Majorana-Oppenheimer approach to Maxwell electrodynamics in Riemannian space-time

TL;DR

This paper develops a matrix formalism for Maxwell's equations in Riemannian space-time, demonstrating symmetry properties and relating the formalism to Dirac matrices, with applications to media and complex 3-vector fields.

Contribution

It extends the Majorana-Oppenheimer approach to Maxwell equations into Riemannian space-time using tetrad formalism, revealing symmetry and algebraic relations with Dirac matrices.

Findings

Matrix form of Maxwell equations exhibits SO(3.C) symmetry.

Relation between Maxwell matrices and Dirac matrices established.

Extension to arbitrary Riemannian space-time achieved.

Abstract

The Riemann -- Silberstein -- Majorana -- Oppengeimer approach to the Maxwell electrodynamics in presence of electrical sources and arbitrary media is investigated within the matrix formalism. The symmetry of the matrix Maxwell equation under transformations of the complex rotation group SO(3.C) is demonstrated explicitly. In vacuum case, the matrix form includes four real $4 \times 4$ matrices $\alpha^{b}$. In presence of media matrix form requires two sets of $4 \times 4$ matrices, $\alpha^{b}$ and $\beta^{b}$ -- simple and symmetrical realization of which is given. Relation of $\alpha^{b}$ and $\beta^{b}$ to the Dirac matrices in spinor basis is found. Minkowski constitutive relations in case of any linear media are given in a short algebraic form based on the use of complex 3-vector fields and complex orthogonal rotations from SO(3.C) group. The matrix complex formulation in the Esposito's form, based on the use of two electromagnetic 4-vector, is studied and discussed. Extension of the 3-vector complex matrix formalism to arbitrary Riemannian space-time in accordance with tetrad method by Tetrode-Weyl-Fock-Ivanenko is performed.