Constitutive relations in optics in terms of geometric algebra

Constitutive relations for electromagnetic field in a form of $6\times 6$ matrices derived from the geometric algebra

“…[16] To sum up, the Clifford geometric algebra provides a coherent picture of how the constitutive relations for a homogeneous EM media originate from all possible linear transformations between time-like and space-like bivectors of relativistic Cl 1,3 algebra and automorphisms (involutions) of this algebra without any need for additional assumptions on space-time properties. In the paper [13] the obtained set of 36 independent real coefficients generated by Cl 1,3 algebra coincides with that found by E. J. Post [4] from space-time symmetry consideration using the tensorial calculus.…”

“…In its turn, the individual 3 × 3 submatrices may be decomposed into sum of even and odd parts, consequently the transformation G γ = χ γ (F ) can be divided into sum of matrices χγ = χs γ + χa γ , where the superscripts s and a indicate the symmetric and antisymmetric parts. Thus we find that the most general transformation which is allowed by GA can be rewritten in the following matrix form [13],…”

“…Both the Faraday electric and magnetic transformations can be rearranged as a single matrix [13] χn,m =…”

“…Post [4] from space-time symmetry consideration using the tensorial calculus. All possible transformations that follow from internal GA structure here were cast into a form of 6 × 6 matrices (6), (7), (13) and (15) which in turn were separated into 3 × 3 susceptibility submatrices according to physical effects they describe. The obtained constitutive 3 × 3 matrices were found to be either symmetric (represented by ellipsoids) or skew symmetric (represented by vectors).…”

“…General forms of CR's formulated in terms of classical Cl 3,0 and relativistic Cl 1,3 algebras were deduced in papers [12,13]. Since the GA is not widely known to physicists the aim of this paper is to display the constitutive relations that follow from space-time structure encoded in the relativistic GA in a form of 6 × 6 matrices which can be easily transformed to Gibbs-Heaviside form and applied to investigate properties of EM waves in various media.…”