Spin statistics

Question

1. I have a very intrinsic question about quantum field theory and even more general, why in 3+1-dimensional spacetime, we have only two statistics for particles to obey? Therefore why we have only two commutator or anticommutator relationships.

2. How we can test something like that in other dimensions and extract the relationships for every dimension of spacetime?

Answer 1

Expanding @ACuriousMind comment as an answer:

By the Spin-Statistics Theorem the symmetry group of Lorentz-Invariant QFT is the lorentz group $SO^+(1,3)$ of which $SL_2(C)$ is the universal covering group. The irreducible representations of the symmetry group of the theory, are related naturaly to the way the wavefunction changes under exchange of the positions of the particles that make up the state $\psi$ (Statistics).

As such the connection between the symmetry group (and its representations) and the wave-function statistics is made.

The rest is based on the irreducible representations of the $SL_2(C)$ group (which includes the $SO^+(1,3)$ group)

A preliminary representation (matrix Lie Algebra) is given for $SU(2)$ (the cover of $SO(3)$) also in wikipedia.

For anyons in 2D theories:

Just as the fermion and boson wavefunctions in a three-dimensional space are just 1-dimensional representations of the permutation group ($S_N$ of $N$ indistinguishable particles), the anyonic wavefunctions in a two-dimensional space are just $1$-dimensional representations of the braid group ($B_N$ of $N$ indistinguishable particles)