Physical properties of elementary particles: Inertia and Interaction

This paper proposes a classical model of elementary particles where distinguishing between the center of mass and the center of interaction leads to a description of spinning particles whose dynamics, when the center of interaction moves at light speed, yield Dirac's equation upon quantization.

The Gist

Here is an explanation of Martin Rivas's paper, translated into simple language with everyday analogies.

The Big Idea: Two Centers, One Particle

Imagine you are trying to describe a tiny, invisible speck of matter (like an electron). In standard physics, we usually treat this speck as a single, perfect dot. We assume that its weight (inertia) and its ability to grab onto other things (interaction/charge) are all located in that exact same spot.

Rivas's paper asks a simple "What if?" question:
What if those two things aren't in the same spot?

Think of it like a spinning coin:

• Inertia (Mass): Imagine the coin's weight is concentrated in the very center. That's where it "wants" to go if you push it.
• Interaction (Charge): Now, imagine the coin has a tiny, glowing sticker on its very edge. When other magnets come near, they grab the sticker, not the center.

Rivas suggests that elementary particles (like electrons) are more like this spinning coin. They have a Center of Mass (where the weight is) and a Center of Charge (where the "grabbing" happens), and these two points are actually different.

The "Zippy" Charge

Here is the wildest part of the theory: The Center of Charge is moving at the speed of light.

Imagine the electron is a tiny, heavy planet (the Center of Mass) that is sitting still. But orbiting that planet at the speed of light is a tiny, glowing satellite (the Center of Charge).

• Because this satellite is zooming around so fast, it creates a magnetic field (just like a spinning magnet).
• This explains why electrons have "spin" and magnetic properties even when they aren't moving forward through space.
• The paper argues that if you do the math on this "heavy planet with a light-speed satellite," it perfectly matches the famous Dirac Equation (the math that describes how electrons behave in quantum mechanics).

The "Rigid" Rule (The Atomic Principle)

Why can't the electron just have three or four different centers for different forces?

Rivas introduces a rule called the Atomic Principle. Think of an elementary particle like a perfectly rigid Lego brick.

• If you bump into it gently, it might bounce or stick to another brick.
• But if you hit it hard, it doesn't change shape, break apart, or get a "scratch." It remains exactly the same brick.

Because the particle is so rigid and unchangeable, it can only have one "interaction point" for all its forces (electric, magnetic, etc.). If it had two different points for two different forces, the math would break, and the particle wouldn't stay "elementary" (indivisible). So, even though the electron has electric charge, color charge (for strong force), and weak charge, they all act through that single, fast-moving "satellite" point.

The Dance of the Two Points

The paper describes the motion of the electron as a dance between two partners:

1. The Heavy Partner (Center of Mass): Moves slowly, obeys normal Newtonian laws, and represents where the particle "is" in a general sense.
2. The Light-Speed Partner (Center of Charge): Zooms around the heavy partner at the speed of light in a perfect circle.

The Magic Connection:
Even though the "Light-Speed Partner" is doing a crazy, fast dance, the "Heavy Partner" moves smoothly. The paper shows that the chaotic motion of the charge actually forces the mass to move in a way that looks exactly like the predictions of quantum mechanics.

Why This Matters

In standard physics, we often say, "Electrons are weird point particles that spin." But we don't really know how a point can spin.

Rivas's model gives us a classical picture (a visualizable story) for that weirdness:

• The electron isn't a spinning dot.
• It's a heavy center with a charge zipping around it at light speed.
• When you turn this picture into quantum math, it solves the Dirac Equation naturally.

Summary Analogy

Imagine a hula hoop (the charge) spinning around a heavy pole (the mass).

• The pole stays relatively still or moves slowly.
• The hoop spins incredibly fast.
• To an outsider, the whole thing looks like a single object with a "spin" and a "magnetic pull."
• This paper argues that the electron is exactly this: a heavy center with a charge spinning around it at the speed of light.

This theory bridges the gap between the "smooth" world of classical physics (things moving in circles) and the "weird" world of quantum physics (Dirac's equation), suggesting that the mystery of the electron's spin is just a very fast, very small orbit.

Technical Summary

Based on the provided paper "Physical properties of elementary particles: Inertia and Interaction" by Martín Rivas, here is a detailed technical summary.

1. Problem Statement

The paper addresses a fundamental conceptual gap in the description of elementary particles within classical physics. Standard models typically treat elementary particles as point-like objects where the Center of Mass (CM) (associated with inertia) and the Center of Charge/Interaction (CC) (associated with interaction) are coincident.

However, experimental evidence (such as the electron possessing intrinsic angular momentum and a magnetic moment even at rest) suggests that these properties might arise from internal dynamics. The author posits that if inertia and interaction are distinct physical properties, they should correspond to distinct geometric points. The core problem is to formulate a classical kinematical model of an elementary spinning particle where the CM and CC are distinct, and to demonstrate that this classical model, when quantized, naturally yields Dirac's equation.

