It is known that to specify a finite-dimensional irreducible representation of the Lorentz group, one needs to specify two half-integers, $(j_1,j_2)$. For instance, the left-handed and right-handed Weyl spinors are non-equivalent representations, despite both having spin half. There is the $(1/2,1/2)$ 4-vector representation, and the $(0,1)$ 2-form representation.
In QFT, in classifying the possible types of particle that can exist, we look for unitary (infinite-dimensional) irreducible representations of the Poincaré group. Here, we need to specify a mass $m$ and a single 'spin' $j$. Fixing the mass to some given number, then, it appears that there are fewer types of particle allowed than the finite-dimensional representations would suggest. For instance, there is only one type of spin half particle, $j = 1/2$. Is this a left-handed Weyl fermion or a right-handed Weyl fermion, or something else?
More generally, how should we interpret the 'particle types' given to us by the infinite-dimensional Poincaré group representations in terms of the 'particle types' given to us by the finite-dimensional Lorentz group representations?
My thoughts: one can show that the Casimirs used in classifying representations of the Lorentz group don't commute with the $P^2$ Poincaré Casimir. This might suggest that when we enlarge our group from Lorentz to Poincaré, we get new transformations that can mix states within the old representations, so that the new irreducible representations must also be enlarged. But I cannot see how a translation (followed by any combination of boosts or rotations) could turn a left-handed Weyl spinor into a right-handed one.