The relations of the homogeneous Maxwell's equations to the theory of functions

Abstract

The thesis developed by Cornelius Lanczos in his doctoral dissertation is that electrodynamics is a pure field theory which is hyperanalytic over the algebra of biquaternions. In this theory Maxwell's homogeneous equations correspond to a generalization of the Cauchy-Riemann regularity conditions to four complex variables, and electrons to singularities in the Maxwell field. Since there are no material particles in Lanczos electrodynamics, the same action principle applies to both regular and singular Maxwell fields. Therefore, the usual action integral of classical electrodynamics is {not} an input in that theory, but rather a consequence which {derives} from the application of Hamilton's principle to a superposition of two or more homogeneous Maxwell fields. This leads to a fully consistent electrodynamics which, moreover, can be shown to be finite. As byproducts to this remarkable thesis Lanczos anticipated the Moisil-Fueter theory of quaternion-analytic functions by more than ten years; showed that Maxwell's equations are invariant in both spin-1 and spin-1/2 Lorentz transformations; that displacing a singularity into imaginary space adds an intrinsic magnetic-like field to its electric field; and that his theory does even include gravitation -- although not in the general relativistic form of Einstein to whom Lanczos dedicated his dissertation.

arXiv comment59 pages. 5 figures. Capture and typesetting by Jean-Pierre Hurni. Abstract and preface by Andre Gsponer

References in corpus (6)

• The tensor analytical relationships of Dirac's equation
• On the physical interpretation of singularities in Lanczos-Newman electrodynamics
• The physical heritage of Sir W.R. Hamilton
• Lanczos's equation to replace Dirac's equation ?
• Lanczos's functional theory of electrodynamics: A commentary on Lanczos's PhD dissertation
• Cornelius Lanczos's derivation of the usual action integral of classical electrodynamics

Cited by in corpus (8)

• Revisiting Quaternionic Dual Electrodynamics
• Quaternions in mathematical physics (1): Alphabetical bibliography
• Lanczos - Einstein - Petiau: From Dirac's equation to nonlinear wave mechanics
• The tensor analytical relationships of Dirac's equation
• On the physical interpretation of singularities in Lanczos-Newman electrodynamics
• On the electromagnetic momentum of static charge and steady current distributions
• Cornelius Lanczos's derivation of the usual action integral of classical electrodynamics
• More on the early interpretation of the Schwarzschild solution