The spin of the electron would reflect a path which spirals very fast. The circular motion would be the origin of the spin and the magnetic moment.
The energy-momentum relation and the Klein-Gordon and Schrödinger equations
The energy-momentum relation of special relativity is
E² = p² + m².
We assume just one spatial coordinate x and that c = 1 and ħ = 1.
Let us use the usual recipe to transform it into a wave equation.
The energy operator is
i d / dt
and the momentum operator is
-i d / dx.
The wave function Ψ is complex-valued. Furthermore, we use the metric signature (- + + +) to decide the sign of the square of an operator:
d²/dt² Ψ = -d²/dx² Ψ + m² Ψ.
The equation is the massive Klein-Gordon equation.
As explained in the Wikipedia article about the Dirac equation, a second order wave equation has "too much freedom". It is hard to conserve the particle number. Charge conservation requires that the number of electrons must stay constant.
We want to reduce the equation to a first order equation. The problem is the square root in
E = sqrt(p² + m²).
How to get rid of the ugly square root?
Erwin Schrödinger in 1925 devised a workaround in the case where p² << m²:
E ≅ p² / (2 m) + m,
i d/dt Ψ = -1 / (2 m) * d²/dx² Ψ + m Ψ.
This is equivalent to the usual Schrödinger equation if we have the potential V(x, t) set to zero. The term m Ψ can be removed. It does not affect the physics.
The Schrödinger trick works if the possible momenta p have very small absolute values |p|.
Also, if we can switch coordinates to make |p| small, then we can solve the equation.
However, if the possible momenta p differ from each other a lot, what to do then?
The ugly energy-momentum wave equation may force curved paths on particles
We could try writing a wave equation like
i dΨ/dt = sqrt( -d²/dx² Ψ + m² Ψ ),
but that is hard to solve.
A simple linear wave equation allows beautiful sine wave solutions:
Ψ(t, x) = exp( -i (E t - p x) ).
An ugly equation like the one above does not allow them.
A beautiful sine wave corresponds to a particle moving along a straight path at a constant velocity.
An ugly wave equation may force the particle to move along a curved path!
That may be the origin of the spin of the electron.
The Dirac equation is somewhat ugly. Its solution for a general wave packet makes the electron to do the zitterbewegung. The electron does not move along a straight line. This probably comes from the ugly nature of the energy-momentum relation.
Hypothesis 1. The Dirac equation somehow simulates the general energy-momentum wave equation and is able to isolate the relevant features of the particle motion: a linear motion and a circular motion.
The underlying nature in various wave equations may be the path integral: they describe path integral values for a set of possible paths of a particle. The circular motion of the electron may be an interference pattern. The phases of various paths cause constructive interference in various locations in time and space.
Hypothesis 2. The spin and magnetic moment of the electron are not a result of its interaction with its own electric field. Rather, the field is dragged along the circular motion (zitterbewegung?) of the electron.
The interference pattern idea solves the question that has troubled us for a long time: what is the force which keeps the electron in the zitterbewegung loop? There is no force. The loop is just an interference pattern.
This is analogous to the double-slit experiment. What is the force which moves photons to the locations of constructive interference? There is no force.
The path integral aspect
In a path integral, a "lagrangian density" is integrated over "all" paths and then summed. The integral determines the phase at the endpoint of the path:
exp(i S),
where S is the integral of the lagrangian density L over the path. The lagrangian density is typically the energy of the particle.
If the formula for the lagrangian density (energy) L is simple and beautiful, then we presumably end up with a beautiful wave equation which can be solved with a standard plane wave.
The lagrangian density which we get from the energy-momentum relation is ugly. Thus, the wave equation is ugly, and the solutions are ugly.
The zitterbewegung shows that the Dirac equation is incorrect?
In the Dirac equation, the spin-z of the electron is "hard-coded" in the components of the 4-component (spinor) wave function ψ.
If we make a standard wave packet and it exhibits the zitterbewegung, then we should see another spin-like motion in the electron.
No second spin has been observed. This suggests that the Dirac equation actually gives incorrect solutions in the case of a general wave packet.
The Dirac equation does seem to work in the restricted case where the electron is described as a single plane wave plus the spin-z value.
The energy-momentum wave equation is nonlinear. The success of the Dirac equation proves that one can estimate a solution with a linear motion of the electron plus the spin motion. The nonlinearity is not pathologically complex if one can make such an estimate.
What is a "free particle"?
We usually think that a free particle is something which is under no interaction and moves along a straight line.
In the double-slit experiment, a photon which has passed the slits is not under any interaction - but it does not make much sense to say that the photon moves along a straight line. Rather, the final position of the photon on a photographic plate depends on the interference pattern.
We conjecture that a "free" electron moves along a curved line. The curved path produces the spin and the magnetic moment of the electron.
Conclusions
This is by far the best idea that we have come up with to explain the electron spin and magnetic moment.
If our idea holds, then the "natural motion" of a particle in spacetime is generally not a linear motion. The natural motion depends on the wave equation of the particle. The natural motion is an interference pattern which arises from various paths that the particle may take.
If the Klein-Gordon equation describes a massless particle, then those particles naturally travel along a straight line. (However, from what does their integer spin come from?)
But a massive particle where the number of particles has to be conserved, naturally "moves", or its interference pattern moves, along a curved path. This is the origin of the spin for fermions.
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