# Relation between representations/classifications

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!

• What are "the continuous $j$ reps"? Of which group? – ACuriousMind Feb 6 '15 at 14:07
• For the massless case there are two distinct classes of reps for the Poincaré group (or better the little group $ISO(2)$): the discrete helicity $h$ rep and the continuous spin $j$ rep. For a comprehensive review, see for instance: arXiv:1302.1198v2 – quantumorsch Feb 6 '15 at 16:23
• Ah, I don't think there is something corresponding to the continuous spin reps. They are theoretically allowed, but do not (yet) occur in the Standard Model or other QFTs describing actual physics, so the corresponding particles do not have a name other than "continuous spin particles". (Also, please edit your comment into the question - it is kind of relevant that these are little group and not Poincare reps) – ACuriousMind Feb 6 '15 at 16:47
• Well, the classification of particles by Wigner (1939) via unitary reps is done using Poincaré reps as the title "On unitary representations of the inhomogeneous Lorentz group" clearly states. – quantumorsch Feb 6 '15 at 16:57
• You need the Casimir invariant $P^2$ to get the first quantum number $m$ and then you transform to a different frame (rest frame in the massive case) to get the reps of the little group to get the second quantum number $j$ or $h$ to achieve a complete classification of particles. Or did I get this wrong? – quantumorsch Feb 6 '15 at 17:06