Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or anti-unitary) for the transition amplitudes between those states to be invariant under the respective symmetry and should be irreducible in order to describe a basic "building block" (multiplet).
In the case of elementary particles, the irreducible unitary representations of the Poincaré/Lorentz group give a complete classification via the eigenvalues of the two Casimir operators $P^2$ and $W^2$ as quantum numbers: mass $m$ and spin $j$ (integer or half-integer $j$ reps for $m>0$ and continuous $j$ or integer helicity $h$ reps for $m=0$).
In the case of multiparticle systems, the unitary representations of the symmetric/braid group give a complete classification via the state change for particle exchange: complex phase $exp(i\theta)$ (bosons: $\theta=0$, fermions: $\theta=\pi$, anyons: $\theta \in [0,2\pi)$) for 1-dim reps and matrix for higher-dim reps (paraparticles).
- Question 1: Is above statement flawed? If yes, where?
- Question 2: The spin-statistics theorem relates the two classifications. Does that mean that the reps of the Lorentz group are somehow related to the reps of the symmetric group? What have the two groups to do with each other?
- Question 3: The boson and fermion reps are the equivalent of the integer $j$ ($m>0$) or integer $h$ ($m=0$) and half-integer $j$ reps, the anyon reps are the equivalent of the non-quantized $j$ reps in 2D (rotations commute) - but what are the equivalent of the continuous $j$ reps??
Thank you very much for your help!