Spin statistics

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?

• The title of your question implies you are familiar with the Spin-statistics theorem. Since, by pure representation theory, there are only half-integer and integer spins in 3D and 4D (for 2D, anyons are an exception), this means that there are only the two statistics the spin-statistics theorem uses (given that our QFT fulfills the assumption of the theorem). It's not about (anti-)commutators, it's about representation theory. Nov 7, 2014 at 20:46
• Related: physics.stackexchange.com/q/27595/2451 and links therein. Nov 7, 2014 at 22:12

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)

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)