Poincare invariant interaction between two Dirac particles

TL;DR

This work develops and compares two classical interaction schemes for spinning Dirac particles: an instantaneous Coulomb coupling and a Poincaré‑invariant, action‑at‑a‑distance interaction previously formulated. Using a two‑point center‑of‑charge description and a synchronous lab frame, the authors derive coupled CC/CM equations of motion and analyze conservation laws, natural units, and boundary conditions. A key finding is that the Poincaré interaction strengthens CM‑to‑CM attraction, enabling stable spin‑1 bound pairs at Compton‑scale separations, while simultaneously not satisfying action–reaction at the CC level. The framework provides a relativistically invariant analytic and numerical approach to high‑energy scattering and bound‑state formation, with clear distinctions from the Coulomb case and potential applications to polarized beams and scattering analysis.

Abstract

The spinning electron-electron interaction is described in classical terms by means of two possible classical interactions: The instantaneous Coulomb interaction between the charge centers of both particles and the Poincaré invariant interaction developed in a previous work. The numerical integrations are performed with several Mathematica notebooks that are available for the interested readers in the reference Section. One difference of these interactions is that the Poincaré invariant interaction does not satisfy the action-reaction principle in the synchronous description and, therefore, there is no conservation of the mechanical linear momentum. It is the total linear momentum of the system what is conserved. In this synchronous description the interaction is not mediated by the retarded fields but is described in terms of the instantaneous positions, velocities and accelerations of the center of charge of both particles. In the Poincaré invariant description the net binding force that holds linked two Dirac particles is stronger than in the Coulomb case, thus forming a stable spin 1 system of 2 Dirac particles. This bosonic state of spin 1 does not correspond to a Cooper pair because the separation between the centers of mass of the Dirac particles is below Compton's wavelength, smaller than the correlation distance of the Cooper pair. Since the Poincaré invariant interaction is relativistically invariant it can be used for analyzing high energy scattering processes.

Figures (10)

• Figure 1: This model represents the circular motion, at the speed of light, of the center of charge of the electron in the center of mass frame, as described by the dynamical equation (\ref{['constS']}). The center of mass is always a different point than the center of charge. The spin ${\bi S}$ has the opposite direction to the angular velocity $\bomega$. The radius of this motion is $R_0=\hbar/2mc$, in this frame. The angular velocity is $\omega=2mc^2/\hbar$.
• Figure 2: Dirac particle in the center of mass reference frame, with the spin along the OZ axis. The initial position and velocity of the CC on the $XOY$ plane is determined by the phase $\psi$. The radius of this motion is $R_0=1/2$, in natural units.
• Figure 3: Initial position of the CM when the velocity of the CM ${\bi v}$ and the spin direction form an angle $\alpha$. It is perpendicular to the vectors ${\bi v}$ and ${\bi u}^*_0\times{\bi r}^*_0$, of distance to the center $O-CM=(v/2)\sin\alpha$, in natural units, and is independent of the initial position and velocity of the CC. In general, the separation between both centers CC and CM, is in the interval $|{\bi r}-{\bi q}|\in[1/2-(v/2)\sin\alpha,1/2+(v/2)\sin\alpha]$, in natural units, so that this separation is not a constant of the motion.
• Figure 4: Three instants of the Poincaré interaction of two Dirac particles with different spin orientations. It is depicted the initial velocity of both particles and in some places the CM spin to visualize the spin dynamics. It is analyzed in the center of mass of the two particles. The initial velocity is $v=0.08$, the initial location of particle 1 is $(3,0.3,0)$ and the spin orientations $\theta_1=30^\circ, \phi_1=60^\circ, \psi_1=0$ and $\theta_2=120^\circ, \phi_2=180^\circ, \psi_2=180^\circ$. We see that the motion is almost free before and after the interaction section that is a region of a few Compton's wavelentgh. The spin $1/2$ has been reescaled and we see a slight change of orientation of both spins during the interaction.
• Figure 5: Poincaré invariant interaction of two Dirac particles with boundary conditions in natural units $v=0.058$, $\beta=90^\circ$, $\lambda=210^\circ$, initial position of the CM of particle 1, $dx=5,dy=3,dz=0$, spin orientation $\theta_1=20^\circ,\phi_1=0^\circ,\psi_1=30^\circ$ and for particle 2, $\theta_2=30^\circ,\phi_2=180^\circ,\psi_2=180^\circ$. The second and third pictures correspond only to the change of $\psi_2=90^\circ$, and $\psi_2=30^\circ$, respectively. The spin orientation of both particles has changed during the interaction.