The two dimensionality of quantum theory:
In Stern-Gerlach experiment there are only splitting of atoms into 2 parts! Why there are not 3 or more splitting parts.
The Stern-Gerlach-Splitting of atoms in 2 parts is due to 2 directions of magnetization related with the spin structure of atom electrons: 1/2. h/2π – (-1/2. h/2 π) = 1 h/2π and is a result of the fact that spin is a representation of SO(2).
The question is why is it so and not a SO(3) representation with:1 . h/2 π, 0. h/2 π, -1. h/2 π, where the 3 values are also spaced by 1 h/2 π, etc.
Thus, also Dirac equation is nothing else than two two component Spinor equations and the allegedly 4-component Dirac equation is decomposed always in two two-component spin equations.
The spin s representation of SU(2) has dimension 2s+1. So, for s=1/2, 2s+1=2. SO(2) is abelian, so it only has one-dimensional representations and SO(3) doesn’t have representations with s half-integer.
None of this is new snd is described in all textbooks on group theory and reviewed in all textbooks on quantum mechanics.
And the possible values of the eigenvalue of one of the generators, that can be diagonalized, are spaced by 1, not 1/2. So for a spin-1/2 the possible values are -hbar/2 and +hbar/2.
For a spin-1 the Stern-Gerlach would show three groups, corresponding to the values of -hbar, 0,+hbar.
Dear colleague,
QM spin is a representation of SO(2) related with the "real" space-time symmetry SO(4) as it is elaborated by Penrose Twistor theory. Unitary groups were not involved.
My point is why just two different states and that numbers like +,- 1/2 are settings and the only quantum aspect here is their difference 1. So the only reasonable argument should be as it is well known that spin is a SO(2) representation. But then one may ask why just SO(2) and not higher group representation in view of the fact that space-time is assumed to be 4D and its Lorentz-group in SRT is SO(4).
My remarks concern the fact that there are several known aspects in QM that confirm a two degrees of freedom structure for QM and QFT. Among them the square integrability of Hilbert space functions as the state functions of quantum theory at all. Then square integrability is geometrically a 2D surface integrability as I proved in my works available also in RG.
See: "Basic equivalence between classical and quantum physics including probability aspects." and "Can we see the shape of probability:
On the two-dimensional topology of probability".
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