} iF_{3} &{} -iF_{2}\\ G_{2} &{} -iF_{3} &{} 0 &{} iF_{1}\\ G_{3} &{} iF_{2} &{} -iF_{1} &{} 0 \end{array} \right) \end{aligned}$$

(5.1)

and use the standard four-vectors, $x^{\mu }=\left( ct,\mathbf {r}\right)$ and $j^{\mu }=\left( c\rho ,\mathbf {j}\right)$ for contravariant coordinates and current, respectively.

Maxwell’s equations then take the covariant form [47], [54]:

$$\begin{aligned} \frac{\partial }{\partial x^{\nu }}Q^{\mu \nu }=-\frac{\partial }{\partial x^{\nu } }Q^{\nu \mu }=-\frac{4\pi }{c}j^{\mu } \end{aligned}$$

(5.2)

with summation over two repeated indices. Indeed, in block form, we have

$$\begin{aligned} \frac{\partial Q^{\mu \nu }}{\partial x^{\nu }}=\frac{\partial }{\partial x^{\nu } }\left( \begin{array} [c]{cc} 0 &{} -G_{q} \\ G_{p} &{} ie_{pqr}F_{r} \end{array} \right) =\left( \begin{array} [c]{c} -{\text {div}}\,\mathbf {G}=-4\pi \rho \\ \dfrac{1}{c}\dfrac{\partial \mathbf {G}}{\partial t}+i{\text {curl}}\, \mathbf {F}=-\dfrac{4\pi }{c}\mathbf {j} \end{array} \right) , \end{aligned}$$

(5.3)

which verifies this fact. The continuity equation,

$$\begin{aligned} 0\equiv \frac{\partial ^{2}Q^{\mu \nu }}{\partial x^{\mu }\partial x^{\nu }} =-\frac{4\pi }{c}\frac{\partial j^{\mu }}{\partial x^{\mu }}, \end{aligned}$$

(5.4)

or in the 3D-form,

$$\begin{aligned} \frac{\partial \rho }{\partial t}+{\text {div}}\,\mathbf {j}=0, \end{aligned}$$

(5.5)

describes conservation of the electrical charge. The latter equation can also be derived in the complex 3D-form from (2.5)–(2.6).

Note. In vacuum, when $\mathbf {G}=\mathbf {F}$ and $\rho =0,$ $\mathbf {j}=0,$ one can write due to (B.5)–(B.6):

$$\begin{aligned} Q^{\mu \nu }=F^{\mu \nu }-\frac{i}{2}e^{\mu \nu \sigma \tau }F_{\sigma \tau },\qquad F^{\mu \nu }=g^{\mu \sigma }g^{\nu \tau }F_{\sigma \tau },\qquad g_{\mu \sigma } g_{\nu \tau }Q^{\sigma \tau }=Q_{\mu \nu }. \end{aligned}$$

(5.6)

As a result, the following self-duality property holds

$$\begin{aligned} e_{\mu \nu \sigma \tau }Q^{\sigma \tau }=2iQ_{\mu \nu },\qquad 2iQ^{\mu \nu }=e^{\mu \nu \sigma \tau }Q_{\sigma \tau } \end{aligned}$$

(5.7)

(see, for example, [8], [48] and appendix B). Two covariant forms of Maxwell’s equations are given by

$$\begin{aligned} \partial _{\nu }Q^{\mu \nu }=0,\qquad \partial ^{\nu }Q_{\mu \nu }=0, \end{aligned}$$

(5.8)

where $\partial ^{\nu }=g^{\nu \mu }\partial _{\mu },$ $\partial _{\mu } =\partial /\partial x^{\mu }$ and $g_{\mu \nu }=g^{\mu \nu }=$diag$\left( 1,-1,-1,-1\right) .$ The last equation can be derived from a more general equation, involving a rank three tensor,

$$\begin{aligned} g^{\alpha \alpha }e_{\alpha \mu \nu \tau }\partial ^{\nu }Q^{\tau \beta }-g^{\beta \beta }e_{\beta \mu \nu \tau }\partial ^{\nu }Q^{\tau \alpha }=-i\partial _{\mu }Q^{\alpha \beta } \end{aligned}$$

(5.9)

($\alpha ,\beta =0,1,2,3$ are fixed; no summation is assumed over these two indices), which is related to the Pauli–Lubański vector from the representation theory of the Poincaré group [47]. Different spinor forms of Maxwell’s equations are analyzed in [48] (see also the references therein).

6 Dual Electromagnetic Field Tensors

Two dual anti-symmetric field tensors of complex fields, $\mathbf {F} =\mathbf {E}+i\mathbf {H}$ and $\mathbf {G}=\mathbf {D}+i\mathbf {B},$ are given by

$$\begin{aligned} Q^{\mu \nu }&=\left( \begin{array} [c]{cccc} 0 &{} -G_{1} &{} -G_{2} &{} -G_{3}\\ G_{1} &{} 0 &{} iF_{3} &{} -iF_{2}\\ G_{2} &{} -iF_{3} &{} 0 &{} iF_{1}\\ G_{3} &{} iF_{2} &{} -iF_{1} &{} 0 \end{array} \right) =R^{\mu \nu }+iS^{\mu \nu }\\&=\left( \begin{array} [c]{cccc} 0 &{} -D_{1} &{} -D_{2} &{} -D_{3}\\ D_{1} &{} 0 &{} -H_{3} &{} H_{2}\\ D_{2} &{} H_{3} &{} 0 &{} -H_{1}\\ D_{3} &{} -H_{2} &{} H_{1} &{} 0 \end{array} \right) +i\left( \begin{array} [c]{cccc} 0 &{} -B_{1} &{} -B_{2} &{} -B_{3}\\ B_{1} &{} 0 &{} E_{3} &{} -E_{2}\\ B_{2} &{} -E_{3} &{} 0 &{} E_{1}\\ B_{3} &{} E_{2} &{} -E_{1} &{} 0 \end{array} \right) \nonumber \end{aligned}$$

(6.1)

and

$$\begin{aligned} P_{\mu \nu }&=\left( \begin{array} [c]{cccc} 0 &{} F_{1} &{} F_{2} &{} F_{3}\\ -F_{1} &{} 0 &{} iG_{3} &{} -iG_{2}\\ -F_{2} &{} -iG_{3} &{} 0 &{} iG_{1}\\ -F_{3} &{} iG_{2} &{} -iG_{1} &{} 0 \end{array} \right) =F_{\mu \nu }+iG_{\mu \nu }\\&=\left( \begin{array} [c]{cccc} 0 &{} E_{1} &{} E_{2} &{} E_{3}\\ -E_{1} &{} 0 &{} -B_{3} &{} B_{2}\\ -E_{2} &{} B_{3} &{} 0 &{} -B_{1}\\ -E_{3} &{} -B_{2} &{} B_{1} &{} 0 \end{array} \right) +i\,\left( \begin{array} [c]{cccc} 0 &{} H_{1} &{} H_{2} &{} H_{3}\\ -H_{1} &{} 0 &{} D_{3} &{} -D_{2}\\ -H_{2} &{} -D_{3} &{} 0 &{} D_{1}\\ -H_{3} &{} D_{2} &{} -D_{1} &{} 0 \end{array} \right) .\nonumber \end{aligned}$$

(6.2)

The real part of the latter represents the standard electromagnetic field tensor in a medium [6], [72], [91]. As for the imaginary part of (6.1), which, ironically, Pauli called an “artificiality” in view of its non-standard behavior under spatial inversion [72], the use of complex conjugation restores this important symmetry for our complex field tensors.

The dual tensor identities are given by

$$\begin{aligned} e_{\mu \nu \sigma \tau }Q^{\sigma \tau }=2iP_{\mu \nu },\qquad 2iQ^{\mu \nu }=e^{\mu \nu \sigma \tau }P_{\sigma \tau }. \end{aligned}$$

(6.3)

Here $e^{\mu \nu \sigma \tau }=-e_{\mu \nu \sigma \tau }$ and $e_{0123}=+1$ is the Levi-Civita four-symbol [27]. Then

$$\begin{aligned} 6i\frac{\partial Q^{\mu \nu }}{\partial x^{\nu }}=e^{\mu \nu \lambda \sigma }\left( \frac{\partial P_{\lambda \sigma }}{\partial x^{\nu }}+\frac{\partial P_{\nu \lambda }}{\partial x^{\sigma }}+\frac{\partial P_{\sigma \nu }}{\partial x^{\lambda }}\right) \end{aligned}$$

(6.4)

and both pairs of Maxwell’s equations can also be presented in the form [47]

$$\begin{aligned} \frac{\partial P_{\mu \nu }}{\partial x^{\lambda }}+\frac{\partial P_{\nu \lambda }}{\partial x^{\mu }}+\frac{\partial P_{\lambda \mu }}{\partial x^{\nu }} =-\frac{4\pi i}{c}e_{\mu \nu \lambda \sigma }j^{\sigma } \end{aligned}$$

(6.5)

in addition to the one given above

$$\begin{aligned} \frac{\partial Q^{\mu \nu }}{\partial x^{\nu }}=-\frac{4\pi }{c}j^{\mu }. \end{aligned}$$

(6.6)

The real part of the first equation traditionally represents the first (homogeneous) pair of Maxwell’s equation and the real part of the second one gives the remaining pair. In our approach, both pairs of Maxwell’s equations appear together (see also [6], [8], [9], [54], and [87] for the case in vacuum). Moreover, a generalization to complex-valued four-current may naturally represent magnetic charge and magnetic current not yet observed in nature [79].

An important cofactor matrix identity,

$$\begin{aligned} P_{\mu \nu }Q^{\nu \lambda }=\left( \mathbf {F}\cdot \mathbf {G}\right) \delta _{\mu }^{\lambda }=\frac{1}{4}\left( P_{\sigma \tau }Q^{\tau \sigma }\right) \delta _{\mu }^{\lambda }, \end{aligned}$$

(6.7)

was originally established, in a general form, by Minkowski [65]. Once again, the dual tensors are given by

$$\begin{aligned} P_{\mu \nu }=\left( \begin{array} [c]{cc} 0 &{} F_{q} \\ -F_{p} &{} ie_{pqr}G_{r} \end{array} \right) ,\quad Q^{\mu \nu }=\left( \begin{array} [c]{cc} 0 &{} -G_{q}\\ G_{p} &{} ie_{pqr}F_{r} \end{array} \right) , \end{aligned}$$

(6.8)

in block form. A complete list of relevant tensor and matrix identities is given in appendix B.

