# The Haag-Lopuszanski-Sohnius theorem

1. Jul 9, 2006

### Perturbation

Hey, guys. I recently bought Weinberg's QFT Vol. III on Supersymmetry and I'm a bit stuck with part of the proof he gives for the Haag-Lopuszanski-Sohnius theorem in chapter 25.2. He starts off by giving the usual way of classifying representations of the Homo' Lorentz group by a pair of integers (A, B) according to

$$\mathbf{A}=\tfrac{1}{2}\left(\mathbf{J}+i\mathbf{K}\right)$$
$$\mathbf{B}=\tfrac{1}{2}\left(\mathbf{J}-i\mathbf{K}\right)$$

Where J and K are the generators of rotations and boost respectively. This I'm familiar with. Then he introduces a set of (2A+1)(2B+1) fermionic operators $Q_{ab}^{AB}$ (with a=-A...A and b=-B...B) that furnish an (A, B) representation of the Homo' Lorentz group, ok. But what I don't get is the commutation relations he gives for these operators with A and B as above

$$[\mathbf{A}, Q_{ab}^{AB}]=-\sum_{a'}\mathbf{J}^{(A)}_{aa'}Q_{a'b}^{AB}$$.
$$[\mathbf{B}, Q_{ab}^{AB}]=-\sum_{b'}\mathbf{J}^{(B)}_{bb'}Q_{ab'}^{AB}$$

Where $\mathbf{J}^{(j)}$ is the spin j three-vector matrix. The commutation relations make sense intuitively: the commutator of A and Q should be a sum of Q's that belong to the A rep', likewise with the commutator with B. But I don't quite get the introduction of J, does anyone have a proof they can give or link me to? I follow the rest of the proof of the theorem, but these commutation relations are quite important to establish a starting point of the theorem, namely the relation between the Hermitian adjoint of an (A, B) operator and a (B, A) operator. I'd skip over it and just accept it but it's bothering me and I'm not usually the sort to assume important results.

I was thinking I could write $Q_{ab}^{AB}$ as a tensor product of A and B spinor operators and work it through like that, and given that A and B satisfy the usual commutation relations of angular momentum it makes sense that J should pop out at the end, but I'm not sure. Perhaps I'm not looking at it right and the Q's are just adjusted so that they obey said relations...if so I wasted five minutes writing this. All Weinberg says in relation to them is "Moreover the Q's satisfy the following commutation relations [the ones referenced above]", or something like that.

Any help would be appreciated, this damn thing is stopping me from progressing through the topic, something I've been interested in for a while, but haven't had the money to buy a book on.

Cheers, folk

Last edited: Jul 9, 2006
2. Jul 10, 2006

### Perturbation

Nobody...?

3. Jul 10, 2006

Staff Emeritus
Not I, said selfy-welfy.

4. Jul 10, 2006

### Perturbation

Bummer

Perhaps in the "Beyond the Standard Model" forum?

5. Jul 10, 2006

Staff Emeritus
Could be. That's the proper home for supersymmetry anyway. I'm going to move the thread.
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