The electron spin: the classical origin

We have previously introduced the rubber string model for the electron electric field. The field lines are rubber strings. The electron is "suspended" from rubber strings which repel each other.

                              |

                              | e-

                     ------- ● -------

                              |

                              |  field lines

Since the field lines cannot move faster than light, they might form a "wire cage" which tries to keep the electron static.

Suppose then that we could make the electron to move at the speed of light. The field lines will bend very tightly at the electron. The force against the electron might be able to keep it in a circular orbit.

The circular orbit would be the zitterbewegung orbit, and that would explain why the electron has a magnetic moment.

Quantum mechanics then would dictate why the orbit has a constant size: the spin and magnetic moment are constant. It is like the quantized orbits of the electron in the hydrogen atom.

However, we still do not understand why the electron spin is only 1/2 ħ and not ħ.

We can move the electron linearly at a speed less than light. There is no circular orbit in that case. The situation may be different if the electron moves at the speed of light.

The mass of the electron in this model is in the deformation of its electric field. Since the electron does the circle at the speed of light, its rest mass must be zero.

If the field lines somehow are able to "confine" the electron in a "box", then quantum mechanics says that the electron must move. Classical mechanics says that it is able to do a circle if the centripetal force is strong enough.

The mystery of the Dirac equation

The Dirac equation predicts the spin 1/2 and the magnetic moment of the electron. The equation probably does not know anything about the wire cage around the electron. How is the equation then able to predict these things?

The Dirac equation actually contains four fields: each component of the spinor is associated with field. These four fields interact in a very complex way in the Dirac equation.

Let us form a standard wave packet for the Dirac equation. Erwin Schrödinger showed that the expectation value of the electron position does circular motion at the speed of light. This is zitterbewegung.

Thus, the "natural motion" of a particle in the Dirac equation is zitterbewegung. In the Schrödinger equation the natural motion is a linear motion.

The Dirac equation does NOT describe a point particle which moves freely and independently in space.

The massless Klein-Gordon equation nicely and in a very simple way describes an independent particle which moves at the speed of light. The Schrödinger equation does the same for a massive particle which moves slower than light.

If we want an equation which describes a particle doing zitterbewegung, the equation probably must be more complex.

Why the complex equation should be one which we obtain by taking a "square root" of the Klein-Gordon equation? That is, the Dirac equation.

And why would that equation describe the behavior of the electron in its wire cage?

A clue: the Klein-Gordon equation describes the electromagnetic field nicely. And the electron is an electric charge. That probably is the connection. It remains to show that the electron in its wire cage must satisfy the Dirac equation.

A second clue: the Feynman propagator for the Dirac field calculates much the same thing as our classical model in the previous blog post about Compton scattering.

The static electric field of a charge as an optimization problem: the drum skin model

We have previously suggested that the static electric field around a spherical, non-pointlike, charge assumes a minimum energy configuration. The potential around the charge is reduced as long as the gain from the lower potential can cover the energy cost of creating the electric field.

This is analogous to putting a heavy metal sphere on a drum skin. It creates a pit and assumes a minimum energy configuration with the skin.

Can the sphere do a circular motion in the pit that it created? That is only possible if the sphere moves very fast compared to the movements of the skin. Otherwise, the pit follows the sphere.

Conclusions

In the hydrogen atom, the orbits of the electron have classical counterparts, as the Sommerfeld atom model proves.

Quantum mechanics restricts the permitted orbits to those which are stationary and do not self-destruct in destructive interference.

If the combined system electron plus its electric field is analogous to the hydrogen atom, then we should find a classical model where the electron does zitterbewegung inside its own electric field. Quantum mechanics would then somehow restrict the spin to 1/2.