7 Covariant Derivation of Energy-Momentum Balance Equations

7.1 Preliminaries

As has been announced in [47] (see also [48]), the covariant form of the differential balance equations can be presented as follows³

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left[ \frac{1}{16\pi }\left( P_{\mu \lambda }^{*}Q^{\lambda \nu }+P_{\mu \lambda }\overset{*}{\left. Q^{\lambda \nu }\right. }\right) \right] \\&\ +\frac{1}{32\pi }\left( P_{\sigma \tau }^{*}\frac{\partial Q^{\tau \sigma }}{\partial x^{\mu }}+P_{\sigma \tau }\frac{\partial \overset{*}{\left. Q^{\tau \sigma }\right. }}{\partial x^{\mu }}\right) \ =-\frac{1}{c} F_{\mu \lambda }j^{\lambda }=\left( \begin{array} [c]{c} -\mathbf {j}\cdot \mathbf {E}/c\\ \rho \mathbf {E}+\mathbf {j}\times \mathbf {B}/c \end{array} \right) .\nonumber \end{aligned}$$

(7.1)

In our complex form, when $\mathbf {F}=\mathbf {E}+i\mathbf {H}$ and $\mathbf {G}=\mathbf {D}+i\mathbf {B},$ the energy-momentum tensor is given by

$$\begin{aligned}&16\pi T_{\mu }{}^{\nu }=P_{\mu \lambda }^{*}Q^{\lambda \nu }+P_{\mu \lambda }\overset{*}{\left. Q^{\lambda \nu }\right. } \\&=\left( \begin{array} [c]{cc} \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G} &{} 2i\left( \mathbf {F\times F}^{*}\right) _{q} \\ -2i\left( \mathbf {G\times G}^{*}\right) _{p} &{} \quad 2\left( F_{p} G_{q}^{*}+F_{p}^{*}G_{q}\right) -\delta _{pq}\left( \mathbf {F} \cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G}\right) \end{array} \right) .\nonumber \end{aligned}$$

(7.2)

Here, we point out for the reader’s convenience that

$$\begin{aligned}&i\left( \mathbf {F\times F}^{*}\right) =2\left( \mathbf {E\times H}\right) ,\qquad i\left( \mathbf {G\times G}^{*}\right) =2\left( \mathbf {D\times B}\right) ,\\&\quad \mathbf {F}\cdot \mathbf {G}^{*}+\mathbf {F}^{*}\cdot \mathbf {G} =2\left( \mathbf {E}\cdot \mathbf {D}+\mathbf {H}\cdot \mathbf {B}\right) \nonumber \end{aligned}$$

(7.3)

and, in real form,

$$\begin{aligned} 4\pi T_{\mu }{}^{\nu }=\left( \begin{array} [c]{cc} \left( \mathbf {E}\cdot \mathbf {D}+\mathbf {H}\cdot \mathbf {B}\right) /2 &{} \left( \mathbf {E\times H}\right) _{q}\\ -\left( \mathbf {D\times B}\right) _{p} &{} \quad E_{p}D_{q}+H_{p}B_{q} -\delta _{pq}\left( \mathbf {E}\cdot \mathbf {D}+\mathbf {H}\cdot \mathbf {B} \right) /2 \end{array} \right) . \end{aligned}$$

(7.4)

The covariant form of the differential balance equation allows one to clarify the physical meaning of different energy-momentum tensors. For instance, it is worth noting that the non-symmetric Maxwell and Heaviside form of the 3D-stress tensor [72],

$$\begin{aligned} \widetilde{T}_{pq}=\frac{1}{4\pi }\left( E_{p}D_{q}+H_{p}B_{q}\right) -\frac{1}{8\pi }\delta _{pq}\left( \mathbf {E}\cdot \mathbf {D}+\mathbf {H} \cdot \mathbf {B}\right) , \end{aligned}$$

(7.5)

appears here in the corresponding “momentum” balance equation [86]:

$$\begin{aligned}&-\frac{\partial }{\partial t}\left[ \frac{1}{4\pi c}\left( \mathbf {D} \times \mathbf {B}\right) \right] _{p}+\frac{\partial \widetilde{T}_{pq} }{\partial x_{q}}-\left( \rho \mathbf {E}+\frac{1}{c}\mathbf {j}\times \mathbf {B}\right) _{p}\\&\quad +\frac{1}{8\pi }\left( \mathbf {E}\cdot \frac{\partial \mathbf {D} }{\partial x_{p}}-\mathbf {D}\cdot \frac{\partial \mathbf {E}}{\partial x_{p} }+\mathbf {H}\cdot \frac{\partial \mathbf {B}}{\partial x_{p}}-\mathbf {B} \cdot \frac{\partial \mathbf {H}}{\partial x_{p}}\right) =0.\nonumber \end{aligned}$$

(7.6)

At the same time, in view of (3.16), use of the form (7.5) differs from Hertz’s symmetric tensors in (3.1) and (3.17) only in the case of anisotropic media (crystals) [72], [85].

Indeed,

$$\begin{aligned} 8\pi \frac{\partial }{\partial x_{q}}\left( \widetilde{T}_{pq}-T_{pq}\right) =\left[ {\text {curl}}\,\left( \mathbf {E\times D}+\mathbf {H} \times \mathbf {B}\right) \right] _{p}. \end{aligned}$$

(7.7)

Moreover, with the help of elementary identities,

$$\begin{aligned} \left[ {\text {curl}}\,\left( \mathbf {A\times B}\right) \right] _{p}=\frac{\partial }{\partial x_{q}}\left( A_{p}B_{q}-A_{q}B_{p}\right) \end{aligned}$$

(7.8)

and

$$\begin{aligned} 2\frac{\partial }{\partial x_{q}}\left( A_{p}B_{q}\right) =\frac{\partial }{\partial x_{q}}\left( A_{p}B_{q}+A_{q}B_{p}\right) +\left[ {\text {curl}}\,\left( \mathbf {A\times B}\right) \right] _{p}, \end{aligned}$$

(7.9)

one can transform the latter balance equation into its “symmetric” form, which provides an independent proof of (3.16).

(For further discussion of symmetric and non-symmetric forms of the energy-momentum and stress tensors, the interested reader is referred to the classical accounts [32], [62], [72], [85]; see also the references therein.)

7.2 Proof

The fact that Maxwell’s equations can be united with the help of a complex second rank (anti-symmetric) tensor allows us to utilize the standard Sturm–Liouville type argument in order to establish the energy-momentum differential balance equations in covariant form. Indeed, by adding matrix equation

$$\begin{aligned} P_{\mu \lambda }^{*}\left( \frac{\partial Q^{\lambda \nu }}{\partial x^{\nu } }=-\frac{4\pi }{c}j^{\lambda }\right) \end{aligned}$$

(7.10)

and its complex conjugate

$$\begin{aligned} P_{\mu \lambda }\left( \frac{\partial \overset{*}{\left. Q^{\lambda \nu }\right. }}{\partial x^{\nu }}=-\frac{4\pi }{c}j^{\lambda }\right) \end{aligned}$$

(7.11)

one gets

$$\begin{aligned} P_{\mu \lambda }^{*}\frac{\partial Q^{\lambda \nu }}{\partial x^{\nu }} +P_{\mu \lambda }\frac{\partial \overset{*}{\left. Q^{\lambda \nu }\right. } }{\partial x^{\nu }}=-\frac{8\pi }{c}F_{\mu \lambda }j^{\lambda }. \end{aligned}$$

(7.12)

A simple decomposition,

$$\begin{aligned} f\frac{\partial g}{\partial x}=\frac{1}{2}\frac{\partial }{\partial x}\left( fg\right) +\frac{1}{2}\left( f\frac{\partial g}{\partial x}-\frac{\partial f}{\partial x}g\right) \end{aligned}$$

(7.13)

with $f=P_{\mu \lambda }^{*}$ and $g=Q^{\lambda \nu }$ (and their complex conjugates), results in

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left[ \frac{1}{16\pi }\left( P_{\mu \lambda }^{*}Q^{\lambda \nu }+P_{\mu \lambda }\overset{*}{\left. Q^{\lambda \nu }\right. }\right) \right] \\&\ +\frac{1}{16\pi }\left[ \left( P_{\mu \lambda }^{*}\frac{\partial Q^{\lambda \nu }}{\partial x^{\nu }}-\frac{\partial P_{\mu \lambda }}{\partial x^{\nu }}\overset{*}{\left. Q^{\lambda \nu }\right. }\right) +\left( \text {c.c.}\right) \right] =-\frac{1}{c}F_{\mu \lambda }j^{\lambda }.\nonumber \end{aligned}$$

(7.14)

By a direct substitution, one can verify that

$$\begin{aligned}&Z_{\mu }=P_{\mu \lambda }^{*}\frac{\partial Q^{\lambda \nu }}{\partial x^{\nu }}-\frac{\partial P_{\mu \lambda }}{\partial x^{\nu }}\overset{*}{\left. Q^{\lambda \nu }\right. }=\frac{1}{2}P_{\sigma \tau }^{*} \frac{\partial Q^{\tau \sigma }}{\partial x^{\mu }} \\&\quad \ =-\frac{1}{2}\overset{*}{\left. Q^{\sigma \tau }\right. } \frac{\partial P_{\tau \sigma }}{\partial x^{\mu }}=\mathbf {F}^{*}\cdot \frac{\partial \mathbf {G}}{\partial x^{\mu }}-\mathbf {G}^{*}\cdot \frac{\partial \mathbf {F}}{\partial x^{\mu }}.\nonumber \end{aligned}$$

(7.15)

(An independent covariant proof of these identities is given in appendix C.) Finally, introducing

$$\begin{aligned} 16\pi X_{\mu }=Z_{\mu }+Z_{\mu }^{*}, \end{aligned}$$

(7.16)

we obtain (7.1) with the explicitly covariant expression for the ponderomotive force (3.19), which completes the proof.

