# Mathematical Rationale for Fermion and Boson Spin Representations

I am beginning with the statement that: All physical states occur as one dimensional representations of $\mathfrak{S}_n$; they are either bosonic or fermionic. Where a fermionic state of n identical particles is defined as one that lies in the embedding of $\bigwedge^n\mathcal{H}$ in $\mathcal{H}^{\otimes n}$, and a bosonic state is one that lies in $\text{Sym}^n\mathcal{H}$, so basically they are defined to be those things with anti-symmetric or symmetric wavefunctions.

I was wondering if there was a mathematical way to rationalize modeling fermion spins with the half-integer reps of $SU(2)$ and bosons with integer reps using these defintions. Are these just experimental facts that we must simply state about fermions and bosons, or are there ways to derive them?

• spinor irreps vs true irreps under rotations by $2\pi$. BTW this only holds in dim=3; in 2d anyons are possible. – ZeroTheHero Apr 11 '17 at 16:16
• Do you mean we want fermions to have the spinor reps because we want them to gain a minus sign after a rotation by $2\pi$? If so, is there a rationale for that too? Or is this something completely different. – zzz Apr 11 '17 at 16:27
• basically yes, we want fermions to pick up a - sign under rotations by $2\pi$. This is tied to the spin statistics theorem en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem – ZeroTheHero Apr 11 '17 at 16:29