[Paper Review] Matrix-based approach to electrodynamics in media

A. A. Bogush, В. М. Редьков+2 more|ArXiv.org|Aug 5, 2008

Geophysics and Sensor Technology60 references3 citations

TL;DR

This paper develops a matrix-based formulation of Maxwell's electrodynamics in media using complex 3-vector fields and the SO(3,C) complex rotation group, demonstrating Lorentz symmetry through 4×4 matrices αᵇ and βᵇ. It establishes a unified matrix form for Maxwell's equations in arbitrary linear media, relates the formalism to Dirac matrices, and shows that Esposito’s representation arises from a trivial identity, challenging claims of its fundamental status in moving frames.

ABSTRACT

The Riemann -- Silberstein -- Majorana -- Oppenheimer 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 $4 imes 4$ matrices $α^{b}$. In presence of media matrix form requires two sets of $4 imes 4$ matrices, $α^{b}$ and $β^{b}$ -- simple and symmetrical realization of which is given. Relation of $α^{b}$ and $β^{b}$ 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-vectors, $e^α(x) = u_β F^{αβ}(x), b^α (x) = u_β ilde{F}^{αβ}(x) $ is studied and discussed. It is argued that Esposito form is achieved trough the use of a trivial identity $I=U^{-1}(u)U(u)$ in the Maxwell equation.

Motivation & Objective

• To reformulate Maxwell's electrodynamics in linear media using a matrix formalism based on complex 3-vectors and SO(3,C) symmetry.
• To demonstrate the Lorentz invariance of the matrix Maxwell equations through explicit transformation properties under SO(3,C).
• To provide a simple, symmetric realization of the αᵇ and βᵇ matrices for media, linking them to Dirac matrices.
• To clarify the relationship between the Riemann–Silberstein–Majorana–Oppenheimer formalism and Esposito’s matrix representation.
• To challenge the claim that Esposito’s form represents a fundamental Maxwell equation in moving frames, showing it arises from a trivial identity.

Proposed method

• Introduces a complex 3-vector ψᵏ = Eᵏ + icBᵏ to unify electric and magnetic fields into a single complex entity.
• Derives a matrix form of Maxwell’s equations using 4×4 matrices αᵇ and βᵇ, with αᵇ in vacuum and βᵇ in media.
• Establishes the matrix equation (−iα⁰∂₀ + αʲ∂ⱼ)Ψ = J/ε₀, where Ψ is a 4-component vector combining the complex field and current.
• Uses the identity I = U⁻¹(u)U(u) to derive Esposito’s form from the standard matrix form, showing equivalence.
• Relates the matrices αᵇ and βᵇ to the Dirac matrices via a transformation matrix β, enabling spinor-like algebraic treatment.
• Applies the Minkowski constitutive relations in a compact algebraic form using complex orthogonal rotations from SO(3,C).

Experimental results

Research questions

• RQ1 How can Maxwell’s equations in linear media be expressed in a unified matrix form using complex 3-vector fields?
• RQ2 What is the role of the SO(3,C) complex rotation group in preserving Lorentz symmetry in the matrix formulation?
• RQ3 How are the matrices αᵇ and βᵇ related to the Dirac matrices in a spinor basis?
• RQ4 What is the mathematical and physical significance of Esposito’s matrix representation in the context of moving reference frames?
• RQ5 Does Esposito’s form represent a deeper physical law or merely a trivial reparameterization of the standard matrix Maxwell equation?

Key findings

• The matrix form of Maxwell’s equations in vacuum is explicitly invariant under SO(3,C) transformations, confirming its Lorentz symmetry.
• A symmetric and simple realization of the 4×4 matrices αᵇ and βᵇ is provided, with βᵇ being essential for media and related to Dirac matrices via β.
• The Riemann–Silberstein–Majorana–Oppenheimer formalism is embedded in a matrix framework using αᵇ, yielding a Dirac-like equation for the photon.
• Esposito’s representation is shown to be equivalent to the standard matrix form through the identity I = U⁻¹(u)U(u), implying no new physics.
• The Minkowski constitutive relations for linear media are expressed in a compact algebraic form using complex 3-vectors and SO(3,C) rotations.
• The claim that Esposito’s form is fundamental in moving frames is refuted, as it arises from a trivial identity and does not represent a distinct physical law.