Dirac, Schroedinger, and Maxwell equations in scalar and vector field quantum mechanics

This paper proposes a reinterpretation of relativistic quantum mechanics by deriving the Dirac equation from a photon-like dispersion relation and introducing a vector field framework that redefines wave-particle duality as electromagnetic wave-particle duality.

The Gist

The Cosmic Radio: A New Way to See the Building Blocks of Reality

Imagine you are trying to understand how a musical instrument works.

Most of modern physics treats a particle (like an electron) as if it were a tiny, solid marble that also happens to have a "ghostly" wave following it around. This is the standard way we teach quantum mechanics: you have the "particle" (the marble) and the "wave function" (the ghostly cloud of where the marble might be).

In this paper, physicist Boris Chichkov suggests a much more elegant idea. He proposes that there is no "marble" and "ghost" separately. Instead, everything is just a wave.

Here is the breakdown of his idea using everyday analogies.

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1. The "Universal Recipe" (The Dispersion Relation)

Every wave has a "recipe" that tells it how fast it moves based on its energy. For light (photons), this recipe is very simple.

Chichkov says: “What if we treat every particle—even heavy ones like electrons—as if they were following the light's recipe?”

To make this work, he imagines that every particle is traveling through a special, invisible "medium" (like a thick syrup or a clear jelly) that changes depending on the particle's energy. By pretending the particle is just a specialized type of light traveling through this "quantum jelly," he can use the math of light to explain everything else.

2. The "Two-Sided Coin" (Scalar vs. Vector Fields)

In standard physics, we usually talk about Scalar Fields.

• Analogy: Think of a temperature map of a room. At every point, there is just a number (e.g., 72°F). It doesn't point anywhere; it just is. This is how we usually describe the "probability" of finding a particle.

Chichkov introduces Vector Field Quantum Mechanics.

• Analogy: Think of a wind map. At every point, the wind doesn't just have a strength; it has a direction. It pushes and pulls.

He shows that the famous Dirac Equation (which describes how electrons behave) and Maxwell’s Equations (which describe how light/electricity behaves) are actually two sides of the same coin.

3. The Big Reveal: "Electromagnetic Wave-Particle Duality"

This is the most exciting part of the paper.

For a century, scientists have been puzzled by "Wave-Particle Duality." We see electrons act like little bullets (particles) sometimes, and like ripples in a pond (waves) other times. It feels like the particle is "switching modes."

Chichkov’s math suggests a different story: The particle is an electromagnetic wave.

Instead of a "marble" accompanied by a "ghostly wave," he suggests the particle is actually a tiny, swirling, oscillating electromagnetic field—much like a microscopic version of the radio waves that carry music to your car.

The Metaphor:
Imagine you see a whirlpool in a river.

• The old view says: "There is a tiny spinning pebble (the particle) that creates a circular ripple (the wave) around it."
• Chichkov’s view says: "There is no pebble. The whirlpool is the movement of the water itself. The 'particle' is just the concentrated center of the wave."

Why does this matter?

If this theory holds up, it simplifies our map of the universe. It suggests that the "stuff" of the universe—from the light hitting your eyes to the electrons in your brain—is all made of the same fundamental "stuff": oscillating electromagnetic fields.

It turns the universe from a collection of "things and ghosts" into a grand, cosmic symphony of waves, all playing by the same rules.

Technical Summary

Technical Summary: Dirac, Schrödinger, and Maxwell Equations in Scalar and Vector Field Quantum Mechanics

Author: Boris Chichkov (Leibniz University Hannover)

1. Problem Statement

The paper addresses the fundamental theoretical connection between the primary equations of quantum mechanics (Schrödinger and Dirac) and classical electromagnetism (Maxwell). While these equations are often treated as distinct mathematical frameworks—one describing probability amplitudes (scalar/spinor wave functions) and the other describing physical electromagnetic fields—the author seeks to demonstrate that they can be unified through a single physical principle: the photon-like dispersion relation derived from Einstein’s special relativity.

2. Methodology

The author employs the first quantization technique applied to the relativistic energy conservation equation:
$$E^2 = (pc)^2 + (mc^2)^2$$
The methodology follows these logical steps:

• Effective Medium Analogy: The author treats a relativistic particle moving in a potential $V$ as if it were a photon moving through a dispersive medium. By comparing the particle's energy equation to the photon dispersion relation ($E = pc/n$), the author defines an effective refractive index ($n$), permittivity ($\epsilon$), and permeability ($\mu$) that depend on the particle's energy and potential.
• Scalar Field Quantization: By replacing classical observables ($E, p$) with operators ($\hat{E}, \hat{p}$) acting on scalar functions ($\phi, \chi$), the author derives the Dirac equation.
• Vector Field Quantization: Instead of acting on scalars, the author applies the same dispersion relation to transversal vector fields ($\mathbf{E}, \mathbf{H}$) that are perpendicular to the particle's momentum ($\mathbf{p} \cdot \mathbf{\Psi} = 0$). This allows for the derivation of a "quantum version" of Maxwell's equations.
• Isomorphism Mapping: The author uses Pauli matrices and vector identities to establish a formal mathematical mapping (isomorphism) between the components of the Dirac spinor and the components of the electromagnetic vector fields.

3. Key Contributions

• Unified Derivation: A highly simplified, "physically justified" derivation of the Dirac equation that bypasses more complex traditional methods by using the photon-like dispersion relation.
• Introduction of Vector Field Quantum Mechanics: The paper proposes a framework where relativistic particles are associated with specific transversal electromagnetic vector fields, rather than just abstract probability amplitudes.
• Derivation of Quantum Maxwell Equations: The author derives a set of source-free Maxwell-like equations (Eqs. 31 & 32) that describe the oscillating electromagnetic fields surrounding an arbitrary quantum particle.
• Formal Isomorphism: The paper provides an explicit mathematical link showing how the four components of a Dirac spinor can be represented by the components of the Riemann–Silberstein (Weber) vector.

4. Results

• Dirac and Schrödinger Limits: The author successfully recovers the Dirac equation, the non-relativistic Lévy-Leblond equations, and the standard Schrödinger equation as limits of the proposed framework.
• Electromagnetic Representation: The results show that the Dirac spinor $\Psi$ can be mapped to the electromagnetic fields $\mathbf{E}$ and $\mathbf{H}$. Specifically, the "large" and "small" components of the Dirac spinor correspond to the transversal electric and magnetic field components.
• Non-Relativistic Maxwell-Schrödinger Equations: In the non-relativistic limit, the author derives "mixed" equations (Eqs. 35 & 36) where the electromagnetic fields $\mathbf{E}$ and $\mathbf{H}$ satisfy the Schrödinger equation.

5. Significance

The paper proposes a paradigm shift in the interpretation of quantum mechanics:

• Redefinition of Duality: It redefines "wave-particle duality" as "electromagnetic wave-particle duality." This suggests that the de Broglie wave of a particle is not merely a mathematical probability wave, but a physical transversal electromagnetic wave.
• Physical Interpretation of the Wave Function: It provides a concrete physical identity to the wave function, suggesting that the "probability" described by quantum mechanics is rooted in the behavior of oscillating electromagnetic fields.
• Theoretical Unification: By showing that fermions (electrons) and bosons (photons) can be described using the same underlying dispersion-relation logic, the paper offers a conceptual bridge that explains why different types of particles produce similar interference patterns in experiments like the double-slit test.