The gyromagnetic ratio is 2 because the spin lives in a 1+2-dimensional space?

In the Pauli equation, the effect of the vector potential can be modeled with the flow vector of water.

Close to the wire carrying the current, water flows to the same direction as the current. The velocity vector of water is the vector potential A.

          --->      --->        A

        ------>   ----->

      --------------------- wire

          I -->

If we have an electron doing a circle in the plane of the page anti-clockwise, then the flow of the water "helps" it in its movement close to the wire. The water flow will inflate the wavelength of the electron close to the wire.

Suppose that we have a stationary wave solution in a 2-dimensional (the time is the 3rd dimension) round cavity. Suppose that we increase the area of half of the cavity by 1% and decrease the other half by 1%. This corresponds to modifying the wave equation slightly inside the cavity. The wavelength in one half will decrease 0.5%. That corresponds to a kinetic energy increase of 1%.

But if the cavity would be 1-dimensional, and we would decrease the length of one half by 1%, that would correspond to a 2% increase in kinetic energy.

The angular momentum of the translational movement of an electron,

       r × v,

lives in a 1+3-dimensional space. There are three orthogonal axes of rotation. But the spin angular momentum seems to live in a 1+2-dimensional space. The eigenvectors of the Pauli matrices are in a 2-dimensional space.

The difference of dimensionality may be the origin of the strange gyromagnetic ratio g = 2 for the spin. If the effect of a magnetic field is to change the volume (or the area in 2D or the length in 1D) of a half of the the stationary wave system by a ratio which depends on B, then its effect on the kinetic energy depends on the dimensionality of the system.

In our example, an electron circles in a plane close to the wire. There are 1+2 dimensions. If the spin would live in a 1+1-dimensional system, then the effect of B on the kinetic energy of a stationary solution might be 2X in the spin case.

https://en.wikipedia.org/wiki/Pauli_matrices#Eigenvectors_and_eigenvalues

If we stretch the plane by 1 % in the direction of the x axis, then a vector which is at a 45 degree angle to the x axis will get stretched by 0.5 %. Many angles between the eigenvectors of various σ_i are 45 degrees or 135 degrees. If the effect of a magnetic field would be to stretch one eigenvector by 1%, then the volume of a 3D cube might grow by 2 %. This might be the origin of the gyromagnetic ratio g = 2.

Why does the spin live in 1+2 dimensions?

Why does the spin live in one less spatial dimensions than the ordinary angular momentum? Maybe some uncertainty relation drops one spatial dimension from the description of a 1/2 h-bar angular momentum?