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Perturbation
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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
[tex]\mathbf{A}=\tfrac{1}{2}\left(\mathbf{J}+i\mathbf{K}\right)[/tex]
[tex]\mathbf{B}=\tfrac{1}{2}\left(\mathbf{J}-i\mathbf{K}\right)[/tex]
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 [itex]Q_{ab}^{AB}[/itex] (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
[tex][\mathbf{A}, Q_{ab}^{AB}]=-\sum_{a'}\mathbf{J}^{(A)}_{aa'}Q_{a'b}^{AB}[/tex].
[tex][\mathbf{B}, Q_{ab}^{AB}]=-\sum_{b'}\mathbf{J}^{(B)}_{bb'}Q_{ab'}^{AB}[/tex]
Where [itex]\mathbf{J}^{(j)}[/itex] 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 [itex]Q_{ab}^{AB}[/itex] 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
[tex]\mathbf{A}=\tfrac{1}{2}\left(\mathbf{J}+i\mathbf{K}\right)[/tex]
[tex]\mathbf{B}=\tfrac{1}{2}\left(\mathbf{J}-i\mathbf{K}\right)[/tex]
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 [itex]Q_{ab}^{AB}[/itex] (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
[tex][\mathbf{A}, Q_{ab}^{AB}]=-\sum_{a'}\mathbf{J}^{(A)}_{aa'}Q_{a'b}^{AB}[/tex].
[tex][\mathbf{B}, Q_{ab}^{AB}]=-\sum_{b'}\mathbf{J}^{(B)}_{bb'}Q_{ab'}^{AB}[/tex]
Where [itex]\mathbf{J}^{(j)}[/itex] 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 [itex]Q_{ab}^{AB}[/itex] 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
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