# Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am talking about:

Now let us use the fact that $\Phi_a(x) = U_P(-x)\Phi_a(0) = e^{iP_\mu x^\mu}\Phi_a(0)$. We can write the action of the generators as follows:

\begin{align*} G\Phi_a(x) &= Ge^{iP_\mu x^\mu}\Phi_a(0) \\ &= e^{iP_\mu x^\mu}e^{-iP_\mu x^\mu}Ge^{iP_\mu x^\mu}\Phi_a(0) \\ &= e^{iP_\mu x^\mu}\tilde{G}\Phi_a(0) \end{align*}

where have defined $\tilde{G} = e^{-iP_\mu x^\mu}Ge^{iP_\mu x^\mu}$. We can now calculate the $\tilde{G}$ using the Hausdorff formula:

$e^{-A}Be^{A} = B+[B,A] + \frac{1}{2!}[[B,A],A] + \frac{1}{3!}[[[B,A],A],A] + \dots$

We obtain:

\begin{align*} \tilde{\J}_{\mu\nu} &= e^{-ix^\rho P_\rho}\J_{\mu\nu}e^{ix^\rho P_\rho} = \J_{\mu\nu} + ix^\rho[\J_{\mu\nu}, P_\rho] \\ &= \J_{\mu\nu} + x^\rho(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \\ &= \J_{\mu\nu} + x_\mu P_\nu - x_\nu P_\mu \\ &= \J_{\mu\nu} - i(x_\mu \partial_\nu - x_\nu \partial_\mu) \\ \\ \tilde{D} &= e^{-ix^\rho P_\rho}De^{ix^\rho P_\rho} = D + ix^\rho[D,P_\rho] = D - x^\rho P_\rho = D + ix^\rho\partial_\rho \\ \\ \tilde{K}_\mu &= e^{-ix^\rho P_\rho}K_\mu e^{ix^\rho P_\rho} = K_\mu + ix^\rho[K_\mu,P_\rho] - \frac{1}{2}x^\rho x^\nu[[K_\mu,P_\rho], P_\nu] \\ &= K_\mu - 2x^\rho(\eta_{\mu\rho}D - \mathcal{J}_{\mu\rho}) + x^\rho x^\nu(-\eta_{\mu\rho}P_\nu + (\eta_{\nu\mu}P_\rho - \eta_{\nu\rho}P_\mu)) \\ &= K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} + (2 x_\mu x^\nu P_\nu - x_\rho x^\rho P_\mu) \\ &= K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu) \end{align*}

The action of the generators are therefore given by:

\begin{align*} {\J}_{\mu\nu}\Phi_a(x) &= e^{iP_\mu x^\mu}(\J_{\mu\nu} - i(x_\mu \partial_\nu - x_\nu \partial_\mu))\Phi_a(0) \\ &= (\Sigma_{\mu\nu})_a^{\;\;b}\Phi_b(x) + i(x_\mu \partial_\nu - x_\nu \partial_\mu))\Phi_a(x) \\ D\Phi_a(x) &= e^{iP_\mu x^\mu}(D + ix^\rho\partial_\rho)\Phi_a(0) \\ &= D_a^{\;\;b}\Phi_b(x) - ix^\rho\partial_\rho\Phi_a(x) \\ K_\mu &= e^{iP_\mu x^\mu}(K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu))\Phi_a(0) \\ &= (K_\mu)_a^{\;\;b}\Phi_b(x) + 2x_\mu D_a^{\;\;b}\Phi_b(x) - 2x^\rho(\mathcal{\Sigma}_{\mu\rho})_a^{\;\;b}\Phi_b(x) - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu)\Phi_a(x) \end{align*}

As you can see, every time from the first to the second line there is a minus sign appearing precisely where there is an x. This is what happens in Salam's paper Finite component field representations of the conformal group and it has to happen in my case for it to be consistent with what I wrote before. BUT I don't understand why this change of sign appears. $e^{iP}$ and $P$ commute, so why is this happening.