A new approach to Lorentz invariance in electromagnetism with hyperbolic octonions

Abstract

In this study, after introducing the hyperbolic octonionic (counteroctonion) algebra and its properties, Maxwell's equations with magnetic monopole and currents, Lorenz conditions for the electric, magnetic fields and Lorentz invariance are presented in detail with the most acknowledged forms using vectors. In a subsequent step, the differential operator and Laplacian, the hyperbolic octonionic Lorentz invariance of Maxwell's equations with monopole and relevant field equations, which are newly described, and the components of the wave equation are obtained in a compact, useful, simple and elegant form in higher dimensions. This approach demonstrates that the hyperbolic octonionic representations for the Lorentz invariance of Maxwell's equations with magnetic monopole can also contribute to field theories. Maxwell's equations with sources are provided in Gauss units. As a result, the terms of Lorentz invariance for electromagnetism with the newly described field equations, and the wave equation are clearly attained in the hyperbolic octonionic algebra.