Does the Dirac equation work by chance?

Despite three months of hard work we have not been able to find an intuitive physical model which would explain the Dirac equation. Richard P. Feynman wrote that no one has understood the Dirac equation "directly".

Is it possible that the Dirac factorization of the Klein-Gordon operator just by chance adds the necessary degree of freedom that can be used to describe the spin 1/2 of the electron?

Does the Dirac derivation of his equation predict the spin 1/2?

The Klein-Gordon equation is Lorentz covariant, and Dirac derives his own equation in a way which makes it Lorentz covariant, too.

But what properties of the electron does the Dirac equation actually predict?

The eigenfunctions of the equation can be defined as a standard plane wave times a 4-component Dirac spinor.

We then note that if we define an operator S in a way similar to the Pauli equation, S obeys rules which emulate the simple commutation rules we expect from a particle of spin 1/2. That is, if we know the spin projection in the z direction, we know nothing of the projection in the x direction, and so on.

Did the Dirac equation predict the spin 1/2? One might say: no - we just picked an arbitrary operator S which emulates the known rules of spin 1/2. But the Dirac equation made it possible to use a very simple operator S. At least in that sense, the Dirac equation predicted the spin 1/2.

The Dirac hamiltonian H commutes with L + S, where L is the usual orbital angular momentum operator. That fact suggests that S really describes some kind of angular momentum.

But H does not commute with S if the electron is relativistic. Does that make sense? If we have a free electron, why would its spin change during its flight?

Does the Dirac equation predict the gyromagnetic ratio 2?

https://en.m.wikipedia.org/wiki/Pauli_equation

The Pauli equation is the non-relativistic limit of the Dirac equation.

In the general form of the Pauli equation, the interaction with a magnetic field B is hidden in the minimal coupling

       (σ • (p - qA))^2

kinetic term.

But in the standard form, the interaction is shown explicitly as

       σ • B.

The Dirac equation does predict the correct interaction strength and the gyromagnetic ratio 2.

Is it possible that the Dirac equation by chance gets the ratio 2 right, even if the equation does not "really" describe the physical system? That looks unlikely.

The Dirac factorization of the Klein-Gordon operator is a general way to add a spin degree of freedom?

The Klein-Gordon equation with the minimal coupling describes the behavior of a spinless electrically charged massive point particle.

The Dirac factorization trick of the Klein-Gordon operator adds a spin degree of freedom, and furthermore, the minimal coupling gives the right interaction strength with a magnetic field for the spin, too. The minimal coupling term was designed to describe the behavior of a spinless point particle, but it magically produces the right interaction also for the spin angular momentum.

Is it a general rule that factorization of an operator adds a spin-like degree of freedom? If yes, why?

Does the factorization always give a sensible strength for the interaction of the spin with an external field?

The spin 1/2 h-bar of an electron is a remnant of the orbital angular momentum of positronium?

In the hydrogen atom, the electron orbital angular momentum in the z direction is an integer multiple of h-bar. Classically (that is, in Newtonian mechanics), almost the entire orbital angular momentum is in the movement of the electron, and only a tiny fraction in the movement of the proton.

In the positronium "atom", the orbital angular momentum is divided evenly between the electron and the positron. This may be the origin of the strange electron spin 1/2 h-bar. An electron and a positron are always created together. If they form some kind of a primitive positronium atom where the particles do not yet possess a spin, then after flying away, the electron and the positron will evenly share the 1 h-bar of orbital angular momentum of the positronium atom. In this model, the electron and the positron would have parallel spins. In the real world, they seem to have opposite spins.

Classically, the distance between the electron and the positron is 2r in the positronium atom, where r is the Bohr radius of the hydrogen atom. Both particles move around the center of mass in a circular orbit whose radius is r. Since the electric pull on the electron in positronium is only 1/4 of the pull in a hydrogen atom, the orbital velocity of the electron has to be 1/2 of that in a hydrogen atom, so that the centripetal acceleration agrees with the electric pull.

The de Broglie wavelength is defined as

       λ = h / p.

Since the electron momentum p in a positronium atom is just half of the hydrogen atom, the electron only completes half of a de Broglie wavelength in its circular orbit. In previous blog posts we have said that a "natural" periodic movement of a single particle must contain an integer number of wavelengths, to avoid destructive interference. But we noted that if two particles are moving "in unison", then we may consider them as a single system, and it is enough that the system completes a full number of wavelengths in one period. Now we realize that the positronium atom is just such a system.

In a previous blog post we developed the "pipe model" of an electron-positron pair. The particles rotate in unison at the opposite ends of the pipe. A problem is to find a way how the particles can preserve their tandem movement even when one of the particles is accelerated or its spin is rotated.

If we think of a positronium atom in a laboratory, both the electron and the positron have a spin 1/2 h-bar, and they may also have a multiple of h-bar of orbital angular momentum. How do we model the annihilation then?