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Consider a single particle with Hilbert space $L^2(\mathbb{R}^3) \otimes V_\ell$ where $V$ is a vector space of dimension $2\ell + 1$ equipped with a projective unitary representation of $SO(3)$. Physically we say that the particle has spin $\ell$, for example for bosons $\ell$ is an integer and for fermions $\ell$ is an odd multiple of $1/2$.

The above math is fine but I am confused on the physics. For example, suppose we did not know that the spin of an electron is $\ell = 1/2$. How would we determine that the dimension of the vector space used to represent an electron must be 2? Why not 4, 6, 8...etc?

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  • $\begingroup$ $2\times 1/2+1 =2$? $\endgroup$
    – ZeroTheHero
    Commented Jul 26, 2023 at 21:47
  • $\begingroup$ @ZeroTheHero Sorry maybe the wording of my question isn't great. Since an electron is a Fermion we know that its spin must be a half-integer. But why not spin 3/2, 5/2...etc? Why must it be $1/2$? In other words, why must we represent SO(3) on a vector space of dimension 2? $\endgroup$
    – CBBAM
    Commented Jul 26, 2023 at 21:52
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    $\begingroup$ Experirments... $\endgroup$
    – Tobias Fünke
    Commented Jul 26, 2023 at 22:01
  • $\begingroup$ @TobiasFünke Is this an entirely experimental result or can it be shown to be a mathematical consequence? $\endgroup$
    – CBBAM
    Commented Jul 26, 2023 at 22:02
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    $\begingroup$ entirely experiment. The Stern-Gerlach (or its one-electron version by Phipps, T.E.; Taylor, J.B. (1927). "The Magnetic Moment of the Hydrogen Atom". Physical Review. 29 (2): 309–320), show the beam splits only $2$ ways. This also makes the spectroscopy data on the number of electrons per atomic state correct. $\endgroup$
    – ZeroTheHero
    Commented Jul 26, 2023 at 22:41

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