[Paper Review] Quaternion Analyticity of Time-Harmonic Dyon Field Equations

Jivan Singh, P. S. Bisht+1 more|ArXiv.org|Mar 11, 2007

Quantum and Classical Electrodynamics2 references3 citations

TL;DR

This paper develops a quaternionic formulation of time-harmonic Maxwell’s equations for dyons—particles with both electric and magnetic charges—using complex vector fields and generalized four-potentials. It demonstrates that the generalized field equations reduce to standard electromagnetic theory in the absence of magnetic or electric charges, and shows that the system can be diagonalized into decoupled equations via biquaternionic operators, simplifying analysis of dyonic dynamics in isotropic media.

ABSTRACT

Quaternion analysis of time dependent Maxwell's equations in presence of electric and magnetic charges has been developed in unique, simple and consistent manner. It has been shown that this theory is extended consistently to time-harmonic Maxwell's equation for dyons.

Motivation & Objective

• To extend quaternionic analysis to time-harmonic Maxwell’s equations in the presence of both electric and magnetic charges (dyons), ensuring consistency and simplicity.
• To reformulate generalized electromagnetic field equations using complex vector fields and biquaternionic operators for unified treatment of electric and magnetic sources.
• To derive decoupled equations for dyonic current and field vectors through matrix diagonalization, simplifying the solution of generalized Maxwell-Dirac equations.
• To recover standard electromagnetic theory as a limiting case when only electric or magnetic charges are present.
• To provide a covariant, consistent, and compact framework for studying dyons in homogeneous and isotropic media using quaternionic calculus.

Proposed method

• Define a complex vector field $\vec{\psi} = \vec{E} - i v \vec{B}$ to unify electric and magnetic fields into a single quaternionic object.
• Express the generalized electromagnetic potentials using $\phi = \phi_e - i v \phi_m$ and $\vec{V} = \vec{C} - i \frac{\vec{D}}{v}$, forming a generalized four-potential $V_\mu$.
• Apply the quaternionic differential operator $D = \partial_1 e_1 + \partial_2 e_2 + \partial_3 e_3$ to the complex field $\vec{\psi}$, yielding a unified field equation.
• Introduce biquaternionic auxiliary fields $\vec{l}$ and $\vec{m}$ to diagonalize the system, transforming coupled equations into decoupled forms.
• Use matrix representation $B_\alpha$ and its inverse to diagonalize the system, enabling independent analysis of $\vec{J}$ and $\vec{m}$.
• Derive continuity equations for electric and magnetic charge densities under time-harmonic conditions, ensuring charge conservation in the complex framework.

Experimental results

Research questions

• RQ1 How can time-harmonic Maxwell’s equations for dyons be consistently formulated using quaternionic analysis?
• RQ2 What is the role of complex vector fields and generalized four-potentials in unifying electric and magnetic sources in a single quaternionic formalism?
• RQ3 Can the generalized Maxwell-Dirac equations for dyons be decoupled using biquaternionic diagonalization techniques?
• RQ4 How do the derived equations reduce to known results in the limit of pure electric or magnetic monopoles?
• RQ5 What is the structure of the continuity equations for dyonic charge and current densities in the time-harmonic regime?

Key findings

• The generalized field equations for dyons are successfully expressed in a compact, unified form using the complex vector field $\vec{\psi} = \vec{E} - i v \vec{B}$, which satisfies $\vec{\nabla} \cdot \vec{\psi} = \rho / \epsilon$ and $\vec{\nabla} \times \vec{\psi} = -i v (\mu \vec{J} + \frac{1}{v^2} \frac{\partial \vec{\psi}}{\partial t})$.
• The current density $\vec{J} = \vec{j}_e - i v \vec{j}_m$ satisfies the equation $(D - \alpha)\vec{J} = \mu \vec{\nabla} \cdot \vec{J}^* + \alpha \mu \vec{J}^*$, derived via quaternionic operator application.
• The system is diagonalized using the matrix $B_\alpha = \begin{pmatrix} -i\omega/v^2 & \alpha \\ i\omega/v^2 & \alpha \end{pmatrix}$, leading to decoupled equations for $\vec{l}$ and $\vec{m}$, simplifying the solution of the generalized field equations.
• The continuity equations $\vec{\nabla} \cdot \vec{j}_e - i\omega \rho_e = 0$ and $\vec{\nabla} \cdot \vec{j}_m - i\omega \mu \epsilon \rho_m = 0$ are derived, confirming charge conservation in the time-harmonic regime.
• In the absence of magnetic charge ($\rho_m = 0$), the theory reduces to the standard electromagnetic theory as previously derived by Kravchenko, validating consistency.
• The formalism consistently reduces to the theory of electric or magnetic monopoles when one type of charge is absent, confirming its physical coherence and generality.