As a result, the covariant energy-momentum balance equation is given by

$$\begin{aligned} \frac{\partial }{\partial x^{\nu }}T_{\mu }{}^{\nu }+X_{\mu }=-\frac{1}{c} F_{\mu \lambda }j^{\lambda }, \end{aligned}$$

(7.17)

in a compact form. If these differential balance equations are written for a stationary medium, then the corresponding equations for moving bodies are uniquely determined, since the components of a tensor in any inertial coordinate system can be derived by a proper Lorentz transformation [72].

8 Covariant Derivation of Angular Momentum Balance

By definition, $x_{\mu }=g_{\mu \nu }x^{\nu }=\left( ct,-\mathbf {r}\right)$ and $T_{\mu \lambda }=T_{\mu }{}^{\nu }g_{\nu \lambda },$ where $g_{\mu \nu }=\partial x_{\mu }/\partial x^{\nu }$ = diag$\left( 1,-1,-1,-1\right)$. In view of (7.17), we derive

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left( x_{\lambda }T_{\mu }{}^{\nu }-x_{\mu }T_{\lambda }{}^{\nu }\right) =\left( T_{\mu \lambda }-T_{\lambda \mu }\right) \\&-\left( x_{\lambda }X_{\mu }-x_{\mu }X_{\lambda }\right) -\frac{1}{c}\left( x_{\lambda }F_{\mu \nu }-x_{\mu }F_{\lambda \nu }\right) j^{\nu }\nonumber \end{aligned}$$

(8.1)

as a required differential balance equation.

With the help of familiar dual relations (B.4), one can get another covariant form of the angular momentum balance equation:

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left( e^{\mu \lambda \sigma \tau }x_{\sigma }T_{\tau }{}^{\nu }\right) +e^{\mu \lambda \sigma \tau }T_{\sigma \tau }\\&\ +e^{\mu \lambda \sigma \tau }x_{\sigma }X_{\tau }{}+\frac{1}{c}e^{\mu \lambda \sigma \tau }x_{\sigma }F_{\tau \nu }j^{\nu }=0^{\mu \lambda }.\nonumber \end{aligned}$$

(8.2)

In 3D-form, the latter relation can be reduced to (4.3)–(4.5).

Indeed, when $\mu =0$ and $\lambda =p=1,2,3,$ one gets

$$\begin{aligned}&-\frac{1}{4\pi c}\frac{\partial }{\partial t}\left[ e_{pqr}x_{q}\left( \mathbf {D}\times \mathbf {B}\right) _{r}\right] +\frac{\partial }{\partial x_{s}}\left( e_{pqr}x_{q}\widetilde{T}_{rs}\right) \\&\qquad +e_{pqr}\widetilde{T}_{qr}+e_{pqr}x_{q}\left( X_{r}+Y_{r}\right) =0,\nonumber \end{aligned}$$

(8.3)

where $-\mathbf {Y}=\rho \mathbf {E}+\mathbf {j}\times \mathbf {B}/c$ is the familiar Lorentz force. Substitution, $\widetilde{T}_{rs}=T_{rs}+\left( \widetilde{T}_{rs}-T_{rs}\right) ,$ results in (4.3) in view of identity (7.7). The remaining cases, when $\mu ,\nu =p,q=1,2,3,$ can be analyzed in a similar fashion. In 3D-form, the corresponding equations are equivalent to (3.15) and (7.6). Details are left to the reader.

Thus the angular momentum law has the form of a local balance equation, not a conservation law, since in general, the energy-momentum tensor will not be symmetric [34]. A torque, for instance, may occur, which cannot be compensated for by a change in the electromagnetic angular momentum, though not in contradiction with experiment [72].

Fig. 1

Complex electromagnetic fields decomposition.

9 Transformation Laws of Complex Electromagnetic Fields

Let $\mathbf {v}$ be a constant real-valued velocity vector representing uniform motion of one frame of reference with respect to another one. Let us consider the following orthogonal decompositions,

$$\begin{aligned} \mathbf {F}=\mathbf {F}_{\parallel }+\mathbf {F}_{\perp },\qquad \mathbf {G} =\mathbf {G}_{\parallel }+\mathbf {G}_{\perp }, \end{aligned}$$

(9.1)

such that our complex vectors $\left\{ \mathbf {F}_{\parallel },\mathbf {G} _{\parallel }\right\}$ are collinear with the velocity vector $\mathbf {v}$ and $\left\{ \mathbf {F}_{\perp },\mathbf {G}_{\perp }\right\}$ are perpendicular to it (Figure 1). The Lorentz transformation of electric and magnetic fields $\left\{ \mathbf {E,D},\mathbf {H},\mathbf {B}\right\}$ take the following complex form

$$\begin{aligned} \mathbf {F}_{\parallel }^{\prime }=\mathbf {F}_{\parallel },\qquad \mathbf {G} _{\parallel }^{\prime }=\mathbf {G}_{\parallel } \end{aligned}$$

(9.2)

and

$$\begin{aligned} \mathbf {F}_{\perp }^{\prime }=\frac{\mathbf {F}_{\perp }-\dfrac{i}{c}\left( \mathbf {v\times G}\right) }{\sqrt{1-v^{2}/c^{2}}},\qquad \mathbf {G}_{\perp }^{\prime }=\frac{\mathbf {G}_{\perp }-\dfrac{i}{c}\left( \mathbf {v\times F}\right) }{\sqrt{1-v^{2}/c^{2}}}. \end{aligned}$$

(9.3)

(Although this transformation was found by Lorentz, it was Minkowski who realized that this is the law of transformation of the second rank anti-symmetric four-tensors [58], [65] ; a brief historical overview is given in [72].) This complex 3D-form of the Lorentz transformation of electric and magnetic fields was known to Minkowski (1908), but apparently only in vacuum, when $\mathbf {G} =\mathbf {F}$ (see also [88]). Moreover,

$$\begin{aligned} \mathbf {r}_{\parallel }^{\prime }=\frac{\mathbf {r}_{\parallel }-\mathbf {v} t}{\sqrt{1-v^{2}/c^{2}}},\qquad \mathbf {r}_{\perp }^{\prime }=\mathbf {r}_{\perp },\qquad t^{\prime }=\frac{t-\left( \mathbf {v}\cdot \mathbf {r}\right) /c^{2} }{\sqrt{1-v^{2}/c^{2}}}, \end{aligned}$$

(9.4)

in the same notation [72].

The latter equations can be rewritten as follows

$$\begin{aligned} \mathbf {r}^{\prime }=\mathbf {r}+\left[ \left( \gamma -1\right) \frac{\mathbf {v}\cdot \mathbf {r}}{v^{2}}-\gamma t\right] \mathbf {v},\qquad t^{\prime }=\gamma \left( t-\frac{\mathbf {v}\cdot \mathbf {r}}{c^{2}}\right) , \end{aligned}$$

(9.5)

where $\gamma =\left( 1-v^{2}/c^{2}\right) ^{-1/2}.$ In a similar fashion, one gets

$$\begin{aligned} \mathbf {F}^{\prime }=\gamma \left( \mathbf {F}-\dfrac{i}{c}\mathbf {v\times G}\right) -\left( \gamma -1\right) \frac{\mathbf {v}\cdot \mathbf {F}}{v^{2} }\mathbf {v}, \end{aligned}$$

(9.6)

$$\begin{aligned} \mathbf {G}^{\prime }=\gamma \left( \mathbf {G}-\dfrac{i}{c}\mathbf {v\times F}\right) -\left( \gamma -1\right) \frac{\mathbf {v}\cdot \mathbf {G}}{v^{2} }\mathbf {v}, \end{aligned}$$

(9.7)

as a compact 3D-version of the Lorentz transformation for the complex electromagnetic fields.

In complex four-tensor form,

$$\begin{aligned} \left. Q^{\prime }\right. ^{\mu \nu }\left( x^{\prime }\right) =\varLambda _{\ \sigma }^{\mu }\varLambda _{\ \tau }^{\nu }Q^{\sigma \tau }\left( x\right) ,\qquad x^{\prime }=\varLambda x. \end{aligned}$$

(9.8)

Although Minkowski considered the transformation of electric and magnetic fields in a complex 3D-vector form, see Eqs. (8)–(9) and (15) in [65] (or Eqs. (25.5)–(25.6) in [50]), he seems never to have combined the corresponding four-tensors into the complex forms (6.1)–(6.2). In the second article [66], Max Born, who used Minkowski’s notes, didn’t mention the complex fields. As a result, the complex field tensor seems only to have appeared, for the first time, in [54] (see also [87]). The complex identity, $\mathbf {F}\cdot \mathbf {G}=~$invariant under a similarity transformation, follows from Minkowski’s determinant relations (B.23)–(B.25).

Fig. 2

Example of a moving frame.

Example. Let $\left\{ \mathbf {e}_{k}\right\} _{k=1}^{3}$ be an orthonormal basis in $\left. \mathbb {R} \right. ^{3}.$ We choose $\mathbf {v}=v\mathbf {e}_{1}$ and write $x^{\prime \mu }=\varLambda _{\ \nu }^{\mu }x^{\nu }$ with

$$\begin{aligned} \varLambda _{\ \nu }^{\mu }=\left( \begin{array} [c]{cccc} \gamma &{} -\beta \gamma &{} 0 &{} 0\\ -\beta \gamma &{} \gamma &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) ,\qquad \quad \beta =\frac{v}{c},\quad \gamma =\frac{1}{\sqrt{1-\beta ^{2} }} \end{aligned}$$

(9.9)

for the corresponding Lorentz boost (Figure 2).

