In the previous pages, I explained the intuitive approach that led me to postulate the existence of two types of energy and to believe that complex numbers, as well as quaternions, could potentially help substantiate this claim.
I will therefore set to work and attempt to prove the validity of my intuition by applying it to electromagnetic fields.
It therefore seems necessary to recall some essential data concerning them.
J. C. Maxwell's equations are among the oldest known representations of electromagnetic fields [01]. The efforts of O. Heavisid [02], led to favoring a Cartesian vector representation in which differential forms appear [03; section 7.4, p. 355]. These forms (divergence, curl, Laplacian) help to account for field variations in space and over time. The era in which these tools were developed explains why the initial mathematical formulations were implicitly carried out within a three-dimensional Euclidean geometric environment.
The consideration of different geometries in scientific work, notably that of Riemann [04 - D (1854)], occurred after the publication of A. Einstein's theory of gravitation [05 - D]; for example, by É. Cartan in [06 - FR (1922)], [07 - FR (1925)].
The end of the nineteenth century and the beginning of the twentieth correspond to a prolific period for mathematics, which saw the emergence of, among other discoveries of the human mind, tensor calculus, the study of differential bilinear forms and surfaces [08 - FR (1889)], and the theory of spinors [09- USA]. This list does not claim to be exhaustive.
Regarding electromagnetism specifically, [10 - FR], these bursts of human thought led to the realization of the advantages of representing electromagnetic fields using exterior forms; this is the approach followed in [11 - FR ; § 8, p.15]:
Expression in which (i) the indices are 0, 1, 2, 3; (ii) the classical cross product appears. The fact that these components belong to the commutative field of complex numbers, C, proves to be a necessity, allowing, for example, to account for experimental realities concerning string oscillations, alternating currents, and, more generally, wave phenomena [12 - FR].
Exterior quadratic forms can be incorporated into the body of work dedicated to tensor calculus. There is, moreover, a "Que sais-je ?" volume that outlines the rudiments necessary to understand the subject [13]. I drew inspiration from it to construct the deformations of cross products; see the document attached on the page regarding deformed cross products.
The concept of the adjoint tensor is part of the knowledge accompanying this type of calculation. The foundations of its application to the field of electromagnetism are explained in [11; § 8, p.16]; I will return to this.
The electromagnetic field tensor components are traditionally grouped as follows [14; § 23, p.73]:
Or sometimes:
I have so far described the possible representations of an electromagnetic field, in extenso a pair (E, H) traditionally in E^2 (3, R), in an order roughly respecting the chronological evolution of the concepts used to develop them. As a brief reminder: vectors (Maxwell-Heaviside), exterior forms (Pfaff, Cartan, ...), and matrices (as a tool indirectly accounting for the Lorentz law).
To these historical representations, I wish to add others in which quaternions play a central role. An online M.I.T. course [15] explaining the reasons for the birth of quaternions as well as some of their properties reinforces my motivation.
I noticed there that the usual matrix representations of the elements of M(4, R) or their equivalents in a partitioned form allow the product of two quaternions to be represented as follows:
It therefore seems possible to represent any quaternion by an element of M(4, R); the example above suggests:
The formalism of the second matrix in this sum is such that it could represent an electromagnetic field whose two vector branches would be equal and equal to the vector part of a quaternion.
This seemingly trivial remark will allow us to link the sum of the representations of a pair of quaternions with electromagnetic fields, and that is exactly where I wanted to get to.
The previous remarks invite us to formalize the matrix representation of a standard quaternion in the following standardized way:
1) The set of matrix representations of quaternions is denoted by Quat(4, R); it is a subset of M(4, R).
2) The matrix representation of the adjoint of a quaternion is the adjoint of the matrix representation of that quaternion.
3) The matrix representation of the sum of two quaternions is equal to the sum of the matrix representations of each of them.
4) There are at least two ways to decompose the matrix representation of a sum of two quaternions.
