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A biquaternionic reformulation of Maxwell's equations via Fourier analysis

This paper employs biquaternionic analysis and Fourier transforms to characterize the parabolic Dirac operator and construct explicit vectorial solutions for the time-dependent Maxwell system, thereby extending previous biquaternionic approaches to electromagnetic problems.

Original authors: Aarón Guillén-Villalobos, Briceyda B. Delgado, Héctor Vargas Rodríguez

Published 2026-05-21

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CC BY 4.0

Original authors: Aarón Guillén-Villalobos, Briceyda B. Delgado, Héctor Vargas Rodríguez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, tangled knot of equations that describe how electricity and magnetism dance together through space and time. This is the job of Maxwell's equations, the fundamental rules of electromagnetism. Usually, untangling these knots requires heavy mathematical machinery that can be slow and difficult to manage.

This paper introduces a new, streamlined toolkit to untangle these knots faster and more elegantly. Here is how the authors did it, explained in everyday terms:

1. The Magic "Splitter" (Factorization)

Think of the wave equation (the rule that describes how light and radio waves ripple through space) as a heavy, solid block of stone. The authors realized they could crack this block open using a special "splitter."

In their mathematical language, they split the complex wave equation into two simpler, first-order pieces called parabolic Dirac operators (specifically $D \pm i\partial_t$ ).

The Analogy: Imagine you have a complex puzzle. Instead of trying to solve the whole picture at once, they found a way to break the puzzle into two smaller, easier-to-handle halves. If you can solve these two halves, you automatically solve the whole puzzle.

2. The "Translator" (Fourier Analysis)

To solve these two halves, the authors used a powerful tool called Fourier analysis.

The Analogy: Imagine you are listening to a chaotic orchestra. It's hard to understand the music as a whole. Fourier analysis is like a magical pair of glasses that lets you see the orchestra not as a blur of sound, but as individual notes on a sheet of music.
In this paper, they used this "glasses" to translate their difficult equations from the "real world" (where things change over space and time) into a "frequency world" (where the equations become simple multiplication problems). In this frequency world, the complex math becomes as easy as multiplying numbers.

3. The "Recipe" for Solutions (The Right Inverse)

Once the equations were translated into the frequency world, the authors created a specific "recipe" (a mathematical operator they call the parabolic Teodorescu transform) to reverse the process.

The Analogy: Think of this as a "undo" button or a reverse-engineering kit. They figured out exactly how to take the simple frequency data and turn it back into a real-world solution.
They didn't just guess this recipe; they proved it works perfectly for a specific type of mathematical function space (like a specific size of bucket that holds the water). This recipe acts as a right inverse, meaning if you put a problem into this machine, it spits out the exact solution.

4. The Grand Finale: Solving Maxwell's Equations

The ultimate goal was to solve the time-dependent Maxwell equations (the rules for electric and magnetic fields).

The Breakthrough: By using their "splitter" to break the problem down, their "translator" to simplify it, and their "recipe" to solve it, they derived a brand-new, explicit formula for how electric and magnetic fields behave over time.
The Result: They showed that you can construct a solution that is purely "vectorial" (meaning it deals directly with the direction and strength of the fields, without getting bogged down in unnecessary extra math).

Summary

In short, the authors didn't invent new physics; they invented a new mathematical shortcut.

They broke a hard problem into two easier ones.
They used a "frequency translator" to make the math simple.
They built a "reverse-engineering machine" to turn the simple math back into a real solution.
They used this machine to write down a clear, direct formula for how electromagnetic fields evolve, extending previous methods to handle more complex, time-changing situations.

The paper claims this approach offers "analytical efficiency," meaning it provides a cleaner, more direct way to calculate these electromagnetic solutions than previous methods allowed.

Technical Summary: A Biquaternionic Reformulation of Maxwell's Equations via Fourier Analysis

Problem Statement
The paper addresses the analytical challenge of solving the time-dependent Maxwell system and related wave equations within a biquaternionic framework. While Clifford algebras and quaternionic analysis have previously been used to factorize differential operators and analyze time-harmonic Maxwell systems, there is a need for explicit, purely vectorial solutions to the full time-dependent system. The authors aim to characterize the kernel of parabolic Dirac operators ( $D \pm i\partial_t$ ) and construct a well-defined right inverse operator to facilitate the solution of inhomogeneous electromagnetic problems without relying on classical fundamental solutions that may be difficult to compute directly.

Methodology
The authors employ a combination of biquaternionic algebra, Fourier analysis, and operator theory.