In view of (9.8), by matrix multiplication one gets

$$\begin{aligned}&\left( \begin{array} [c]{cccc} \gamma &{} -\beta \gamma &{} 0 &{} 0\\ -\beta \gamma &{} \gamma &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) \left( \begin{array} [c]{cccc} 0 &{} -G_{1} &{} -G_{2} &{} -G_{3}\\ G_{1} &{} 0 &{} iF_{3} &{} -iF_{2}\\ G_{2} &{} -iF_{3} &{} 0 &{} iF_{1}\\ G_{3} &{} iF_{2} &{} -iF_{1} &{} 0 \end{array} \right) \left( \begin{array} [c]{cccc} \gamma &{} -\beta \gamma &{} 0 &{} 0\\ -\beta \gamma &{} \gamma &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) \nonumber \\&\qquad =\left( \begin{array} [c]{cccc} 0 &{} -G_{1} &{} -\gamma G_{2}-i\beta \gamma F_{3} &{} -\gamma G_{3}+i\beta \gamma F_{2}\\ G_{1} &{} 0 &{} \beta \gamma G_{2}+i\gamma F_{3} &{} \beta \gamma G_{3}-i\gamma F_{2}\\ \gamma G_{2}+i\beta \gamma F_{3} &{} -\beta \gamma G_{2}-i\gamma F_{3} &{} 0 &{} iF_{1}\\ \gamma G_{3}-i\beta \gamma F_{2} &{} -\beta \gamma G_{3}+i\gamma F_{2} &{} -iF_{1} &{} 0 \end{array} \right) . \end{aligned}$$

(9.10)

Thus $G_{1}^{\prime }=G_{1}$ and

$$\begin{aligned} G_{2}^{\prime }&=\gamma G_{2}+i\beta \gamma F_{3}=\frac{G_{2}+i\left( v/c\right) F_{3}}{\sqrt{1-v^{2}/c^{2}}}=\frac{G_{2}-\dfrac{i}{c}\left( \mathbf {v\times F}\right) _{2}}{\sqrt{1-v^{2}/c^{2}}},\\ G_{3}^{\prime }&=\gamma G_{3}-i\beta \gamma F_{2}=\frac{G_{3}-i\left( v/c\right) F_{2}}{\sqrt{1-v^{2}/c^{2}}}=\frac{G_{3}-\dfrac{i}{c}\left( \mathbf {v\times F}\right) _{3}}{\sqrt{1-v^{2}/c^{2}}}.\nonumber \end{aligned}$$

(9.11)

In a similar fashion, $F_{1}^{\prime }=F_{1}$ and

$$\begin{aligned} F_{2}^{\prime }&=\gamma F_{2}+i\beta \gamma G_{3}=\frac{F_{2}-\dfrac{i}{c}\left( \mathbf {v\times G}\right) _{2}}{\sqrt{1-v^{2}/c^{2}}} ,\\ F_{3}^{\prime }&=\gamma F_{3}-i\beta \gamma G_{2}=\frac{F_{3}-\dfrac{i}{c}\left( \mathbf {v\times G}\right) _{3}}{\sqrt{1-v^{2}/c^{2}}}.\nonumber \end{aligned}$$

(9.12)

The reader can easily verify that the latter relations are in agreement with the complex field transformations (9.2)–(9.3).

In block form, one gets

$$\begin{aligned} \left( \begin{array} [c]{c} F_{1}^{\prime }\\ F_{2}^{\prime }\\ G_{3}^{\prime }\\ G_{2}^{\prime }\\ F_{3}^{\prime }\\ G_{1}^{\prime } \end{array} \right) =\left( \begin{array} [c]{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} \cos \left( i\theta \right) &{} \sin \left( i\theta \right) &{} 0 &{} 0 &{} 0\\ 0 &{} -\sin \left( i\theta \right) &{} \cos \left( i\theta \right) &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cos \left( i\theta \right) &{} \sin \left( i\theta \right) &{} 0\\ 0 &{} 0 &{} 0 &{} -\sin \left( i\theta \right) &{} \cos \left( i\theta \right) &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array} \right) \left( \begin{array} [c]{c} F_{1}\\ F_{2}\\ G_{3}\\ G_{2}\\ F_{3}\\ G_{1} \end{array} \right) , \end{aligned}$$

(9.13)

where, by definition,

$$\begin{aligned} \cos \left( i\theta \right) =\gamma =\frac{1}{\sqrt{1-\beta ^{2}}},\qquad \sin \left( i\theta \right) =i\beta \gamma =\frac{i\beta }{\sqrt{1-\beta ^{2}} },\qquad \beta =\frac{v}{c}. \end{aligned}$$

(9.14)

As a result, the transformation law of the complex electromagnetic fields $\left\{ \mathbf {F},\mathbf {G}\right\}$ under the Lorentz boost can be thought of as a complex rotation in $\left. \mathbb {C} \right. ^{6},$ corresponding to a reducible representation of the one-parameter subgroup of $SO\left( 3, \mathbb {C} \right) .$ (Cyclic permutation of the spatial indices cover the two remaining cases; see also [88].)

10 Material Equations, Potentials, and Energy-Momentum Tensor for Moving Isotropic Media

Electromagnetic phenomena in moving media are important in relativistic astrophysics, the study of accelerated plasma clusters and high-energy electron beams [15], [16], [26], [91].

10.1 Material equations

Minkowski’s field- and connecting-equations [65], [66] were derived from the corresponding laws for the bodies at rest by means of a Lorentz transformation (see [15], [18], [34], [51], [67], [72], [91]). Explicitly covariant forms, which are applicable both in the rest frame and for moving media, are analyzed in [15], [16], [34], [39], [40], [67], [71], [72], [75], [77], [88], [91] (see also the references therein). In standard notation,

$$\begin{aligned} \beta =v/c,\qquad \gamma =\left( 1-\beta ^{2}\right) ^{-1/2},\qquad v=\left| \mathbf {v}\right| ,\qquad \kappa =\varepsilon \mu -1, \end{aligned}$$

(10.1)

one can write [15], [16], [18], [91]:

$$\begin{aligned} \mathbf {D}&=\varepsilon \mathbf {E}+\frac{\kappa \gamma ^{2}}{\mu }\left[ \beta ^{2}\mathbf {E}-\dfrac{\mathbf {v}}{c^{2}}\left( \mathbf {v\cdot E}\right) +\dfrac{1}{c}\left( \mathbf {v\times B}\right) \right] ,\\ \mathbf {H}&=\frac{1}{\mu }\mathbf {B}+\frac{\kappa \gamma ^{2}}{\mu }\left[ -\beta ^{2}\mathbf {B}+\dfrac{\mathbf {v}}{c^{2}}\left( \mathbf {v\cdot B}\right) +\dfrac{1}{c}\left( \mathbf {v\times E}\right) \right] .\nonumber \end{aligned}$$

(10.2)

In covariant form, these relations are given by

$$\begin{aligned} R^{\lambda \nu }&=\varepsilon ^{\lambda \nu \sigma \tau }F_{\sigma \tau }=\frac{1}{2}\left( \varepsilon ^{\lambda \nu \sigma \tau }-\varepsilon ^{\lambda \nu \tau \sigma }\right) F_{\sigma \tau }\\&=\frac{1}{4}\left( \varepsilon ^{\lambda \nu \sigma \tau }-\varepsilon ^{\lambda \nu \tau \sigma }+\varepsilon ^{\nu \lambda \tau \sigma }-\varepsilon ^{\nu \lambda \sigma \tau }\right) F_{\sigma \tau }\nonumber \end{aligned}$$

(10.3)

(see [14], [15], [16], [39], [40], [75], [77], [91] and the references therein). Here,

$$\begin{aligned} \varepsilon ^{\lambda \nu \sigma \tau }=\frac{1}{\mu }\left( g^{\lambda \sigma }+\kappa u^{\lambda }u^{\sigma }\right) \left( g^{\nu \tau }+\kappa u^{\nu }u^{\tau }\right) =\varepsilon ^{\nu \lambda \tau \sigma } \end{aligned}$$

(10.4)

is the four-tensor of electric and magnetic permeabilities⁴ and

$$\begin{aligned} u^{\lambda }=\left( \gamma ,\gamma \mathbf {v}/c\right) ,\qquad u^{\lambda }u_{\lambda }=1 \end{aligned}$$

(10.5)

is the four-velocity of the medium (a computer algebra verification of these relations is given in [52]). In a complex covariant form,

$$\begin{aligned} \left( Q^{\mu \nu }+\overset{*}{\left. Q^{\mu \nu }\right. }\right) =\varepsilon ^{\mu \nu \sigma \tau }\left( P_{\sigma \tau }+\overset{*}{\left. P_{\sigma \tau }\right. }\right) . \end{aligned}$$

(10.6)

In view of (10.3) and (B.5)–(B.6), we get

$$\begin{aligned} Q^{\mu \nu }=\left( \varepsilon ^{\mu \nu \sigma \tau }-\frac{i}{2}e^{\mu \nu \sigma \tau }\right) F_{\sigma \tau },\qquad P_{\mu \nu }=\left( \delta _{\mu }^{\lambda }\delta _{\nu }^{\rho }-\frac{i}{2}e_{\mu \nu \sigma \tau }\varepsilon ^{\sigma \tau \lambda \rho }\right) F_{\lambda \rho }, \end{aligned}$$

(10.7)

in terms of the real-valued electromagnetic field tensor.

