This blog post is a continuation for the pipe model that we have developed for the spinning of a created electron-positron pair.
The idea is that the wave function (relevant for the rotation) of the system electron & positron has a 360 degree rotation symmetry. The wave function of the system completes one wavelength when we rotate coordinates by 360 degrees.
The wave function of the system is the product of the wave functions of the electron and the positron.
The individual wave function of a particle does not behave in an intuitive way for our 3-dimensional space: the phase of the wave function is inverted when we rotate coordinates by 360 degrees.
That inversion does not make much sense for a complete quantum system, but does happen if we break up the system into parts.
We may require that the wave function of a complete quantum system must return to the original value after we rotate coordinates by 360 degrees.
If we try to treat the electron as an independent quantum system, we end up with the strange 720 degree rotation symmetry of the spin.
The SU(2) geometry of spinors gets an explanation from this.
What about the gyromagnetic ratio 2? In quantum mechanics, we do not have rules about how to determine the momentum of an individual part of a quantum system. The rotation of the electron lives in a strange 720 degree geometry. How should we map it to the ordinary 3D space to determine the velocity v of the electron and calculate the magnetic force it exerts?
It might be that the observer must treat the electron rotation as having a double velocity compared to the "real" velocity, so that he sees the electron wave function having a 360 degree rotation symmetry. Then he will get a double value for v, and will see the electron exert a magnetic force which is 2X of what we would expect based on the spin angular momentum 1/2 h-bar.
What about the spin angular momentum, the 1/2 h-bar? Why he does not see the electron spin as double? One may conjecture that since quantum mechanics conserves angular momentum, one must see the spin to have its correct value. On the other hand, there is no conservation of magnetic moment. Nothing breaks if an observer sees and feels the magnetic moment of an electron as double.
The spin angular momentum has been measured directly by flipping the spin of many electrons in a block of solid material. We know that the angular momentum is 1/2 h-bar. The flipping obviously has to use the magnetic moment of the electron as a handle which we use to flip the orientation of the spin. Why does the magnetic handle have a double strength, compared to what we would expect?
The Dirac equation offers sort of an explanation. We still need to find the link from the model we tried to sketch, to the Dirac equation.
The Dirac equation does not have a reasonable speed operator v. There is also the mystery of zitterbewegung. These are probably connected to the fact that the electron only forms half of the complete quantum system, the electron-positron pair.
The concept of the reduced mass may offer a clue. A two-particle system is represented by the center of mass and the vector from particle 1 to particle 2. The treatment of the complete electron-positron quantum system might involve similar ideas.
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