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Srednicki (ch90)

  1. Mar 21, 2012 #1
    Does anyone know exactly how Srednicki identitifies the electromagnetic gauge field with his [itex]l,r,b [/itex] fields. I know he is trying to match covariant derivatives, i.e.

    [itex] D_{\mu} p=(\partial_{\mu}-il_{\mu})p[/itex] with [itex]D_{\mu} p=(\partial_{\mu}-ieA_{\mu})p[/itex]

    and that he has set [itex]l_{\mu}=l_{\mu}^a T^a+b_{\mu} [/itex]

    and also match [itex] D_{\mu} n=(\partial_{\mu}-ir_{\mu})n[/itex] with [itex]D_{\mu} n=(\partial_{\mu})n[/itex]

    and that he has set [itex]r_{\mu}=r_{\mu}^a T^a+b_{\mu} [/itex]

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

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

    from these.

    I guess it must be quite simple, and I thought maybe I should just expand the [itex]T^{a} [/itex] gens in terms of Pauli then solve simultaneously, but this didn't quite seem to work out..
     
  2. jcsd
  3. Mar 23, 2012 #2
    I seem to be finding:

    [tex]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] [/tex]

    and

    [tex]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] [/tex]

    when what I want to demand consistency with is

    [tex] D_{\mu}p=\partial_{\mu}p-ieA_{\mu}p [/tex]
    and
    [tex] D_{\mu}n=\partial n [/tex]

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

    Anyone tell me where I am going wrong?
     
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