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Complex Form of Classical and Quantum Electrodynamics

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 electromagnetic field will be discussed in this covariant approach elsewhere.


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 beauty 1 , 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.
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.
Introduction of two complex fields allows one to rewrite the phenomenological Maxwell equations in the following compact form 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.

Hertz Symmetric Stress Tensor
We begin from a complex 3D-interpretation of the traditional symmetric energy-momentum tensor [72]. By definition, and the corresponding "momentum" balance equation, 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.) [86]: Taking into account the complex conjugate, we derive 1 2 On the other hand, in view of Maxwell's equations (2.5)-(2.6), one gets and, with the help of its complex conjugate, providing the second important fact. (Up to the constant, the first term in 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 Finally, in the last two lines, one can utilize the following differential vector calculus identity, see (A.5), with A = F, B = 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), due to a familiar vector calculus identity (A.1): In a traditional form, (see, for example, [18], [86]), where one can substitute (3.14) As a result, 3D-differential "energy-momentum" balance equations are given by ∂ ∂t respectively (see also [32], [62]). The real form of the symmetric stress tensor (3.1), namely, 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 R 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 µ = (ct, r) and try to match (3.15)-(3.16) with the expression, 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 nonhomogeneous variable medium (without dispersion and/or compression), when D = ε (r, t) E and B = µ (r, t) H, the "ponderomotive forces" in (3.15) and (3.16) take the form [86]: which may be interpreted as a four-vector "energy-force" acting from an inhomogeneous and timevariable medium. Its covariance is analyzed in section 7.

"Angular Momentum" Balance
The 3D-"linear momentum" differential balance equation (3.16), can be rewritten in a more compact form, with the help of the Hertz symmetric stress tensor T pq = T qp defined by (3.17). A "net force" is given by In this notation, we state the 3D-"angular momentum" differential balance equation as follows where the "field angular momentum density" is defined by and the "flux of angular momentum" is described by the following tensor [37]: 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 ∂M pq ∂x q = e prs T sr + e prs x r ∂T sq ∂x q (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.

Complex Covariant Form of Macroscopic Maxwell's Equations
With the help of complex fields F = E + iH and G = D + iB, we introduce the following anti-symmetric four-tensor, and use the standard four-vectors, x µ = (ct, r) and j µ = (cρ, j) for contravariant coordinates and current, respectively.
Maxwell's equations then take the covariant form [47], [54]: with summation over two repeated indices. Indeed, in block form, we have which verifies this fact. The continuity equation, 4) or in the 3D-form, ∂ρ ∂t + div j = 0, (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).

Dual Electromagnetic Field Tensors
Two dual anti-symmetric field tensors of complex fields, F = E + iH and G = D + iB, are given by 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 Here e µνστ = −e µνστ and e 0123 = +1 is the Levi-Civita four-symbol [27]. Then and both pairs of Maxwell's equations can also be presented in the form [47] ∂P µν ∂x λ + in addition to the one given above 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, was originally established, in a general form, by Minkowski [65]. Once again, the dual tensors are given by in block form. A complete list of relevant tensor and matrix identities is given in appendix B.

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 follows In our complex form, when F = E + iH and G = D + iB, the energy-momentum tensor is given by .
Here, we point out for the reader's convenience that 3) and, in real form, The covariant form of the differential balance equation allows one to clarify the meanings 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], appears here in the corresponding "momentum" balance equation [86]: 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, Moreover, with the help of elementary identities, one can transform the latter balance equation into its "symmetric" form, which provides an independent proof of (3.16).

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 and its complex conjugate A simple decomposition, 13) with f = P * µλ and g = Q λν (and their complex conjugates), results in ∂ ∂x ν 1 16π P * µλ Q λν + P µλ * Q λν (7.14) By a direct substitution, one can verify that (An independent covariant proof of these identities is given in appendix C.) Finally, introducing 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
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].
With the help of familiar dual relations (B.4), one can get another covariant form of the angular momentum balance equation: In 3D-form, the latter relation can be reduced to (4.3)-(4.5).
Indeed, when µ = 0 and λ = p = 1, 2, 3, one gets where −Y = ρE + j × B/c is the familiar Lorentz force. Substitution, T rs = T rs + T rs − T rs , results in (4.3) in view of identity (7.7). The remaining cases, when µ, ν = p, q = 1, 2, 3, can be analyzed in a similar fashion. In 3D-form, the corresponding equations can be reduced 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].

Transformation Laws of Complex Electromagnetic Fields
Let 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, and 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 , but apparently only in vacuum, when G = F (see also [88]). Moreover, in the same notation [72]. The latter equations can be rewritten as follows In a similar fashion, one gets as a compact 3D-version of the Lorentz transformation for the complex electromagnetic fields.
In complex four-tensor form, 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])  Example. Let {e k } 3 k=1 be an orthonormal basis in R 3 . We choose v = ve 1 and write x ′µ = Λ µ ν x ν with for the corresponding Lorentz boost (Figure 2). In view of (9.8), by matrix multiplication one gets  Thus G ′ 1 = G 1 and In a similar fashion, F ′ 1 = F 1 and The latter relations are in agreement with the field transformations (9.2)-(9.3).
In block form, one gets  where, by definition, As a result, the transformation law of the complex electromagnetic fields {F, G} under the Lorentz boost can be thought of as a complex rotation in C 6 , corresponding to a reducible representation of the one-parameter subgroup of SO (3, C) . (Cyclic permutation of the spatial indices cover the two remaining cases; see also [88].)

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].

Potentials. In practice, one can choose
for the real-valued four-vector potential A λ (x) . Then by (10.4). Substitution into Maxwell's equations (6.6) or (6.5) results in In view of an inverse matrix identity, the latter equations take the form 4 Subject to the subsidiary condition, 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).

Hertz's tensor and vectors.
We follow [15], [16], [91] with somewhat different details. The substitution, (a generalization of Hertz's potentials for a moving medium [15], [91]), into the gauge condition (10.12) results in Z λσ = −Z σλ , in view of Then, equations (10.11) take the form Indeed, the left-hand side of (10.11) is given by from which the result follows due to (10.10).
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 with the help of (10.3)-(10.4) and (B.14) (see also [84]).

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 and Then, by formal differentiation, and (11.4) in view of (B.7).
The complex covariant Maxwell equations (5.2) take the forms 11.5) and the covariant energy-momentum balance relations (7.1) are given by in terms of the complex Lagrangians under consideration, respectively.
Finally, with the help of the following densities, one can derive analogs of the Euler-Lagrange equations for electromagnetic fields in media: In the case of a moving isotropic medium, a relation between P νµ and A µ 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 Here, −6∂ ν S µν = e µνλσ (∂ ν F λσ + ∂ σ F νλ + ∂ λ F σν ) ≡ 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 in view of (10.3). Then, for "conjugate momenta" to the four-potential field A µ , one gets (11.12) and the corresponding Euler-Lagrange equations take a familiar form 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.