Octonionic Maxwell's equations for bi-isotropic media

Abstract

In this study, octonions with eight dimensions and their algebra, which are both noncommutative and nonassociative, are presented. Moreover, the general properties of complex octonions with 16 dimensions and the products of basis are defined by using Cayley–Dickson multiplication rules. Maxwell's equations are taken into consideration when it comes to bi-isotropic media in which the electric and magnetic fields are coupled by means of bi-isotropic constitutive relations. The Drude–Born–Fedorov constitutive relations defined with the complex representations of electric and magnetic fields are used for all calculations. In the next stage, the complex octonionic differential operator is introduced and the octonionic field and source equations are defined for bi-isotropic media. As a result, Maxwell's equations for bi-isotropic media, which are fundamental features that serve as the groundwork for electromagnetism, all of them as being unique, more compact, and a convenient form with magnetic monopole.

Representation of electromagnetic and gravitoelectromagnetic Poynting theorems in higher dimensions

“…Here,ê k ≡ e 4+k , k ∈ (1, 2, 3) and e 0 = 1 [22][23][24][25]. These rules can also be summarized as a table.…”

“…Therefore, octonions are also known as "Cayley numbers" in the literature. Octonion algebra has both noncommutative and nonassociative algebraic structures [2,3,[22][23][24][25]. A real octonion, A, is shown as follows:…”

“…(7), the scalar and vectorial products of real octonions A and B are given the following manner, respectively [22][23][24][25]:…”

“…The complex octonionic differential operator and its octonionic conjugate are defined in terms of Cayley-Dickson notation in the literature as below [22][23][24][25]:…”

“…In this paper, energy conservation equations for both EM and GEM are expressed by using complex octonions. Some studies related to complex octonions can be summarized as follows: Tolan et al reformulated classical electromagnetism [22], Tanışlı and Kansu studied electromagnetism for bi-isotropic media [23], Tanışlı and Jancewicz represented electromagnetic Lorenz conditions with magnetic monopoles [24], and Kansu et al showed electromagnetic energy conservation without sources [25].…”