Representation of dilations

Question

I am having some trouble getting some signs right on the representation of the dilation operator on a field. Let us follow the conventions of Joshua D. Qualls https://arxiv.org/abs/1511.04074. According to equation (2.36), we have $$e^{ix\cdot P}De^{-ix\cdot P}=D+x\cdot P.\tag{2.36}$$ Recall that $$P_\mu=-i\partial_\mu.\tag{2.34}$$ We then have for a field of scaling dimension $\Delta$, i.e. $$D\phi(0)=-i\Delta\phi(0),\tag{1}$$ $$D\phi(x)=De^{ix\cdot P}\phi(0)=e^{ix\cdot P}e^{-ix\cdot P}De^{ix\cdot P}\phi(0)=e^{ix\cdot P}(D-x\cdot P)\phi(0)=e^{ix\cdot P}(-i\Delta-x\cdot P)\phi(0)=(-i\Delta-x\cdot P)\phi(x).\tag{2}$$ We conclude that $$D=-i\Delta-x\cdot P=-i\Delta+ix\cdot\partial.\tag{3}$$ However, according (2.37), I go the sign of the momentum part wrong. Can somebody help me understand where my mistake is?

Answer 1

The problem is in the first equality of eq. (2): How is the momentum (2.34) supposed to act on the missing $x$-dependence in $\phi(0)$? Instead use eq. (2.36) to deduce that $$ [D,P_{\mu}] ~=~i P_{\mu}. \tag{2.27}$$ Conclude from eq. (2.27) that $$D~=~x^{\mu}P_{\mu}~\stackrel{(2.34)}{=}~-i x^{\mu}\partial_{\mu}$$ up to a central element.

Answer 2

As explained in Sign error in representation of angular momentum on fields, the problem is actually that the commutations relations should be $$e^{ix\cdot P}De^{-ix\cdot P}=D-x\cdot P,$$ because, the generators on the transformation on fields satisfy the opposite commutators than those acting on spacetime.