Classical Dirac particle I

TL;DR

We introduce a classical Dirac particle: a spinning, elementary mechanical system whose quantization yields the Dirac equation. The model centers on a center-of-charge ${i r}$ moving at $|{i u}|=c$ and a separate center-of-mass ${i q}$, yielding two spins ${i S}$ and ${i S}_{CM}$ and a fourth-order CC dynamics that reduces to a coupled second-order system for ${i r}$ and ${i q}$. Noether analysis provides conserved quantities, including the Pauli-Lubanski structure, while the invariants $m$ and $S$ remain fixed under interactions; the framework accommodates minimal coupling to electromagnetism and specific field configurations, yielding phenomena such as zitterbewegung and time dilation. The approach quantifies internal CC motion with frequency $ u_0=2mc^2/ hbar$ in the CM frame, which Doppler-shifts to $ u(v)= u_0/ ildeeta(v)$, and predicts classical magnetic and electric dipole moments consistent with Dirac theory, highlighting potential experimental probes and natural-unit scaling for high-energy regimes.

Abstract

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 diferential 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.

Figures (14)

• Figure 1: This model represents the circular motion, at the speed of light, of the center of charge of the Dirac particle in the center of mass frame, as described by the dynamical equation (\ref{['eq:Sxur']}). 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 in this frame $\bomega$. The trajectory is flat and the angular velocity has no component along the velocity ${\bi u}$. In general, for an arbitray inertial observer the spin ${\bi S}$ and the angular velocity $\bomega$ are not collinear vectors. The angular velocity has another component along the velocity $\bomega_{u}$ which produces the torsion of the CC motion, and which vanishes if the CC trajectory is flat. When we quantize this model the parameter $S=\hbar/2$. The radius of this motion is $R_0=\hbar/2mc$, and the angular velocity is $\omega_0=2mc^2/\hbar$ in this frame. If the CM is moving this internal frequency decreases. The moving clock goes slowly.
• Figure 2: Free motion of the Dirac particle with a constant CM velocity $v_z=0.12$ along OZ axis, and the CM spin orientation is $\theta=30^\circ$ and $\phi=65^\circ$. We see the zitter motion of the CC (blue) and the CM trajectory (dotted red). Attached to these points it is drawn the comoving cartesian frames. It is also depicted the center of mass spin ${\bi S}_{CM}$ (red) and the center of charge spin ${\bi S}$ (blue). On the right picture both spins are depicted to see that the ${\bi S}_{CM}$ is conserved but ${\bi S}$ is not, as given in (\ref{['eq:spinDyn']}) because the spin dynamics $d{\bi S}/dt={\bi p}\times{\bi u}$, is orthogonal to the constant linear momentum along the velocity ${\bi v}$, and precesses around ${\bi S}_{CM}$. The two components of the angular velocity $\bomega_p$ and $\bomega_u$ (magenta), are also drawn. The component $\bomega_u$ has the opposite direction to the velocity of the CC ${\bi u}$, because the angle $\alpha$ between the vector $\bomega_p$ and the velocity ${\bi v}$ is $\alpha >\pi/2$, formula (24). If $\alpha<\pi/2$, $\bomega_u$ is in the same direction than the CC velocity ${\bi u}$ (see figure \ref{['angular170']}). This component of the angular velocity is different from zero to produce the torsion of the CC trajectory and vanishes when this trajectory is flat.
• Figure 3: Free motion of the Dirac particle with a constant CM velocity $v_z=0.14$ along OZ axis, and the CM spin orientation is $\theta=170^\circ$ and $\phi=270^\circ$. We see the zitter motion of the CC (blue) and the CM trajectory (dotted red). On the right picture both spins are depicted to see that the ${\bi S}_{CM}$ is conserved but ${\bi S}$ is not, as given in (\ref{['eq:spinDyn']}) because the spin dynamics $d{\bi S}/dt={\bi p}\times{\bi u}$, is orthogonal to the constant linear momentum along the velocity ${\bi v}$, and precesses around ${\bi S}_{CM}$. The two components of the angular velocity $\bomega_p$ and $\bomega_u$ (magenta), are also drawn. The component $\bomega_u$ has in this case the same direction than the velocity of the CC ${\bi u}$, because the angle $\alpha$ between the vector $\bomega_p$ and the velocity ${\bi v}$, formula (24), is $\alpha<\pi/2$.
• Figure 4: Variation of the absolute value of the CM spin of (\ref{['absolespinvar']}) with the velocity of the CM and orientation $\phi$ between the velocity ${\bi v}$ and the center of mass spin ${\bi S}_{CM}({\bi v})$.
• Figure 5: Motion of the Dirac particle in a uniform magnetic field $B_z$. The initial velocity along OX axis is $v_x=0.05$, in natural units. We see the zitter motion of the CC (blue) and the cyclotron motion of the CM (red) which is almost a circle with a small oscillation of the same frequency as the CC motion, during a time of flight of around five turns of the CC. The CM velocity $|{\bi v}|$ is not a constant. It is also depicted the CM spin with orientation $\theta=30^\circ$, $\phi=60^\circ$, at different CM points. The red spin is the last value of the spin at the end of the integration time. In natural units the time of a turn is $\pi$ for the particle at rest and $\pi\gamma(v)$ for a moving Dirac particle. The center of mass spin precesses backwards with Larmor frequency, although the trajectory of the CM is a kind of a curly circle. In the right figure we see the free motion of the particle, the zitteberwegung of the trajectory of the CC while the trajectory of the CM is a straight line with no zitter. It is also shown the projection of the CM trajectory and the ${\bi S}_{CM}$ projection on the XOY plane that remains constant.