Roots of Lie Algebra (Jones book)

[LAHLH]

Hi,

If anyone has a copy of this book, I'm just struggling to understand a point in the chapter about Lie Groups. In particular the section on qunatization of the roots, on p177-178, he generates the diagrams in fig 9.2 from those of 9.1. I thought I understand this, but I don't understand how he gets from (c) in 9.1 to (c) in 9.2, I mean we have roots strings of length 2 and 3 for $$\alpha$$ strings through $$\beta$$ and $$\beta$$ strings through $$\alpha$$, then the two original roots themselves. So why don't we have 2+2+3=7 roots in total? the diagram (c) in fig 9.2 appears to show 8 in total?

If we look at going from (b) 9.1 to 9.2, this logic seems to work, we start with 2, each root string is length two, giving a total of 2+2+2=6 roots, sim for (a) going from 9.1 to 9.2, 2 originally, each root string of length 1, giving 2+1+1=4 roots in total.

Not sure if I'm understanding what he is doing correctly, I don't really understand why he is allowed to set p=0 either, or equivalently assume $$[E_{\beta},E_{\alpha}]=0$$

Thanks for any help

 

[metroplex021]

Jones misses a lot of steps (and I *think* there are also a couple of mistakes in there). A better source that is just as approachable (though also quite concise) is Georgi, 'Lie algebras and particle physics'. Your best bet though is probably Cornwell, Group Theory in Physics. This has a nice chapter (in volume 2 of the 3-volume series; otherwise toward the end of the shorter version 'Group theory in physics: an introduction') on su(3) algebras (which if I remember is the example Jones uses). You should get on better with either of them.

 

[LAHLH]

Thanks, I will give those a go.