Maybe this does it:
Translation:Pμ=−i∂μRotation:Mμν=i(xμ∂ν−xν∂μ)Dilation:D=ixμ∂μSpecial Conformal:Cμ=−i(→x⋅→x−2xμ→x⋅∂)
Then the commutation relation gives:
[D,Cμ]=−iCμ
so Cμ acts as raising and lowering operators for the eigenvectors of the dilation operator D. That is, suppose:
D|d⟩=d|d⟩
By the commutation relation:
DCμ−CμD=−iCμ
so
DCμ|d⟩=(CμD−iCμ)|d⟩
and
D(Cμ|d⟩)=(d−i)(Cμ|d⟩)
But given the dilational eigenvectors, it's possible to define the raising and lowering operators from them alone. And so that defines the Cμ.
P.S. I cribbed this from:
http://web.mit.edu/~mcgreevy/www/fall08/handouts/lecture09.pdf
This post imported from StackExchange Physics at 2014-07-28 11:17 (UCT), posted by SE-user Carl Brannen