Let there be two quaternions, and let their sum be:
Indeed, let (E, H) be a pair; it has at least the following two usual matrix representations (see above):
Assuming that each field present in an empty region (absence of electric charge), whether of an electric or magnetic nature, represents a purely imaginary quaternion (or conversely can be represented by a purely imaginary quaternion), ... the decomposition of the sum of the matrix representation of an electric field and the matrix representation of a magnetic field can be represented by the difference between the matrix representation of the twice-contravariant version of the electromagnetic field tensor and the dual matrix representation of this twice-contravariant version:
This statement can be extended to a representation of the sum of two almost arbitrary quaternions:
This observation might lead one to interpret the real part of these electric and magnetic quaternions as equal electric charges with opposite signs; which suggests a link to the notion of particle-antiparticle.
The references listed below were consulted to document the statements made on this page.
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[01] Maxwell, J. C.: A Dynamical Theory of the Electromagnetic Field; Philosophical Transactions of the Royal Society of London, 1865, 155: 459...512.
[02] Heaviside, O.: XI. On the Forces, Stresses, and Fluxes of Energy in Electromagnetic Field (June 1891 - Mai 1892), pp. 423-480 ; [online] https://royalsocietypublishing.org (lien externe) doi/10.1098/rsta.1892.0011/1272264/rsta.1892.001 ; [consulté le 8 June 2026].
[03] Berkeley, Cours de physique, volume 3, ondes, Collection U, © Librairie Armand Colin, Paris, 1972, 603 pages.
[04] Riemann, B.: Über die Hypothesen welche die Geometrie zu Grunde liegen: Habilitationsschrift, 10 June 1854, in Göttingen - Gesammelte Math. Werke, Leipzig, 1872, p. 254-269.
[05] Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie; Annalen der Physik, vierte Folge, Band 49, (1916), N 7.
[06] Cartan, É. : Sur les équations de la gravitation d'Einstein (J. de Math. pures et appliquées, 9e série, t. 1, 1922, p. 141-203).
[07] Cartan, É. : La géométrie des espaces de Riemann ; Mémorial des Sciences Mathématiques (lien externe), fascicule IX, Gauthier-Villars et Cie Éditeurs, Paris, 1925.
[08] Darboux, G. : Leçons sur la théorie des surfaces ; Cours de géométrie de la Faculté des Sciences, deuxième partie : les congruences et les équations aux dérivées partielles des lignes tracées sur les surfaces. Gauthier-Villars et Cie Éditeurs, Paris, 1889.
[09] Cartan, É.: The theory of spinors. First published by Hermann of Paris in 1966; translation of the “Leçons sur la théorie des spineurs (2 volumes)''; Hermann, 1937; Dover Publications, Inc. New York. © 1966 by Hermann, Paris, ISBN 0-486-64070-1.
[10] Berkeley, Cours de physique, volume 2, électricité et magnétisme, Collection U, © Librairie Armand Colin, Paris, 1973, 460 pages.
[11] Lichnerowicz, A.. : Théories relativistes de la gravitation et de l'électromagnétisme, Relativité générale et théories unitaires, Masson et Cie, Éditeurs 1955, 289 pages.
[12] Berkeley, Cours de physique, volume 3, ondes, Collection U, © Librairie Armand Colin, Paris, 1972, 603 pages.
[13] Delachet, A. : Le calcul tensoriel, Collection « Que sais-je ? », numéro 1336, Presse Universitaire de France (P.U.F.), Paris, 1969, 126 pages.
[14] Landau, L. D. et Lifschitz, E. M.: Lehrbuch der theoretischen Physik, II: Klassische Feldtheorie; Akademie Verlag GmbH, Berlin 1992 ISBN 3-05-501550-9, 480 S.
[15] Prof. Berthold Horn: Lecture 18 - Rotation and How to represent it, Unit Quaternions, the Space of Rotations [Vidéo Youtube, 8 juin 2022], M.I.T 6.801 Machine Vision, OpenCourseWare (ocw.mit.edu), printemps 2020.