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Every irreducible massive unitary representation of the Poincaré group is specified by a mass and a non-negative half integer spin. Every massless irreducible unitary representation of the Poincaré group is specified by a half-integer helicity. This was proven by Wigner.

Every finite dimensional irreducible representation of the Poincaré group is given by two non-negative half integers $(j_1, j_2)$.

We care about unitary representations of the Poincaré group when talking about particle states, and we we care about finite dimensional representations of the Lorentz group when talking about field operators and constructing Lagrangians.

I know that there is no 1-1 correspondence between the two. For example, specifying $(j_1, j_2)$ doesn't tell you what the mass of particles in your spectrum is. However, there is still SOME relationship between the two.

For example, take the Dirac representation $(\tfrac{1}{2}, 0) \oplus (0, \tfrac{1}{2})$. If there is no mass term in the Lagrangian, then there will be four particles in the spectrum: massless right and left handed Weyl particles/anti particles, each with a helicity $h = \pm \tfrac{1}{2}$. If there is a mass term, then there will only be two independent particles, the massive spin $\tfrac{1}{2}$ particles and anti-particles.

So clearly there is some relationship between the $(j_1, j_2)$ representation of the Lorentz group and the corresponding unitary representation of the Poincaré group. But that relationship is confusing, and I do not understand how it works in general.

Lets take the $(1, \tfrac{1}{2})$ representation for example. What kind of particles does this correspond to, anyway? I constructed the representation and then looked at it when restricted to the $SO(3)$ subgroup. Once restricted, the $(1, \tfrac{1}{2})$ broke up into $\tfrac{3}{2} \oplus \tfrac{1}{2}$. Lets imagine that these particles are massless. Does it correspond to six massless particles, with helicities $\tfrac{3}{2}, \tfrac{1}{2}, -\tfrac{1}{2}, -\tfrac{3}{2}, \tfrac{1}{2}, -\tfrac{1}{2}$? Can these particles be made massive, by some sort of generalization of the Majorana procedure? Would it then correspond to only two massive particles, one spin $\tfrac{3}{2}$ and one spin $\tfrac{1}{2}$?

It seems very strange to me that you can have some irreducible, non-trivial way for the rotations to interact with the boosts when it comes to the Lorentz group, but this is not the case when it comes the Poincaré group.

I suppose my general question is what are the possible particle representations that can be born from a $(j_1, j_2)$ field representation?

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    $\begingroup$ Weinberg, v1, p232. $\endgroup$ – Cosmas Zachos Nov 5 '18 at 16:07
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    $\begingroup$ A field with Lorentz labels $(j_1,j_2)$ creates particles with Poincaré label $s$ only if $s\in j_1+j_2,j_1+j_2-1,\dots,|j_1-j_2|$. (More succinctly, only if $s\in j_1\otimes j_2$). See Can Poincare representations be embedded in non-standard Lorentz representations?. $\endgroup$ – AccidentalFourierTransform Nov 5 '18 at 16:57
  • $\begingroup$ How does that work in the massless case? Not every spin state is represented by a helicity, for example how photons have no $h = 0$ helicity? Would a "massless spin 3/2" particle only have helicities $h = \pm 3/2$? $\endgroup$ – user1379857 Nov 5 '18 at 17:01
  • $\begingroup$ But I mean, for the spin $3/2$ massless case, do you get particles with helicities $\pm 3/2$, or also $\pm 1/2$ as well? $\endgroup$ – user1379857 Nov 5 '18 at 17:19
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I think that my question wasn't actually well defined. For starters, when you restrict to the $SO(3)$ subgroup of the $(j_1, j_2)$ rep. of the Lorentz group, you get the $j_1 \otimes j_2$ rep of $SO(3)$, which as usual decomposes as $$ j_1 \otimes j_2 = |j_1 + j_2 | \oplus \ldots \oplus |j_1 - j_2|. $$

So your particles reps must be contained in those reps.

However, some of those reps will not be physical particles. For example, the $(\frac{1}{2}, \frac{1}{2})$ rep becomes $0 \oplus 1$, a spin $0$ particle and spin $1$ particle. However, the spin $0$ particle will be "killed off" by the $\partial_\mu A^\mu$ equation derived from the massive Proca Lagrangian. So the spin $0$ particle doesn't actually exist.

The point is that if you want to know what physical particles there are, it is not enough to specify $(j_1, j_2)$ rep of the Lorentz group. You also need to know the exact form of your Lagrangian as well. I still don't know how to find desirable Lagrangians in general, though.

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