10.2 Potentials

In practice, one can choose

$$\begin{aligned} F_{\sigma \tau }=\frac{\partial A_{\tau }}{\partial x^{\sigma }}-\frac{\partial A_{\sigma }}{\partial x^{\tau }}, \end{aligned}$$

(10.8)

for the real-valued four-vector potential $A_{\lambda }\left( x\right) .$ Then

$$\begin{aligned} \partial _{\nu }Q^{\lambda \nu }&=\varepsilon ^{\lambda \nu \sigma \tau }\partial _{\nu }\left( \partial _{\sigma }A_{\tau }-\partial _{\tau }A_{\sigma }\right) -\frac{i}{2}e^{\lambda \nu \sigma \tau }\partial _{\nu }\left( \partial _{\sigma }A_{\tau }-\partial _{\tau }A_{\sigma }\right) \\&=\frac{1}{\mu }\left( g^{\lambda \sigma }+\kappa u^{\lambda }u^{\sigma }\right) \left( g^{\nu \tau }+\kappa u^{\nu }u^{\tau }\right) \partial _{\nu }\left( \partial _{\sigma }A_{\tau }-\partial _{\tau }A_{\sigma }\right) \end{aligned}$$

by (10.4). Substitution into Maxwell’s equations (6.6) or (6.5) results in

$$\begin{aligned}&\left( g^{\lambda \sigma }+\kappa u^{\lambda }u^{\sigma }\right) \left\{ -\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] A_{\sigma }\right. \\&\qquad \left. +\partial _{\sigma }\left( \partial ^{\tau }A_{\tau }+\kappa u^{\nu }u^{\tau }\partial _{\nu }A_{\tau }\right) ^{\ }\!\right\} =-\frac{4\pi \mu }{c}j^{\lambda },\nonumber \end{aligned}$$

(10.9)

where $-\partial ^{\tau }\partial _{\tau }=-g^{\sigma \tau }\partial _{\sigma }\partial _{\tau }=\varDelta -\left( \partial /c\partial t\right) ^{2}$ is the d’Alembert operator. In view of an inverse matrix identity,

$$\begin{aligned} \left( g_{\lambda \rho }-\frac{\kappa }{1+\kappa }u_{\lambda }u_{\rho }\right) \left( g^{\lambda \sigma }+\kappa u^{\lambda }u^{\sigma }\right) =\delta _{\rho }^{\sigma }, \end{aligned}$$

(10.10)

the latter equations take the form⁵

$$\begin{aligned}&\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] A_{\sigma }-\partial _{\sigma }\left( \partial ^{\tau }A_{\tau }+\kappa u^{\nu }u^{\tau }\partial _{\nu }A_{\tau }\right) \\&\qquad =\frac{4\pi \mu }{c}\left( g_{\sigma \lambda }-\frac{\kappa }{1+\kappa }u_{\sigma }u_{\lambda }\right) j^{\lambda }.\nonumber \end{aligned}$$

(10.11)

Subject to the subsidiary condition,

$$\begin{aligned} \partial ^{\tau }A_{\tau }+\kappa u^{\nu }u^{\tau }\partial _{\nu }A_{\tau }=\left( g^{\nu \tau }+\kappa u^{\nu }u^{\tau }\right) \partial _{\nu }A_{\tau }=0, \end{aligned}$$

(10.12)

these equations were studied in detail for the sake of development of the phenomenological classical and quantum electrodynamics in a moving medium (see [13], [14], [15], [16], [71], [75], [76], [77], [91] and the references therein). In particular, Green’s function of the photon in a moving medium was studied in [39], [75], [76] (with applications to quantum electrodynamics).

10.3 Hertz’s tensor and vectors

We follow [15], [16], [91] with somewhat different details. The substitution,

$$\begin{aligned} A^{\mu }\left( x\right) =\left( \frac{\kappa }{1+\kappa }u^{\mu }u_{\lambda }-\delta _{\lambda }^{\mu }\right) \partial _{\sigma }Z^{\lambda \sigma }\left( x\right) \end{aligned}$$

(10.13)

(a generalization of Hertz’s potentials for a moving medium [15], [91]), into the gauge condition (10.12) results in $Z^{\lambda \sigma }=-Z^{\sigma \lambda },$ in view of

$$\begin{aligned}&\left( g_{\nu \mu }+\kappa u_{\nu }u_{\mu }\right) \partial ^{\nu }A^{\mu }\\&\quad =\left( g_{\nu \mu }+\kappa u_{\nu }u_{\mu }\right) \left( \frac{\kappa }{1+\kappa }u^{\mu }u_{\lambda }-\delta _{\lambda }^{\mu }\right) \partial ^{\nu }\partial _{\sigma }Z^{\lambda \sigma }\\&\qquad =-g_{\nu \lambda }\partial ^{\nu }\partial _{\sigma }Z^{\lambda \sigma }=-\partial _{\lambda }\partial _{\sigma }Z^{\lambda \sigma }\equiv 0. \end{aligned}$$

Then, equations (10.11) take the form

$$\begin{aligned} \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] \partial _{\sigma }Z^{\lambda \sigma }=-\frac{4\pi \mu }{c}j^{\lambda }. \end{aligned}$$

(10.14)

Indeed, the left-hand side of (10.11) is given by

$$\begin{aligned}&\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] A_{\sigma }=\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] g_{\sigma \mu }A^{\mu }\\&\quad =\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau } \partial _{\tau }\right) ^{2}\right] g_{\sigma \mu }\left( \frac{\kappa }{1+\kappa }u^{\mu }u_{\lambda }-\delta _{\lambda }^{\mu }\right) \partial _{\rho }Z^{\lambda \rho }\\&\quad =\left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau } \partial _{\tau }\right) ^{2}\right] \left( \frac{\kappa }{1+\kappa }u_{\sigma }u_{\lambda }-g_{\sigma \lambda }\right) \partial _{\rho }Z^{\lambda \rho }\\&\qquad =\frac{4\pi \mu }{c}\left( g_{\sigma \lambda }-\frac{\kappa }{1+\kappa }u_{\sigma }u_{\lambda }\right) j^{\lambda }, \end{aligned}$$

from which the result follows due to (10.10).

Finally, with the help of the standard substitution,

$$\begin{aligned} j^{\lambda }=c\partial _{\sigma }p^{\lambda \sigma },\qquad p^{\lambda \sigma }=-p^{\sigma \lambda } \end{aligned}$$

(10.15)

(in view of $\partial _{\lambda }j^{\lambda }=c\partial _{\lambda }\partial _{\sigma }p^{\lambda \sigma }\equiv 0),$ we arrive at

$$\begin{aligned} \partial _{\sigma }\left\{ \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] Z^{\lambda \sigma }+4\pi \mu p^{\lambda \sigma }\right\} =0. \end{aligned}$$

(10.16)

Therefore, one can choose

$$\begin{aligned} \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] Z^{\lambda \nu }=-4\pi \mu p^{\lambda \nu }. \end{aligned}$$

(10.17)

Here, by definition,

$$\begin{aligned} p^{\lambda \nu }=\left( \begin{array} [c]{cccc} 0 &{} -p_{1} &{} -p_{2} &{} -p_{3}\\ p_{1} &{} 0 &{} m_{3} &{} -m_{2}\\ p_{2} &{} -m_{3} &{} 0 &{} m_{1}\\ p_{3} &{} m_{2} &{} -m_{1} &{} 0 \end{array} \right) \end{aligned}$$

(10.18)

is an anti-symmetric four-tensor [15], [16], [91]. The “electric” and “magnetic” Hertz vectors, $\mathbf {Z} ^{\left( e\right) }$ and $\mathbf {Z}^{\left( m\right) },$ respectively, are also introduced in terms of a single four-tensor,

$$\begin{aligned} Z^{\lambda \nu }=\left( \begin{array} [c]{cccc} 0 &{} Z_{1}^{\left( e\right) } &{} Z_{2}^{\left( e\right) } &{} Z_{3}^{\left( e\right) }\\ -Z_{1}^{\left( e\right) } &{} 0 &{} -Z_{3}^{\left( m\right) } &{} Z_{2}^{\left( m\right) }\\ -Z_{2}^{\left( e\right) } &{} Z_{3}^{\left( m\right) } &{} 0 &{} -Z_{1}^{\left( m\right) }\\ -Z_{3}^{\left( e\right) } &{} -Z_{2}^{\left( m\right) } &{} Z_{1}^{\left( m\right) } &{} 0 \end{array} \right) . \end{aligned}$$

(10.19)

In view of (10.13), for the four-vector potential, $A^{\lambda }=\left( \varphi ,\mathbf {A}\right) ,$ we obtain

$$\begin{aligned} \varphi =-\left( 1-\frac{\kappa \gamma ^{2}}{1+\kappa }\right) {\text {div}}\,\mathbf {Z}^{\left( e\right) }+\frac{\kappa \gamma ^{2} }{\left( 1+\kappa \right) c}\mathbf {v}\cdot \left( \frac{\partial \mathbf {Z}^{\left( e\right) }}{c\partial t}+{\text {curl}}\, \mathbf {Z}^{\left( m\right) }\right) \end{aligned}$$

(10.20)

and

$$\begin{aligned}&\mathbf {A}=\frac{\partial \mathbf {Z}^{\left( e\right) }}{c\partial t}+{\text {curl}}\,\mathbf {Z}^{\left( m\right) }\\&\quad +\frac{\kappa \gamma ^{2}\mathbf {v}}{\left( 1+\kappa \right) c^{2} }\left[ c{\text {div}}\,\mathbf {Z}^{\left( e\right) }+\frac{\partial }{c\partial t}\left( \mathbf {v}\cdot \mathbf {Z}^{\left( e\right) }\right) +\mathbf {v}\cdot {\text {curl}}\,\mathbf {Z}^{\left( m\right) }\right] .\nonumber \end{aligned}$$

(10.21)

Then, equations (10.17) take the form

$$\begin{aligned} \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] \mathbf {Z}^{\left( e\right) }=4\pi \mu \mathbf {p} ,\qquad \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau } \partial _{\tau }\right) ^{2}\right] \mathbf {Z}^{\left( m\right) }=4\pi \mu \mathbf {m} \end{aligned}$$

(10.22)

and, for the four-current, $j^{\lambda }=\left( c\rho ,\mathbf {j}\right) ,$ one gets

$$\begin{aligned} \rho =-{\text {div}}\,\mathbf {p},\qquad \mathbf {j}=\frac{\partial \mathbf {p} }{\partial t}+c{\text {curl}}\,\mathbf {m} \end{aligned}$$

(10.23)

in terms of the corresponding electric $\mathbf {p}$ and magnetic $\mathbf {m}$ moments, respectively (see [15], [16], [91] for more details).

Moreover, in 3D-complex form,

$$\begin{aligned} \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] \mathbf {W}=4\pi \mu \varvec{\zeta }, \end{aligned}$$

(10.24)

where $\mathbf {W}=\mathbf {Z}^{\left( e\right) }+i\mathbf {Z}^{\left( m\right) }$ and $\varvec{\zeta }=\mathbf {p}+i\mathbf {m},$ by definition. In a similar fashion,

$$\begin{aligned} \left[ \partial ^{\tau }\partial _{\tau }+\kappa \left( u^{\tau }\partial _{\tau }\right) ^{2}\right] W^{\lambda \nu }=-4\pi \mu \zeta ^{\lambda \nu }, \end{aligned}$$

(10.25)

for the corresponding (self-dual) four-tensors:

$$\begin{aligned} W^{\lambda \nu }=Z^{\lambda \nu }+\frac{i}{2}e^{\lambda \nu \sigma \tau }Z_{\sigma \tau },\qquad \zeta ^{\lambda \nu }=p^{\lambda \nu }+\frac{i}{2}e^{\lambda \nu \sigma \tau }p_{\sigma \tau }. \end{aligned}$$

(10.26)

The Hertz vector and tensor potentials, for a moving medium and at rest, were utilized in [15], [16], [28], [41], [86], [91], [96] (see also the references therein). Many classical problems of radiation and propagation can be consistently solved by using these potentials.

