Srednicki Ch90: How to Identify Electromagnetic Gauge Field

[LAHLH]

Does anyone know exactly how Srednicki identitifies the electromagnetic gauge field with his $l,r,b$ fields. I know he is trying to match covariant derivatives, i.e.

$D_{\mu} p=(\partial_{\mu}-il_{\mu})p$ with $D_{\mu} p=(\partial_{\mu}-ieA_{\mu})p$

and that he has set $l_{\mu}=l_{\mu}^a T^a+b_{\mu}$

and also match $D_{\mu} n=(\partial_{\mu}-ir_{\mu})n$ with $D_{\mu} n=(\partial_{\mu})n$

and that he has set $r_{\mu}=r_{\mu}^a T^a+b_{\mu}$

But I don't seem to be able to work out the fine print of arriving at (90.20):

$$eA_{\mu}=l^3_{\mu}+r_{\mu}^3+1/2b_{\mu}$$

from these.

I guess it must be quite simple, and I thought maybe I should just expand the $T^{a}$ gens in terms of Pauli then solve simultaneously, but this didn't quite seem to work out..

 

[LAHLH]

I seem to be finding:

$$D_{\mu} p=\partial_{\mu} p-\frac{i}{2}\left[(l_{\mu}^2+r_{\mu}^1)n-i(l_{\mu}^2+r_{\mu}^2)n+(l_{\mu}^3+r_{\mu}^3+2b_{\mu}) p\right]$$

and

$$D_{\mu} n=\partial_{\mu} n-\frac{i}{2}\left[(l_{\mu}^2+r_{\mu}^1)p+i(l_{\mu}^2+r_{\mu}^2)p+(l_{\mu}^3+r_{\mu}^3+2b_{\mu}) n\right]$$

when what I want to demand consistency with is

$$D_{\mu}p=\partial_{\mu}p-ieA_{\mu}p$$
and
$$D_{\mu}n=\partial n$$

and Srednicki says to do this I need to demand $eA_{\mu}=l^3_{\mu}+r_{\mu}^3+1/2b_{\mu}$

Anyone tell me where I am going wrong?