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

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.

32pages. Proccedings of the 14th Conference-School "Foundation & Advances in Nonlinear Science", Minsk, September 22-25, 2008. P. 20-49; ed. V.I. Kuvshinov, G.G. Krylov, Minsk, 2009

References in corpus (15)

• Photon wave function
• Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence
• Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium
• Covariant Majorana Formulation of Electrodynamics
• The Proof that the Standard Transformations of E and B are not the Lorentz Transformations. Clifford Algebra Formalism
• Axiomatic geometric formulation of electromagnetism with only one axiom: the field equation for the bivector field F with an explanation of the Trouton-Noble experiment
• Extended Electrodynamics: A Brief Review
• Historical Note on the Relativistic Theories of Electromagnetism
• Maxwell Optics: II. An Exact Formalism
• Lorentz Invariant Majorana Formulation of the Field Equations and Dirac-like Equation for the Free Photon
• Maxwell Optics: III. Applications
• Complex Structures in Electrodynamics
• From Electromagnetic Duality to Extended Electrodynamics
• Speculations on the Neutrino Theory of Light
• Invariant Relativistic Electrodynamics. Clifford Algebra Approach

Cited by in corpus (3)

• Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions
• A new matrix representation of the Maxwell equations based on the Riemann-Silberstein-Weber vector for a linear inhomogeneous medium
• Some remarks on relativistic zero-mass wave equations and supersymmetry