2. Methodology

The author employs a Lagrangian formalism based on three fundamental principles:

1. Restricted Relativity Principle: Invariance under the Poincaré group.
2. Variational Principle: Derivation of equations of motion via Euler-Lagrange equations.
3. Atomic Principle: The requirement that an elementary particle's internal structure remains unmodified during interactions (unless annihilation occurs). This implies the boundary variables manifold must be a homogeneous space of the Poincaré group.

Key Theoretical Steps:

• Variable Definition: The model defines the particle's state using a set of variables $x \equiv (t, r, u, \alpha)$, where:
  • $r$: Position of the Center of Charge (CC), moving at the speed of light ($u = c$).
  • $u$: Velocity of the CC.
  • $\alpha$: Orientation of a comoving orthogonal frame.
  • $t$: Time.
• Lagrangian Construction: The free Lagrangian $L_0$ depends on velocity, acceleration, and angular velocity. Because the Lagrangian depends on acceleration, the resulting Euler-Lagrange equations are fourth-order differential equations.
• Noether Analysis: The author applies Noether's theorem to identify constants of motion (energy, linear momentum, angular momentum) and defines the Center of Mass ($q$) as a point distinct from the CC ($r$).
• Decoupling: The fourth-order equations for $r$ are decoupled into a system of second-order equations for both the CC ($r$) and the CM ($q$).
• Interaction Modeling: The interaction Lagrangian is constructed to be independent of acceleration and angular velocity (to preserve the internal structure/mass/spin), leading to a general form $L_I = A_0 + u \cdot A$. The paper specifically analyzes the minimal coupling case (electromagnetic interaction).

3. Key Contributions

• Distinction of Centers: The paper rigorously establishes a classical model where the Center of Mass ($q$) and Center of Charge ($r$) are distinct. The CC moves at the speed of light ($c$), while the CM moves at a sub-luminal velocity ($v < c$).
• Geometric Interpretation of Spin: The model provides a geometric origin for spin. The spin vector $\mathbf{S}$ is defined relative to the CC, while the spin relative to the CM ($\mathbf{S}_{CM}$) is conserved. The separation vector between the two centers is proportional to the acceleration of the CC.
• Classical-Quantum Correspondence: The author demonstrates that the classical dynamics of this spinning particle, specifically the evolution of the spin vector, satisfy the same dynamical equations as the Dirac spin operator in quantum mechanics.
• Unified Interaction Center: The paper argues that despite multiple interaction types (electromagnetic, strong, weak), an elementary particle must possess a single unique interaction center to satisfy the Atomic Principle and the requirement that the boundary manifold be a homogeneous space of the Poincaré group.

4. Results

• Equations of Motion:
  • The CM ($q$) obeys the standard relativistic equation of motion: $dp/dt = F$, where $F$ is the Lorentz force.
  • The CC ($r$) executes a circular motion around the CM in the rest frame of the CM, moving at the speed of light.
  • The separation vector $\mathbf{q} - \mathbf{r}$ is aligned with the acceleration of the CC.
• Spin Dynamics:
  • The spin relative to the CC ($\mathbf{S}$) satisfies $d\mathbf{S}/dt = \mathbf{p} \times \mathbf{u}$, which is the classical equivalent of the Dirac equation's spin evolution.
  • The spin relative to the CM ($\mathbf{S}_{CM}$) is a constant of motion for a free particle.
• Intrinsic Properties: The mass $m$ and the magnitude of the spin in the CM frame $S^{(0)}$ are identified as the two Casimir invariants of the Poincaré group, fully characterizing the particle.
• Interaction Generalization: The paper derives that while minimal coupling (linear in velocity) is a specific case, the formalism allows for more general interaction Lagrangians dependent on the velocity of the CC, provided they satisfy specific symmetry constraints.

5. Significance

• Bridging Classical and Quantum Mechanics: The paper offers a compelling classical mechanical description of the Dirac particle. It suggests that the "weird" properties of the electron (spin, magnetic moment, Zitterbewegung) are not purely quantum anomalies but can be understood as the result of a classical point charge moving at the speed of light in a specific geometric configuration.
• Reinterpretation of Elementary Particles: It challenges the standard point-particle model (where CM = CC) by proposing that the internal structure of elementary particles is dynamic, involving a separation between where the particle "is" (inertia/CM) and where it "interacts" (charge/CC).
• Foundation for Further Research: The model provides a framework for analyzing interactions beyond minimal coupling, including strong and weak interactions, and offers a classical basis for studying scattering and bound states of Dirac particles without invoking full quantum field theory immediately.
• Numerical Validation: The author references numerical solutions (via Mathematica) for interactions with uniform and oscillating fields, as well as interactions between two Dirac particles, validating the solvability of the derived fourth-order differential equations.

In summary, Rivas presents a classical Dirac particle model where the separation of the center of mass and center of charge, driven by a charge moving at the speed of light, naturally reproduces the dynamics and intrinsic properties (mass and spin) of the quantum Dirac equation.