UPDATE March 23, 2022: In the Pauli equation, the potential of the magnetic moment associated with the spin-z in a magnetic field B to the z direction is given by the σ • B term. The Pauli equation calculates the same phase shift in the wave function as we calculated below with our simplified "scalar electron" model.
The Feynman lectures do not mention at all how the electron beam is turned through an angle, and do not analyze what the turning force or potential does to the wave function. Feynman just assumes that the beam mysteriously turns.
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Frame-dragging in a pool of water
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<--- ω
Suppose that we have a round pool of water rotating with an angular velocity ω. Let there be a wave which circles the pool to the same direction, so that the angular velocity of the wave is 2 ω relative to a static observer. The pool of water is the frame which is being dragged.
The "rotation of a spinor" through an angle α means that the wave moves through the angle α relative to a static observer. The wave only moves an angle 1/2 α relative to the pool.
The angular position of the pool is considered relevant in the system. If we only specify the angle of the wave, we do not know the full state of the system. We have to know if the pool has been rotated less than 180 degrees or more than 180 degrees.
The rotation of a spinor through an angle α means that the wave has moved the angle α relative to a static observer.
The rotation is not a coordinate rotation. If it were, the system would return to its original state after a rotation of 360 degrees.
We can specify the state of the system by giving the angle 0 ≤ α < 360 of the wave and the angle 0 ≤ β < 360 of the pool.
The state is of the form
(α, β).
Let us define that the rotation of both started from 0 degrees. Then
β = α / 2,
or (1)
β = α / 2 + 180.
What is the state if we rotate coordinates through an angle γ?
The new state is
( (α - γ) mod 360, (β - γ) mod 360 ).
The rotations of the pool and the wave now started from -γ degrees. Formula (1) above no longer holds.
A rotation of coordinates is a different operation from watching the wave to move over an angle α.
Feynman lectures: a "rotation" of the spinor for an electron means that the electron moves along a physical loop
Richard P. Feynman (1965) explains what we mean by a "rotation" of a spinor. He has several diagrams which show a beam of electrons turning, for example, 90 degrees to the left.
If the beam makes a full 360 degree turn, then it is claimed that the electron spinor changes its sign.
If the 360 degree loop adds an integer number of wavelengths to the path of the electron, the flip of the spinor sign will cause destructive interference with another beam which did not do the 360 degree loop.
Feynman writes that the "physics" does not change if the phase of the beam wave function changes. For example, the sign flip of the beam which made a 360 degree loop does not change anything, unless we let it interfere with a beam which did not do the loop.
It would be better to say that the phase shift does change the physics. In this blog we have claimed that a phase change of a wave function can only happen if it interacts with some external system. It is a "physical" interaction and it does change the "physics".
The spin-statistics "theorem"
In our blog post on March 5, 2021 we criticized the presented proof of the theorem in Wikipedia.
Let us analyze the Wikipedia proof again. There
R(π)
denotes a 180 degree "rotation" of the "spin polarization" of a particle annihilated by φ(x) at a location x. The text says that
R(2 π) = -1.
Thus, R has to mean a physical rotation of the spinor, not of the coordinates.
But under the headline "Suggestive bogus argument", there is a coordinate rotation through 180 degrees. The Wikipedia author seems to confuse the two types of rotations:
1. A spinor rotation is a real physical process where, in our water pool example, the wave travels through an angle α as seen by a static observer. In the Feynman lectures, a beam of electrons is physically turned the angle α.
2. A coordinate rotation is just a change of coordinates. It is not a physical process. It a change of notation.
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