# Spin in Lorentz group and Poincare group

I am currently learning representations of Lorentz group and Poincare group by Harald's Introduction to Supersymmetry Chap.1. I have 2 questions about the definition of spin.

1. Provided that finite dimensional representations of Lorentz group labelled by two half-integers $$(m,n)$$, each of which is related to a $$su(2)$$ subalgebra, why does the spin of the representation is defined by $$(n+m)$$? In angular momentum theory, addition of two $$SU(2)$$ should result in spin with values from $$|n-m|$$ to $$n+m$$, so why there is only one value defined in the case of Lorentz group?

2. The second Casimir operator of Poincare group in the massive case has eigenvalues $$-m^2 s(s+1)$$, where $$m$$ here is the mass of the particle and $$s$$ is also interpreted as spin. I am wondering if this spin $$s$$ can be related to $$(m,n)$$ representation of Lorentz group by $$s=m+n$$? If so, can anyone give some details or some references that I can refer to?

• I think it's a bit confusing (in your second question) to use the same $m$ for two different meanings (mass and half-integer). Mar 21, 2021 at 0:27
• @DescheleSchilder A common complaint in the field, unfortunately.
– rob
Mar 21, 2021 at 1:01
• @DescheleSchilder Sorry, I just follow the notation used in the textbook. I have edited it to make it clear
– ZHC
Mar 21, 2021 at 3:20

I think this is a matter of convention because $$(m+n)$$ will be the highest spin irrep. present. What is certainly true, however, is that the sum is insufficient to specify all the information about how the particle transforms. For example the $$(1,0)$$ and $$(1/2,1/2)$$ representations are certainly not the same, though the sum matches.

It's still interesting, however, to ask about the relationship between these numbers and the Casimir you mention, which for reference is given by the square of the Pauli-Lubanski pseudovector, $$W_\mu=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}J^{\nu\rho}P^\sigma$$ where $$J^{\mu\nu}$$ is the generator of the Lorentz group. The wiki article on this is actually fairly reasonable.

The most important point to note about $$W_\mu$$ is that it generates the little group, who's representations are what actually determine a particle's spin content. As described, for example, in the first volume of Weinberg's QFT series, the Wigner classification of particles essentially comes down to working out the representations of the little group. Massive and massless particles behave differently because the little group changes when the mass is zero, but I'll point out that the expression OP has quoted, $$-m^2s(s+1)$$ is for the case of a massive particle. Furthermore, we should keep in mind here that the $$m$$ in this expression is the mass of the particle, not the half-integer $$m$$ used in specifying the particle's Poincare representation...this $$m$$ comes out of squaring $$W_\mu$$, which produces something like $$\sim P^\mu P_\mu \boldsymbol{J}^2$$ so the square of the 4-momentum produces the particle's mass squared while the rest of the expression becomes the square of the generator of spatial rotations.

Now, to see the relationship between the sum of the pair of $$SU(2)$$'s and spin as defined by the little group, recall that the map between the generators $$\boldsymbol A$$ and $$\boldsymbol B$$ of the $$SU(2)\times SU(2)$$ and the Lorentz group generators, which I will now write in terms of the generators of boosts $$\boldsymbol K$$ and rotations $$\boldsymbol J$$, is given by $$\boldsymbol A= \frac{1}{2}(\boldsymbol J+i\boldsymbol K)$$ and $$\boldsymbol B= \frac{1}{2}(\boldsymbol J-i\boldsymbol K)$$.

The next statement I would like to make is: this implies $$\boldsymbol J=\boldsymbol A+\boldsymbol B$$ and hence $$\boldsymbol J^2 = (\boldsymbol A+\boldsymbol B)^2$$, and hence the spin content as defined by the little group and spin content as defined by this pair of $$SU(2)$$'s must be the same (the Casimir's are the same after all).

This statement is really the one that I wanted to make here since, as far as I'm aware, this is the precise relationship between the $$SU(2)$$ representations and the little group representations: the little group's Casimir matches that of the total $$SU(2)\times SU(2)$$ spin (up to a factor of the particles mass, and only in the case of massive particles, though I imagine a similar statement holds for massless particles).

The only lose end I think I've really left here is that I've written $$(\boldsymbol A+\boldsymbol B)^2$$, called it the total spin squared and then called it a day, glossing over the fact that the total spin operator should properly look like $$(A\otimes 1+1\otimes B)$$, as we know from the usual story of spin addition in quantum mechanics. The sum of the vectors $$\boldsymbol A+\boldsymbol B$$ doesn't quite look this way as written. Essentially, to see that it takes the usual form, we would need to play some games about changing how we write our indices. If you're studying SUSY as the OP indicates, this is similar to exchanging Lorentz indices for spinor indices. The details of this would, I think distract from the main point I have been trying to make though, so I will instead refer the reader back to chapter 5.6 in Weinberg's book.

• Thank you very much for your answer! May I ask if there is any restriction to the values that $J$ can take? Since $A$ and $B$ in your answer are related by Hermitian conjudgate (or parity). To be more precise, if $A^2$ and $B^2$ have eigenvalues $a(a+1)$ and $b(b+1)$, can $j$ still take all possible values from $|a-b|$ to $a+b$?
– ZHC
Mar 21, 2021 at 3:34
• @ZHC this $J$ generates the $SO(3)$ subgroup of the Lorentz group as I mentioned. $SO(3)$ is isomorphic to $SU(2)$ and therefore all the representation theory of spin you are used to applies to it. Mar 21, 2021 at 5:18