10.4 Energy-momentum tensor

In the case of the covariant version of the energy-momentum tensor given by (7.2), the differential balance equations under consideration are independent of the particular choice of the frame of reference. Therefore, our relations (10.7) are useful for derivation of the expressions for the energy-momentum tensor and the ponderomotive force for moving bodies from those for bodies at rest which were extensively studied in the literature. For example, one gets

$$\begin{aligned} 4\pi T_{\mu }{}^{\nu }=F_{\mu \lambda }\varepsilon ^{\lambda \nu \sigma \tau } F_{\sigma \tau }+\frac{1}{4}\delta _{\mu }^{\nu }F_{\sigma \tau }\varepsilon ^{\sigma \tau \lambda \rho }F_{\lambda \rho } \end{aligned}$$

(10.27)

with the help of (10.3)–(10.4) and (B.14) (see also [84]).

10.5 Tamm’s problem and Cherenkov radiation

Let a stationary point charge q be located at the origin of laboratory frame in a moving dispersionless medium with the velocity $\mathbf {v}.$ The time-independent potentials can be written in terms of piecewise defined functions as follows [15], [16], [91]:

$$\begin{aligned} \varphi \left( \mathbf {r}\right) =\frac{qf}{\varepsilon }\frac{\alpha \gamma ^{2}}{\left( r_{\parallel }^{2}+\alpha \gamma ^{2}r_{\perp }^{2}\right) ^{1/2}},\qquad \mathbf {A}\left( \mathbf {r}\right) =-\frac{qf}{\varepsilon }\frac{\kappa \gamma ^{2}}{\left( r_{\parallel }^{2}+\alpha \gamma ^{2}r_{\perp }^{2}\right) ^{1/2}}\frac{\mathbf {v}}{c}, \end{aligned}$$

(10.28)

where $\mathbf {r}=\mathbf {r}_{\parallel }+\mathbf {r}_{\perp }$ and $\alpha =1-\varepsilon \mu \beta ^{2}.$ Here, in the “slower-than-light” case, when $\alpha >0,$ one gets $f\left( \mathbf {r}\right) =1;$ while in the “faster-than-light” case, $\alpha <0,$ we should substitute:

$$\begin{aligned} f\left( \mathbf {r}\right) =\left\{ \begin{array} [c]{cc} 2, &{} \text {when }\mathbf {r}_{\parallel }\text { is parallel to }\mathbf {v} \text { and }r_{\parallel }^{2}>\left| \alpha \right| \gamma ^{2} r_{\perp }^{2};\\ 0, &{} \text {otherwise; if }\mathbf {r}_{\parallel }\text { is anti-parallel to }\mathbf {v},\text { or }r_{\parallel }^{2}<\left| \alpha \right| \gamma ^{2}r_{\perp }^{2} \end{array} \right. \end{aligned}$$

(10.29)

(see [15], [16] for more details and [52] for a direct Mathematica verification). The corresponding (static) electric and magnetic fields are given by

$$\begin{aligned} \mathbf {E}\left( \mathbf {r}\right)&=qf\frac{\alpha \gamma ^{2}}{\left( r_{\parallel }^{2}+\alpha \gamma ^{2}r_{\perp }^{2}\right) ^{3/2}}\left( \mathbf {r}_{\parallel }+\alpha \gamma ^{2}\mathbf {r}_{\perp }\right) ,\quad \mathbf {H}\left( \mathbf {r}\right) =\mathbf {0} ,\\ \mathbf {D}\left( \mathbf {r}\right)&=\varepsilon qf\frac{\alpha \gamma ^{2}}{\left( r_{\parallel }^{2}+\alpha \gamma ^{2}r_{\perp }^{2}\right) ^{3/2} }\ \mathbf {r},\qquad \mathbf {r}=\mathbf {r}_{\parallel }+\mathbf {r}_{\perp },\nonumber \\ \mathbf {B}\left( \mathbf {r}\right)&=\frac{\kappa }{\alpha }\left( \mathbf {E}\times \frac{\mathbf {v}}{c}\right) =qf\frac{\alpha \kappa \gamma ^{4} }{\left( r_{\parallel }^{2}+\alpha \gamma ^{2}r_{\perp }^{2}\right) ^{3/2} }\left( \mathbf {r}\times \frac{\mathbf {v}}{c}\right) .\nonumber \end{aligned}$$

(10.30)

On the other hand, if a charge q is moving with constant velocity $\mathbf {v}$ and the medium is at rest, by a Lorentz transformation, one gets

$$\begin{aligned} \varphi \left( \mathbf {r},t\right) =\frac{qf^{\prime }\left( \mathbf {r} ,t\right) }{\varepsilon \left( \left( \mathbf {r}_{\parallel }-\mathbf {v} t\right) ^{2}+\alpha \mathbf {r}_{\perp }^{2}\right) ^{1/2}},\quad \mathbf {A}\left( \mathbf {r},t\right) =\frac{\mu qf^{\prime }\left( \mathbf {r},t\right) }{\left( \left( \mathbf {r}_{\parallel }-\mathbf {v} t\right) ^{2}+\alpha \mathbf {r}_{\perp }^{2}\right) ^{1/2}}\frac{\mathbf {v} }{c}, \end{aligned}$$

(10.31)

provided

$$\begin{aligned} {\text {div}}\,\mathbf {A}+\frac{\varepsilon \mu }{c}\frac{\partial \varphi }{\partial t}=0. \end{aligned}$$

(10.32)

Here, $f^{\prime }\left( \mathbf {r},t\right) =1,$ if $\alpha >0$ and

$$\begin{aligned} f^{\prime }\left( \mathbf {r},t\right) =\left\{ \begin{array} [c]{cc} 2, &{} \text {when }r_{\parallel }<vt-r_{\perp }\left| \alpha \right| ^{1/2};\\ 0, &{} \text {otherwise,} \end{array} \right. \end{aligned}$$

(10.33)

if $\alpha <0$ (see [70], [91] for the vacuum case). Properties of the Cherenkov radiation, when $\alpha <0$ (the charge velocity is greater than the speed of light in the medium under consideration), are discussed in detail in [3], [15], [16], [91] following the original article [85]. (At every given moment of time, the field is confined to the cone with a vertex angle defined by $\sin \theta =c/v\sqrt{\varepsilon \mu }.)$

11 Real versus Complex Lagrangians

In modern presentations of the classical and quantum field theories, the Lagrangian approach is usually utilized.

11.1 Complex forms

We introduce two quadratic “Lagrangian” densities

$$\begin{aligned} \mathscr {L}_{0}&=\mathscr {L}_{0}^{*}=\frac{1}{2}\left( P_{\sigma \tau }Q^{\tau \sigma }+P_{\sigma \tau }^{*}\overset{*}{\left. Q^{\tau \sigma }\right. }\right) \\&=\frac{i}{4}e^{\sigma \tau \kappa \rho }\left( P_{\sigma \tau }P_{\kappa \rho }-P_{\sigma \tau }^{*}P_{\kappa \rho }^{*}\right) \nonumber \\&=F_{\sigma \tau }R^{\tau \sigma }-G_{\sigma \tau }S^{\tau \sigma }=2F_{\sigma \tau }R^{\tau \sigma }\nonumber \\&=4\left( \mathbf {E}\cdot \mathbf {D-H}\cdot \mathbf {B}\right) \nonumber \end{aligned}$$

(11.1)

and

$$\begin{aligned} \mathscr {L}_{1}&=-\mathscr {L}_{1}^{*}=P_{\sigma \tau }^{*} Q^{\tau \sigma }=\frac{1}{2}\left( P_{\sigma \tau }^{*}Q^{\tau \sigma }-P_{\sigma \tau }\overset{*}{\left. Q^{\tau \sigma }\right. }\right) \\&=\frac{i}{2}e^{\sigma \tau \kappa \rho }P_{\sigma \tau }P_{\kappa \rho }^{*}=4i\left( \mathbf {E}\cdot \mathbf {B-H}\cdot \mathbf {D}\right) .\nonumber \end{aligned}$$

(11.2)

Then, by formal differentiation,

$$\begin{aligned} \frac{\partial \mathscr {L}_{0}}{\partial P_{\alpha \beta }}=Q^{\beta \alpha },\qquad \frac{\partial \mathscr {L}_{0}}{\partial P_{\alpha \beta }^{*} }=\overset{*}{\left. Q^{\beta \alpha }\right. } \end{aligned}$$

(11.3)

and

$$\begin{aligned} \frac{\partial \mathscr {L}_{1}}{\partial P_{\alpha \beta }^{*}}=Q^{\beta \alpha },\qquad \frac{\partial \mathscr {L}_{1}^{*}}{\partial P_{\alpha \beta } }=\overset{*}{\left. Q^{\beta \alpha }\right. } \end{aligned}$$

(11.4)

in view of (B.7).

