All courses dedicated to the study of electromagnetic fields introduce an important information regarding them: each pair (E, H) is characterized by two invariant quantities.
A) The first one is nothing but the Euclidean scalar product:
Inv1(E, H) = < E, H > = Ex. Hx + Ey. H y + Ez. Hz
B) The second is defined by the expression:
Inv2(E, H)
=
ψ
=
< H, H > - < E, E >
=
(H2x + H2y + H2z) - (E2x + E2y + E2z)
This statement can be verified in various academic works; for example: [01; Lichnerowicz; § 8, p. 17].
The simplicity of the second invariant formalism masks part of the history of electromagnetism and its true complexity.
Historically speaking, electromagnetic fields were described using so-called exterior differential forms (derivatives). And the expression of the second invariant within this particular mathematical context is formulated [01 ; § 8, p. 17]:
Ψ = 1/2. < F, F > = 1/2. Fαβ. Fαβ
It is difficult to correctly decode the symbolism of this expression without referring to work slightly prior to the cited reference; see [02]. Thanks to this aid, the exterior form of electromagnetism used by A. Lichnerowicz is akin to an alternating bilinear form by E. Cartan, albeit with one crucial technical detail. This chapter of mathematics concerns bilinear forms involving two sets of variables in equal number [02; chapter I, I, 1].
However, the symbolic notation of the exterior form of electromagnetism only manipulates one set of four variables, namely the:
dx0, dx1, dx2, dx3
It should - in theory - introduce a second series, for example the:
δx0, δx1, δx2, δx3
It would then effectively be possible to construct the quadratic version of this exterior form [02; chapter I, I, 3, p. 10]. But what is it really worth? How should it be calculated?
The concept of bilinear covariant was introduced by Darboux [03] in 1882.
The concept of bilinear covariant has since been renamed as: exterior differentiation; it is linked to the study of the variations of a generic differential form:
Θd(X) = ∑α Xα. dxα
An exterior differentiation of this form is:
δΘd(X) = δ(∑α Xα. dxα)
The usefulness of this concept lies in its connections to an old integration problem (volumes in a space of arbitrary dimension). By using the Leibniz rule:
δΘd(X) = ∑α δXα + ∑α δdxα
Expanding the first term to the right of the equals sign presents no difficulty. This is not the case for the second, as we do not know how to calculate δd. Darboux proposes bypassing this difficulty by noting the following:
1) By swapping the symbols d and δ:
dΘδ(X) = ∑α dXα + ∑α dδxα
2)And assuming without proof that:
δd = dδ
... it then becomes possible to obtain:
It is easy to understand the origin of the adjective "exterior" attached to the word differentiation by observing the formalism of this expression. It looks exactly like any component of an exterior product performed between two vectors!
This is the precise place to note that by repeating the Darboux maneuver with D different differential forms (Pfaffian forms):
This results in the following interesting information which connects Darboux's work to the concept of the alternating deformed tensor product.
The existence of a set of D differential forms generates that of a deformed tensor product built on a cube A which, in the (x, Ω) frame, is worth:
[01] Lichnerowicz, A. : Théories relativistes de la gravitation et de l'électromagnétisme, Relativité générale et théories unitaires, Masson et Cie, Éditeurs 1955, 289 pages.
[02] Cartan, E. : Exposés de géométrie, XIV : Les systèmes différentiels extérieurs et leurs applications géométriques ; Actualités scientifiques et industrielles, numéro 994, Hermann et Cie, Éditeurs Paris 1945.
[03] Darboux, G. : Sur le problème de Pfaff, Bulletin des sciences mathématiques et astronomiques, VI (1882), numéro 1, 14-36 et 49-68.