I understand that spin is derived from combining special relativity and quantum mechanics, but I don't know the details of how.

I know that the Dirac Equation takes spin into account, but only works on spin 1/2 particles. I did find though that the pairs of matrices are anti commutative, meaning that $a*b=-b*a$, and the square of each matrix equals the identity matrix.

I also found the Spin Matrices for different spin particles, and tried seeing if the pairs of spin matrices for spin 1 particles anti commute, and it seems they don't, nor are the squares of each spin matrix of a spin 1 particle an identity matrix, so it seems that 2 of the requirements for the matrices of spin 1/2 particles do not generalize to matrices of particles of other spins.

It looks like in 3 spatial dimensions the size of the spin matrices is $(|spin|+1/2)*2$ by $(|spin|+1/2)*2$

and in the differential equations analogous to the Dirac Equation it seems that the size of each matrix is $((|spin|+1/2)*2)^2$ by $((|spin|+1/2)*2)^2$

so it looks as if the reason particles can only have spins that are whole number multiples of 1/2 in 3 spatial dimensions might be because a matrix can only have a size that is a whole number by a whole number, but then I'm not sure why this wouldn't be the case in 2 spatial dimensions as in 2 spatial dimensions there are anyons can have any spin.

I'm not sure if the spin matrices are a postulate or if they are derived.

Also I understand that in higher dimensions particles don't have a single spin state, but multiple spin states or analogs of spin states, and I'm not sure what the matrix analog would look like for a particle with multiple spin states.

So how is spin derived mathematically, and are the matrices related to spin a postulate derived or are they a postulate?

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    $\begingroup$ This post (v1) seems very broad. It asks about higher dimensions, spinor representation theory, spin-statistics, anyons & possible higher spin at the same time. $\endgroup$
    – Qmechanic
    Dec 1, 2021 at 9:20