The complex covariant Maxwell equations (5.2) take the forms

$$\begin{aligned} \frac{\partial }{\partial x^{\nu }}\left( \frac{\partial \mathscr {L}_{0} }{\partial P_{\nu \mu }}\right) =-\frac{4\pi }{c}j^{\mu },\qquad \frac{\partial }{\partial x^{\nu }}\left( \frac{\partial \mathscr {L}_{1}}{\partial P_{\nu \mu } }\right) =\frac{4\pi }{c}j^{\mu } \end{aligned}$$

(11.5)

and the covariant energy-momentum balance relations (7.1) are given by

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left[ \frac{1}{16\pi }\left( P_{\mu \lambda }^{*}\frac{\partial \mathscr {L}_{0}}{\partial P_{\nu \lambda } }+P_{\mu \lambda }\frac{\partial \mathscr {L}_{0}}{\partial P_{\nu \lambda }^{*} }\right) \right] \\&\quad +\frac{1}{32\pi }\left[ P_{\sigma \tau }^{*}\frac{\partial }{\partial x^{\mu }}\left( \frac{\partial \mathscr {L}_{0}}{\partial P_{\sigma \tau } }\right) +P_{\sigma \tau }\frac{\partial }{\partial x^{\mu }}\left( \frac{\partial \mathscr {L}_{0}}{\partial P_{\sigma \tau }^{*}}\right) \right] =-\frac{1}{c}F_{\mu \lambda }j^{\lambda }\nonumber \end{aligned}$$

(11.6)

and

$$\begin{aligned}&\frac{\partial }{\partial x^{\nu }}\left[ \frac{1}{16\pi }\left( P_{\mu \lambda }\frac{\partial \mathscr {L}_{1}}{\partial P_{\nu \lambda }} +P_{\mu \lambda }^{*}\frac{\partial \mathscr {L}_{1}^{*}}{\partial P_{\nu \lambda }}\right) \right] \\&\quad +\frac{1}{32\pi }\left[ P_{\sigma \tau }\frac{\partial }{\partial x^{\mu }}\left( \frac{\partial \mathscr {L}_{1}}{\partial P_{\sigma \tau }}\right) +P_{\sigma \tau }^{*}\frac{\partial }{\partial x^{\mu }}\left( \frac{\partial \mathscr {L}_{1}^{*}}{\partial P_{\sigma \tau }^{*}}\right) \right] =\frac{1}{c}F_{\mu \lambda }j^{\lambda }\nonumber \end{aligned}$$

(11.7)

in terms of the complex Lagrangians under consideration, respectively.

Finally, with the help of the following densities,

$$\begin{aligned} L_{0}=\mathscr {L}_{0}-\frac{4\pi }{c}j^{\nu }A_{\nu },\qquad L_{1}=\mathscr {L} _{1}+\frac{4\pi }{c}j^{\nu }A_{\nu }, \end{aligned}$$

(11.8)

one can derive analogs of the Euler–Lagrange equations for electromagnetic fields in media:

$$\begin{aligned} \frac{\partial }{\partial x^{\nu }}\left( \frac{\partial L_{0,1}}{\partial P_{\nu \mu }}\right) -\frac{\partial L_{0,1}}{\partial A_{\mu }}=0. \end{aligned}$$

(11.9)

In the case of a moving isotropic medium, a relation between $P_{\nu \mu }$ and $A_{\mu }$ is given by our equations (10.7)–(10.8).

11.2 Real form

Taking the real and imaginary parts, Maxwell’s equations (6.6) can be written as follows

$$\begin{aligned} \partial _{\nu }R^{\mu \nu }=-\frac{4\pi }{c}j^{\mu },\qquad \partial _{\nu }S^{\mu \nu }=0. \end{aligned}$$

(11.10)

Here,

$$ -6\partial _{\nu }S^{\mu \nu }=e^{\mu \nu \lambda \sigma }\left( \partial _{\nu }F_{\lambda \sigma }+\partial _{\sigma }F_{\nu \lambda }+\partial _{\lambda } F_{\sigma \nu }\right) \equiv 0, $$

with the help of (6.4) and (10.8). Thus the second set of equations is automatically satisfied when we introduce the four-vector potential. For the inhomogeneous pair of Maxwell’s equations, the Lagrangian density is given by

$$\begin{aligned} L&=\frac{1}{4}F_{\sigma \tau }R^{\tau \sigma }-\frac{4\pi }{c}j^{\sigma }A_{\sigma }\\&=\frac{1}{4}F_{\sigma \tau }\varepsilon ^{\tau \sigma \lambda \rho }F_{\lambda \rho }-\frac{4\pi }{c}j^{\sigma }A_{\sigma },\nonumber \end{aligned}$$

(11.11)

in view of (10.3). Then, for “conjugate momenta” to the four-potential field $A_{\mu },$ one gets

$$\begin{aligned} \frac{\partial L}{\partial \left( \partial _{\nu }A_{\mu }\right) } =\frac{\partial L}{\partial F_{\sigma \tau }}\frac{\partial F_{\sigma \tau } }{\partial \left( \partial _{\nu }A_{\mu }\right) }=R^{\mu \nu } \end{aligned}$$

(11.12)

and the corresponding Euler–Lagrange equations take a familiar form

$$\begin{aligned} \partial _{\nu }\left( \frac{\partial L}{\partial \left( \partial _{\nu }A_{\mu }\right) }\right) -\frac{\partial L}{\partial A_{\mu }}=0. \end{aligned}$$

(11.13)

The latter equation can also be derived with the help of the least action principle [72], [88], [90]. The corresponding Hamiltonian and quantization are discussed in [35], [39], [75] among other classical accounts.

In conclusion, it is worth noting the role of complex fields in quantum electrodynamics, quadratic invariants and quantization (see, for instance, [2], [8], [9], [20], [39], [40], [44], [46], [53], [55], [56], [75], [76], [77], [90], [97]). The classical and quantum theory of Cherenkov radiation is reviewed in [3], [11], [13], [29], [31], [81], [85]. For paraxial approximation in optics, see [28], [43], [45], [60], [61] and the references therein. Maxwell’s equations in the gravitational field are discussed in [17], [27]. One may hope that our detailed mathematical consideration of several aspects of macroscopic electrodynamics will be useful for future investigations and pedagogy.

Appendices

Appendix A: Formulas from Vector Calculus

Among useful differential relations are

$$\begin{aligned} \nabla \cdot \left( \mathbf {A}\times \mathbf {B}\right) =\mathbf {B}\cdot \left( \nabla \times \mathbf {A}\right) -\mathbf {A}\cdot \left( \nabla \times \mathbf {B}\right) . \end{aligned}$$

(A.1)

$$\begin{aligned} \nabla \cdot \left( f\mathbf {A}\right) =\left( \nabla f\right) \cdot \mathbf {A}+f\left( \nabla \cdot \mathbf {A}\right) . \end{aligned}$$

(A.2)

$$\begin{aligned} \nabla \times \left( f\mathbf {A}\right) =\left( \nabla f\right) \times \mathbf {A}+f\left( \nabla \times \mathbf {A}\right) . \end{aligned}$$

(A.3)

$$\begin{aligned}&\mathbf {A}\cdot \left( \nabla \times \left( f\nabla \times \mathbf {B}\right) \right) -\mathbf {B}\cdot \left( \nabla \times \left( f\nabla \times \mathbf {A}\right) \right) \\&\quad =\nabla \cdot \left( f\left( \mathbf {B}\times \left( \nabla \times \mathbf {A}\right) -\mathbf {A}\times \left( \nabla \times \mathbf {B} \right) \right) \right) .\nonumber \end{aligned}$$

(A.4)

$$\begin{aligned}&\mathbf {A}\left( \nabla \cdot \mathbf {B}\right) -\mathbf {B}\left( \nabla \cdot \mathbf {A}\right) +\mathbf {A}\times \left( \nabla \times \mathbf {B}\right) -\mathbf {B}\times \left( \nabla \times \mathbf {A}\right) \\&\quad -\nabla \times \left( \mathbf {A}\times \mathbf {B}\right) =\sum _{\alpha =1}^{3}A_{\alpha }^{2}\nabla \left( \frac{B_{\alpha }}{A_{\alpha } }\right) =-\sum _{\alpha =1}^{3}B_{\alpha }^{2}\nabla \left( \frac{A_{\alpha } }{B_{\alpha }}\right) .\nonumber \end{aligned}$$

(A.5)

(See also [1], [79] and [90].) Here, ${\text {div}}\,\mathbf {A}=\nabla \cdot \mathbf {A}$ and ${\text {curl}}\,\mathbf {A}=\nabla \times \mathbf {A}.$

Appendix B: Dual Tensor Identities

In this article, $e^{\mu \nu \sigma \tau }=-e_{\mu \nu \sigma \tau }$ and $e_{0123}=+1$ is the Levi-Civita four-symbol [27] with familiar contractions:

$$\begin{aligned} e^{\mu \nu \sigma \tau }e_{\mu \kappa \lambda \rho }=-\left| \begin{array} [c]{ccc} \delta _{\kappa }^{\nu } &{} \delta _{\lambda }^{\nu } &{} \delta _{\rho }^{\nu }\\ \delta _{\kappa }^{\sigma } &{} \delta _{\lambda }^{\sigma } &{} \delta _{\rho }^{\sigma }\\ \delta _{\kappa }^{\tau } &{} \delta _{\lambda }^{\tau } &{} \delta _{\rho }^{\tau } \end{array} \right| , \end{aligned}$$

(B.1)

$$\begin{aligned} e^{\mu \nu \sigma \tau }e_{\mu \nu \lambda \rho }=-2\left| \begin{array} [c]{cc} \delta _{\lambda }^{\sigma } &{} \delta _{\rho }^{\sigma }\\ \delta _{\lambda }^{\tau } &{} \delta _{\rho }^{\tau } \end{array} \right| =-2\left( \delta _{\lambda }^{\sigma }\delta _{\rho }^{\tau } -\delta _{\rho }^{\sigma }\delta _{\lambda }^{\tau }\right) , \end{aligned}$$

(B.2)

$$\begin{aligned} e^{\mu \nu \sigma \tau }e_{\mu \nu \sigma \rho }=-6\delta _{\rho }^{\tau },\qquad e^{\mu \nu \sigma \tau }e_{\mu \nu \sigma \rho }=-24. \end{aligned}$$

(B.3)

Dual second rank four-tensor identities are given by [27]:

$$\begin{aligned} e^{\mu \nu \sigma \tau }A_{\sigma \tau }=2B^{\mu \nu },\qquad e_{\mu \nu \sigma \tau }B^{\sigma \tau }=A_{\nu \mu }-A_{\mu \nu }. \end{aligned}$$

(B.4)

In particular,

$$\begin{aligned} Q^{\mu \nu }&=R^{\mu \nu }+iS^{\mu \nu }=R^{\mu \nu }-\frac{i}{2}e^{\mu \nu \sigma \tau }F_{\sigma \tau },\end{aligned}$$

