I am trying to obtain the polchinski's equation 4.3.16
which is following
Q2B=12{QB,QB}=−12gKIJgMKLcIcJcLbM=0
Where
QB=CI(GmI+12GgI)
and CI, bJ are anticommuting(ghosts)
and
[GI,GJ]=igKIJGK,
GgI=−igKIJCJbK are ghost parts and GmI are matter part and they satisfy above commutation relations
What I have done are
{QB,QB}={CI(GmI+12GgI),CJ(GmJ+12GgJ)}={CIGmI,CJGmJ}+12{CIGmI,CJGgJ}+12{CIGgI,CJGmJ}+14{CIGgI,CJGgJ}=CICJ[GmI,GmJ]+12CICJ[GmI,GgJ]+12CICJ[GgI,GmJ]+14CICJ[GgI,GgJ]=CICJ[GmI,GmJ]+14CICJ[GgI,GgJ]=CICJigKIJGmK+14CICJigKIJGgK=CICJigKIJGmK+14CICJgKIJgMKLCLbM
compare with the textbook
{QB,QB}=−gKIJgMKLcIcJcLbM
My calculation is something wrong. How can I fix it?
This post imported from StackExchange Physics at 2014-08-26 10:53 (UCT), posted by SE-user phy_math