If angular momentum is stored in a field, the wave function collapse requires the hamiltonian to have global information

Suppose that an electron-positron pair is born from a rotating source, and the source stores angular momentum to the electron wave function.

           rotor
           -----------
               |
             _|_
           |      | motor

The wave function far away may look almost like a plane wave. But since it contains rotation, it should react to a magnetic field like a magnetic needle. How can the hamiltonian at a point far away know that the wave contains rotation?

Classically, there is no problem because the information of the rotation is stored in the global wave function. But in quantum mechanics, the wave function collapse will erase the global wave.

The hamiltonian in quantum mechanics should be able to calculate the energy at a spacetime point. It cannot use global information of the field.

The mysterious spin of particles may be due to the fact that angular momentum is not a local phenomenon, and it has to be added artificially to the local wave function of the particle as an extra quantum number.

The above reasoning does not yet explain the gyromagnetic ratio g = 2 of the electron, though. We need some further insight.

We are currently studying the detailed structure of the Pauli equation. It contains a trick of factoring the kinetic energy operator, which reminds us of the factoring of the Klein-Gordon operator in the Dirac equation.

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

The kinetic energy operator

     p^2 = (d^2/dx^2 + d^2/dy^2 + d^2/dz^2)
                / (2m)

can be factored into a product of sums by using the Pauli matrices:

       (σ_1 d/dx + σ_2 d/dy + σ_3 d/dz)^2
       = I d^2/dx^2 + I d^2/dy^2 + I d^2/dz^2.

The factorization can be seen as introducing a generalized momentum operator P, such that P^2 will react to a magnetic field B through cross terms.

The wave - particle duality "explains" the momentum and the spin of a particle

A particle is like a concentrated packet representation of a wave.

We can store a linear momentum p into a wave, and can associate p with the momentum of a particle. A particle in free space can have any momentum. The momentum is not quantized.

We can store angular momentum in a wave. If the wave just has one particle, then that angular momentum must be its spin. Since rotation happens in a restricted 360 degree space, the values of the spin are quantized.