(B.5)

$$\begin{aligned} P_{\mu \nu }&=F_{\mu \nu }+iG_{\mu \nu }=F_{\mu \nu }-\frac{i}{2}e_{\mu \nu \sigma \tau }R^{\sigma \tau }. \end{aligned}$$

(B.6)

$$\begin{aligned} e_{\mu \nu \sigma \tau }Q^{\sigma \tau }=2iP_{\mu \nu },\qquad 2iQ^{\mu \nu }=e^{\mu \nu \sigma \tau }P_{\sigma \tau }. \end{aligned}$$

(B.7)

$$\begin{aligned} 2R^{\mu \nu }=e^{\mu \nu \sigma \tau }G_{\sigma \tau },\qquad -2S^{\mu \nu }=e^{\mu \nu \sigma \tau }F_{\sigma \tau }. \end{aligned}$$

(B.8)

$$\begin{aligned} 2G_{\mu \nu }=-e_{\mu \nu \sigma \tau }R^{\sigma \tau },\qquad 2F_{\mu \nu }=e_{\mu \nu \sigma \tau }S^{\sigma \tau }. \end{aligned}$$

(B.9)

$$\begin{aligned} P_{\mu \nu }Q^{\mu \nu }=2F_{\mu \nu }R^{\mu \nu }-\frac{i}{2}\left( e^{\mu \nu \sigma \tau }F_{\mu \nu }F_{\sigma \tau }+e_{\mu \nu \sigma \tau }R^{\mu \nu } R^{\sigma \tau }\right) . \end{aligned}$$

(B.10)

By direct calculation,

$$\begin{aligned} F_{\mu \nu }R^{\mu \nu }=2\left( \mathbf {H}\cdot \mathbf {B}-\mathbf {E} \cdot \mathbf {D}\right) , \end{aligned}$$

(B.11)

$$\begin{aligned} e^{\mu \nu \sigma \tau }F_{\mu \nu }F_{\sigma \tau }=8\mathbf {E}\cdot \mathbf {B},\qquad e_{\mu \nu \sigma \tau }R^{\mu \nu }R^{\sigma \tau }=8\mathbf {H}\cdot \mathbf {D}. \end{aligned}$$

(B.12)

As a result,

$$\begin{aligned} \frac{1}{4}P_{\mu \nu }Q^{\mu \nu }=\mathbf {H}\cdot \mathbf {B}-\mathbf {E} \cdot \mathbf {D}-i\left( \mathbf {E}\cdot \mathbf {B}+\mathbf {H}\cdot \mathbf {D}\right) . \end{aligned}$$

(B.13)

An important decomposition,

$$\begin{aligned} P_{\mu \lambda }^{*}Q^{\lambda \nu }+P_{\mu \lambda }\overset{*}{\left. Q^{\lambda \nu }\right. }&=2\left( F_{\mu \lambda }R^{\lambda \nu } +G_{\mu \lambda }S^{\lambda \nu }\right) \\&=4F_{\mu \lambda }R^{\lambda \nu }+\delta _{\mu }^{\nu }F_{\sigma \tau } R^{\sigma \tau }\nonumber \\&=4F_{\mu \lambda }R^{\lambda \nu }-2\delta _{\mu }^{\nu }\left( \mathbf {E} \cdot \mathbf {D}-\mathbf {H}\cdot \mathbf {B}\right) ,\nonumber \end{aligned}$$

(B.14)

is complemented by an identity,

$$\begin{aligned} P_{\mu \lambda }Q^{\lambda \nu }+P_{\mu \lambda }^{*}\overset{*}{\left. Q^{\lambda \nu }\right. }&=\frac{1}{4}\left( P_{\sigma \tau }Q^{\tau \sigma }+P_{\sigma \tau }^{*}\overset{*}{\left. Q^{\tau \sigma }\right. }\right) \delta _{\mu }^{\nu }\\&=\frac{1}{2}\left( \mathbf {E}\cdot \mathbf {D}-\mathbf {H}\cdot \mathbf {B}\right) \delta _{\mu }^{\nu }.\nonumber \end{aligned}$$

(B.15)

In matrix form,

$$\begin{aligned} PQ&=\left( F+iG\right) \left( R+iS\right) =\left( FR-GS\right) +i\left( FS+GR\right) ,\end{aligned}$$

(B.16)

$$\begin{aligned} P^{*}Q&=\left( F-iG\right) \left( R+iS\right) =\left( FR+GS\right) +i\left( FS-GR\right) . \end{aligned}$$

(B.17)

Here,

$$\begin{aligned} FS&=\frac{1}{4}\text {Tr}\left( FS\right) I=\left( \mathbf {E} \cdot \mathbf {B}\right) I,\end{aligned}$$

(B.18)

$$\begin{aligned} GR&=\frac{1}{4}\text {Tr}\left( GR\right) I=\left( \mathbf {H} \cdot \mathbf {D}\right) I. \end{aligned}$$

(B.19)

$$\begin{aligned} FR-GS=\frac{1}{2}\text {Tr}\left( FR\right) I=\left( \mathbf {E} \cdot \mathbf {D}-\mathbf {H}\cdot \mathbf {B}\right) I, \end{aligned}$$

(B.20)

$$\begin{aligned} FR+GS&=2FR-\frac{1}{2}\text {Tr}\left( FR\right) I\\&=2FR-\left( \mathbf {E}\cdot \mathbf {D}-\mathbf {H}\cdot \mathbf {B}\right) I.\nonumber \end{aligned}$$

(B.21)

$$\begin{aligned} \text {Tr}\left( FR+GS\right) =0, \end{aligned}$$

(B.22)

where $I=\mathrm{diag}\left( 1,1,1,1\right)$ is the identity matrix.

Also,

$$\begin{aligned} PQ&=QP=\left( \mathbf {F}\cdot \mathbf {G}\right) I,\end{aligned}$$

(B.23)

$$\begin{aligned} \det P&=\det Q=-\left( \mathbf {F}\cdot \mathbf {G}\right) ^{2} \end{aligned}$$

(B.24)

and

$$\begin{aligned} \mathbf {F}\cdot \mathbf {G}&=\left( \mathbf {E}+i\mathbf {H}\right) \cdot \left( \mathbf {D}+i\mathbf {B}\right) \\&=\left( \mathbf {E}\cdot \mathbf {D}-\mathbf {H}\cdot \mathbf {B}\right) +i\left( \mathbf {E}\cdot \mathbf {B}+\mathbf {H}\cdot \mathbf {D}\right) .\nonumber \end{aligned}$$

(B.25)

Other useful dual four-tensor identities are given by [27]:

$$\begin{aligned} e^{\mu \nu \sigma \tau }A_{\nu \sigma \tau }=6B^{\mu },\qquad A_{\mu \nu \lambda } =e_{\mu \nu \lambda \sigma }B^{\sigma }. \end{aligned}$$

(B.26)

In particular,

$$\begin{aligned} 6i\frac{\partial Q^{\mu \nu }}{\partial x^{\nu }}=e^{\mu \nu \lambda \sigma }\left( \frac{\partial P_{\lambda \sigma }}{\partial x^{\nu }}+\frac{\partial P_{\nu \lambda }}{\partial x^{\sigma }}+\frac{\partial P_{\sigma \nu }}{\partial x^{\lambda }}\right) , \end{aligned}$$

(B.27)

and

$$\begin{aligned} \frac{\partial P_{\mu \nu }}{\partial x^{\lambda }}+\frac{\partial P_{\nu \lambda }}{\partial x^{\mu }}+\frac{\partial P_{\lambda \mu }}{\partial x^{\nu }} =ie_{\mu \nu \lambda \sigma }\frac{\partial Q^{\sigma \tau }}{\partial x^{\tau }} \end{aligned}$$

(B.28)

(see also [47]).

Appendix C: Proof of Identities

In view of (6.3), or (B.7), and (B.28), we can write

$$\begin{aligned} \left( \frac{\partial P_{\mu \nu }}{\partial x^{\lambda }}+\frac{\partial P_{\nu \lambda }}{\partial x^{\mu }}+\frac{\partial P_{\lambda \mu }}{\partial x^{\nu }}=ie_{\mu \nu \lambda \sigma }\frac{\partial Q^{\sigma \tau }}{\partial x^{\tau }}\right) \overset{*}{\left. Q^{\lambda \nu }\right. }, \end{aligned}$$

(C.1)

or

$$\begin{aligned}&2\overset{*}{\left. Q^{\lambda \nu }\right. }\frac{\partial P_{\mu \nu } }{\partial x^{\lambda }}+\overset{*}{\left. Q^{\lambda \nu }\right. } \frac{\partial P_{\nu \lambda }}{\partial x^{\mu }}\\&\quad =i\left( e_{\mu \nu \lambda \sigma }\overset{*}{\left. Q^{\lambda \nu }\right. }\right) \frac{\partial Q^{\sigma \tau }}{\partial x^{\tau }} =-2P_{\mu \sigma }^{*}\frac{\partial Q^{\sigma \tau }}{\partial x^{\tau }} \end{aligned}$$

by (B.7). Therefore,

$$\begin{aligned} P_{\mu \lambda }^{*}\frac{\partial Q^{\lambda \nu }}{\partial x^{\nu }} -\frac{\partial P_{\mu \lambda }}{\partial x^{\nu }}\overset{*}{\left. Q^{\lambda \nu }\right. }=-\frac{1}{2}\overset{*}{\left. Q^{\sigma \tau }\right. }\frac{\partial P_{\tau \sigma }}{\partial x^{\mu }}. \end{aligned}$$

(C.2)

In addition, with the help of (B.7) one gets

$$\begin{aligned}&2i\left( P_{\sigma \tau }^{*}\frac{\partial Q^{\tau \sigma }}{\partial x^{\mu }}\right) =P_{\sigma \tau }^{*}e^{\tau \sigma \lambda \nu }\frac{\partial P_{\lambda \nu }}{\partial x^{\mu }}\\&\quad =e^{\sigma \tau \nu \lambda }P_{\sigma \tau }^{*}\frac{\partial P_{\lambda \nu }}{\partial x^{\mu }}=-2i\left( \overset{*}{\left. Q^{\sigma \tau }\right. }\frac{\partial P_{\tau \sigma }}{\partial x^{\mu } }\right) , \end{aligned}$$

which completes the proof.