Saturday, August 19, 2017

homework and exercises - SU(2) Yang-Mills EOM


I'm having trouble with some indices on my yang-mills lagrangian. I have a gauge group SU(2) and a field strength tensor Fiab=aAibbAia+ϵijkAjaAkb and a lagrangian L=14FiabFabi I have at the end of the exercise that the correct EOM are aFiab+ϵijkAajFkab=0 I am getting the first term on the left, but the second term I am getting something slightly different, instead, when I hit the lagrangian with L/Aia I get a term ϵijkAkbFabi which leaves a free j index. Now I know summed indices don't matter and can change letters freely, but order and contraction does matter, now I'm not sure how to make it come out with matching indices. When I do aL/(aAib) I come out with the typical aFabi but my free index there is i, not j, and I can't get them to match for the life of me.



Answer




Without seeing the details of how you're taking the derivative, I can't be sure, but my first thought is to wonder whether you're making multiple use of indices. For example, suppose you take the derivative with respect to Aia; you'll get


LAia=14Aia[FiabFabi]=14FiabAiaFabi14FabiAiaFiab


In the process of evaluating FiabAia, the indices in the numerator don't contract with the ones in the denominator, but it's going to be easy to miss that if you don't have the colors to distinguish them.


I'd suggest that when you label the field you're differentiating with respect to, use indices that don't show up anywhere in the Lagrangian, like this:


LAnc=14FiabAncFabi14FabiAncFiab


Then there's no danger of accidentally contracting the wrong indices. When you get down to the level of differentiating individual fields, you'll wind up with some Kronecker deltas,


AiaAnc=δcaδin


If you're differentiating a contravariant field, you can use the metric to raise and lower indices as needed,


AaiAnc=AncδimgadAmd=gacδin


δim is the metric for SU(2) configuration space, if I remember correctly.



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