Biquaternionic Setting: The work is set in the algebra of biquaternions ( $\mathbb{B} = \mathbb{H} + i\mathbb{H}$ ), where functions are decomposed into real and imaginary quaternionic parts ($w = u + iv$). The Moisil-Teodorescu operator (Dirac operator) $D$ is defined as $D = \sum_{i=1}^3 e_i \partial_i$ .
Factorization: The wave operator $-\Delta + \partial_{tt}$ is factored into parabolic Dirac operators: $-\Delta + \partial_{tt} = (D + i\partial_t)(D - i\partial_t)$ .
Fourier Analysis: Instead of seeking a fundamental solution $\Phi_\pm$ in the spatial domain, the authors fully characterize its Fourier transform $\mathcal{F}(\Phi_\pm)$ . They derive that $\mathcal{F}(\Phi_\pm)(k, t) = \exp(\mp 2\pi k t)$ , expressed via a quaternionic exponential function involving cosine and sine terms.
Operator Construction: Using the properties of the Fourier transform (specifically convolution and differentiation rules), the authors define a "parabolic Teodorescu transform" ( $T_{\mathcal{C}, \pm}$ ). This operator acts as a right inverse to the parabolic Dirac operators.
Kernel Characterization: The kernel of $D \pm i\partial_t$ is characterized as a generalized div-curl system linking the scalar and vector parts of the real and imaginary components of the biquaternionic function.

Key Contributions and Results

Characterization of the Kernel (Proposition 4): The authors establish that a biquaternionic function $w = u + iv$ belongs to the kernel of $D \pm i\partial_t$ if and only if its components satisfy a specific system of coupled partial differential equations involving divergence, curl, and time derivatives. This system generalizes the Cauchy-Riemann equations to three dimensions with time dependence.
Construction of Solutions (Propositions 5 & 7): The paper demonstrates how to construct a biquaternionic solution $u \pm iv$ given a scalar solution $u$ to the wave equation. This involves an integration process and the use of the Teodorescu transform to generate the imaginary part, effectively generalizing the concept of harmonic conjugates.
Right Inverse Operator (Theorem 10): The central analytical result is the construction of the operator $T_{\mathcal{C}, \pm}$ , defined via an integral involving the Fourier transform of the source term and the quaternionic exponential kernel. The authors prove that $(D \pm i\partial_t) T_{\mathcal{C}, \pm}[\mp i w] = w$ , establishing $T_{\mathcal{C}, \pm}$ as a valid right inverse for functions in $L^2(\mathcal{C}, \mathbb{B})$ .
Explicit Solutions to Maxwell's Equations (Theorem 13): By combining the right inverse operator with the factorization of the wave operator, the authors derive explicit vectorial solutions for the time-dependent Maxwell system in Gaussian units.
- The electric field $\vec{E}$ and magnetic field $\vec{B}$ are expressed as the real and imaginary parts of a purely vectorial biquaternionic function.
- The solution relies on the compatibility condition (charge conservation: $\text{div}\vec{j} + \partial_t \rho = 0$ ) to ensure the scalar part of the constructed solution satisfies the wave equation, allowing for the removal of scalar components to yield a purely vectorial solution.
Extension to Homogeneous Media (Theorem 16): The methodology is extended to Maxwell's equations in homogeneous, isotropic media with permittivity $\varepsilon$ and permeability $\mu$ . By introducing a scaling parameter $\lambda = \sqrt{\varepsilon\mu}$ , the authors provide explicit solutions for the general case where the speed of light $c \neq 1$ .

Significance and Claims
The paper claims that these tools offer "analytical efficiency" for complex electromagnetic problems. The primary significance lies in:

Explicit Vectorial Solutions: The work provides a direct method to construct purely vectorial solutions to the time-dependent Maxwell system, avoiding the need to solve for scalar potentials separately in the final formulation.
Fourier-Quaternionic Synergy: It demonstrates the utility of combining Fourier analysis with biquaternionic algebra to handle first-order perturbed Dirac operators, offering an alternative to classical integral operators used in previous time-harmonic studies.
Generalization: The approach extends prior biquaternionic methods (which often focused on time-harmonic cases) to the full time-dependent regime, providing a unified framework for analyzing the wave equation and electromagnetic fields.

The authors conclude that their results highlight the synergy between Fourier transforms and quaternionic analysis for tackling complex partial differential equation systems in electrodynamics, while noting that future work could explore nonlinear perturbations, relativistic frameworks, and numerical implementations.