Complex Form of Classical and Quantum Electrodynamics

Abstract

Physical laws should have mathematical beauty. " -P. A. M. DiracAbstract. 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.

Quaternionic electrodynamics

“…The energy flux S is expected to be orthogonal to E and to B. Using the scalar product (6), we obtain…”

“…There exists an old interest to apply various complex structures to electromagnetism, and we mention applications with complex numbers [6,7], quaternions [8][9][10][11][12][13][14][15], bi-quaternions [16][17][18][19][20][21], hyperbolic quaternions [22], octonions [23][24][25][26], hyperbolic octonions [15,27] and sedenions [28].…”

The role of the Pauli–Lubański vector for the Dirac, Weyl, Proca, Maxwell and Fierz–Pauli equations

“…under a Lorentz transformation, thus resembling the transformations of electromagnetic fields in classical electrodynamics [32], [33], [41], [59], [61].…”

“…The covariant field "energy-momentum" tensor and the corresponding differential balance equation, ∂ ∂x ν Q * µλσ Q λνσ + Q µλσ * Q λνσ = 0, (7.22) can be derived in a complete analogy with complex electrodynamics [32], [33].…”

The Pauli–Lubański vector, complex electrodynamics, and photon helicity

“…, are derived in terms of these tensors in [27]. (In the rest of the article, we will be dealing with the case of vacuum only, when G = F, but it's convenient to use both vectors in our calculations anyway in order to emphasize where they are coming from.…”