ABSTRACT (original article): In this work we produce a classical Lagrangian description of an elementary spinning particle which satisfies Dirac equation when quantized. We call this particle a classical Dirac particle. We analyze in detail the way we arrive to this model and how the different observables and constants of the motion can be expressed in terms of the degrees of freedom and their derivatives, by making use of Noether’s theorem. The main feature is that the particle has a center of charge r, moving at the speed of light, that satisfies fourth-order differential equations and all observables can be expressed only in terms of this point and their time derivatives. The particle has also a center of mass q, that is a different point than the center of charge. This implies that two different spin observables can be defined, one S with respect to the point r and another SCM with respect to the point q, that satisfy different dynamical equations. The spin S satisfies the same dynamical equation than Dirac’s spin operator. The fourth-order differential equations for the point r can be transformed into a system of second-order ordinary differential equations for the center of charge and center of mass. The dynamics can be described in terms of dimensionless variables. The possible interaction Lagrangians are described and we devote the main part of the work to the electromagnetic interaction of the Dirac particle with uniform and oscillating electric and magnetic fields. The numerical integrations of the dynamical equations are performed with different Mathematica notebooks that are available for the interested reader. CITATION (original article): Juan Barandiaran, Martin Rivas (2025), Classical Dirac particle I, arXiv:2505.08794. https://doi.org/10.48550/arXiv.2505.08794
Keywords: Electron; Spin; elementary particle; Spinning Particle model; Dirac particle; electron dynamic simulation; zitterbewegung; Larmor angular velocity
1. Introduction to the Spinning Particle Model
1. Introduction to the Spinning Particle Model
We have all learnt about a mysterious and little understood property of the electron and other elementary particles called Spin.
Sometimes it leads us to imagine the electron as a little sphere spinning around itself, but this visualization does not make much sense neither physically nor mathematically .
A more advanced visualization makes us think of elementary particles as a "cloud", a region of space, where there is a probability of finding the particle, but this model is far from our daily experience.
The Spinning Particle Model Model will provide us with a clearer, more realistic and intuitive classical picture of how the electron with spin behaves.
In this series of articles we will derive and provide the equations in Mathematica code to model the dynamics of the electrons, described as classical particles with spin, with angular momentum, and their interactions, both numerically and graphically.
Sometimes it leads us to imagine the electron as a little sphere spinning around itself, but this visualization does not make much sense neither physically nor mathematically .
A more advanced visualization makes us think of elementary particles as a "cloud", a region of space, where there is a probability of finding the particle, but this model is far from our daily experience.
The Spinning Particle Model Model will provide us with a clearer, more realistic and intuitive classical picture of how the electron with spin behaves.
In this series of articles we will derive and provide the equations in Mathematica code to model the dynamics of the electrons, described as classical particles with spin, with angular momentum, and their interactions, both numerically and graphically.
This first article describes the derivation of the Spinning Particle Model.
We will also explain the main steps of the derivation of the Spinning Particle model, with some basic clarifications for the non-experts in theoretical physics, and links to the deeper peer-reviewed papers and videos & books that support it.
In following articles we will describe the interaction of the electron with electric and magnetic fields and then the interaction between two electrons.
These articles are based on the work by Martin Rivas and most of the text and figures are extracts from his papers, detailed references are listed in Section 19.
We will also explain the main steps of the derivation of the Spinning Particle model, with some basic clarifications for the non-experts in theoretical physics, and links to the deeper peer-reviewed papers and videos & books that support it.
In following articles we will describe the interaction of the electron with electric and magnetic fields and then the interaction between two electrons.
These articles are based on the work by Martin Rivas and most of the text and figures are extracts from his papers, detailed references are listed in Section 19.
As the Spinning Particle Model is obtained through a general formalism to describe classical elementary particles with spin and satisfies the Dirac equation when it is quantized, we are modelling a classical Dirac particle.
The key feature of the Spinning Particle Model is that it simplifies the system to tracking just one point, whose motion determines all observable properties.
We only need to describe the evolution of the point,, the Center of Charge (CC), where the external forces are defined, and all observables: energy, linear momentum, angular momentum, Center of Mass position, angular velocity and spins are expressed in terms of this point and its time derivatives.
The Spinning Particle Model also has another characteristic point, the Center of Mass (CM), which is a different point than the Center of Charge, and is a function of .
The key feature of the Spinning Particle Model is that it simplifies the system to tracking just one point, whose motion determines all observable properties.
We only need to describe the evolution of the point,
r
r
The Spinning Particle Model also has another characteristic point
q
r