In case of the gauge-fixed Faddeev-Popov Lagrangian:
L=−14FμνaFμνa+ˉψ(iγμDμ−m)ψ−ξ2BaBa+Ba∂μAμa+ˉca(−∂μDμac)cc
(for example in Peskin and Schrder equation 16.44)
If you expand the last term (for the ghost fields) you get:
ˉca(−∂μDμac)cc=−ˉca∂2ca−gfabcˉca(∂μAμb)cc−gfabcˉcaAμb∂μcc
And so, the Lagrangian has a term proportional to the second derivative of ca.
In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints?
I found the following equations of motion so far:
DβdcFβσc=−gˉψγσtdψ+∂σBd+gfdac(∂σˉca)cc=0
∂σˉψα,iiγσ−∑β∑jˉψβ,j(gAμaγμjitaβα−mδjiδβα)=0
(iγμDμ−m)ψ=0
Bb=1ξ∂μAμb
∂μ(Dμdccc)=0
fabd(∂σˉca)Aσb=0
But it is the last equation that I suspect is false (I saw the equation Dμad∂μˉcd=0 in some exercise sheet and I also saw the equation Dμad∂μBd=igfdbc(∂μˉcb)Dμdccc which I don't understand how they were derived.)
Any help about what are the proper equations of motions and how to get them would be much appreciated.
This post imported from StackExchange Physics at 2014-07-03 18:16 (UCT), posted by SE-user PPR