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 Electron in electromagnetic fields
1. Introduction to the Electron in electromagnetic fields
In the previous article (Clasical Dirac Particle I, called Part I in this article) we have described the derivation of the Spinning Particle Model to model the dynamics of the electron (described as a classical Dirac particle with spin, with angular momentum), both numerically and graphically, providing the equations and code to model the electron.
We will refer to the formula number (XX) in the previous article as (Part I.XX).
In this second article, that completes the scope of the “Classical Dirac Particle I” paper [2], we will focus on modelling the interaction of the electron with electric and magnetic fields, deriving the expressions of the interaction and solving them with the code in the last section of this Mathematica notebook .
We will refer to the formula number (XX) in the previous article as (Part I.XX).
In this second article, that completes the scope of the “Classical Dirac Particle I” paper [2], we will focus on modelling the interaction of the electron with electric and magnetic fields, deriving the expressions of the interaction and solving them with the code in the last section of this Mathematica notebook .
As the previous article, this article is 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 16.
The Spinning Particle Model has been derived, without any assumption about the shape or behavior of the electron, from the three fundamental principles used: Restricted Relativity principle, Variational principle and Atomic principle.
We have already seen that 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 .
We have already seen that 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
q
r
The Center of Charge is moving at the speed of light (| | = c) and the Center of Mass is moving at a speed | | < c.
The complete dynamical equations for the evolution of the Center of Mass and Center of Charge can be expressed in terms of dimensionless classical variables.
r
u
q
v
The complete dynamical equations for the evolution of the Center of Mass
q
r