From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are
$[D,P_{\mu}] = iP_{\mu}$ and $[D,K_{\mu}] = -iK_{\mu}$,
so that $P_{\mu}$ and $K_{\mu}$ are raising and lowering operators, respectively, for the operator $D$.
This is analogous to the operators $\hat a$ and $\hat a^{\dagger}$ being creation and annihilation operators for $\hat H$ when discussing the energy spectra of the $n$ dimensional harmonic oscillator.
My question is, while $\hat a$ and $\hat a^{\dagger}$ raise and lower the energy by one unit $( \pm \hbar \omega)$ for each application of the operator onto eigenstates of $\hat H$, what is being raised and lowered when we apply $P_{\mu}$ and $K_{\mu}$ onto the eigenvectors of $D$?
Secondly, what exactly do we mean by the eigenvectors of $D$? Are they fields in space-time?
Using the notation of Di Francesco in his book 'Conformal Field Theory', the fields transform under a dilation like $F(\Phi(x)) = \lambda^{-\Delta}\Phi(x)$, where $\lambda$ is the scale of the coordinates and $\Delta$ is the scaling dimension of the fields.
Can I write $F(\Phi(x)) = D\Phi(x) = \lambda^{-\Delta}\Phi(x)$ to make the eigenvalue equation manifest?
Thanks for clarity.
Best Answer
The commutation relations $$ [D,P_{\mu}] = +i P_{\mu} , \qquad [D,K_{\mu}] = -i K_{\mu} $$ show that $P_{\mu}$ and $K_{\mu}$ raise and lower the conformal dimension of a state. In other words, if you have a state $|\phi\rangle$ of dimensions $\Delta$, so that $D\, |\phi\rangle = i\Delta |\phi\rangle$, then $$ D \, P_{\mu} \, |\phi\rangle = [D,P_{\mu}]\, |\phi\rangle + P_{\mu}\,D\,|\phi\rangle = i(\Delta + 1) \, P_{\mu} \, |\phi\rangle . \tag{1} $$ While the generators of the conformal group act on the fields (after all they generate a symmetry), I find it easier to think about the action on a state, like the $|\phi\rangle>$ above. According to the state operator correspondence (see for example this question), such states can be obtained by acting by local operators on the vacuum. $P_{\mu}$ is the generator of translations, and hence acts on a local operator $\phi(x)$ as a derivative $$ [ P_{\mu} , \phi(x) ] = i \partial_{\mu} \phi(x) . $$ Equation $(1)$ then tells you that that the derivative carries conformal dimension $1$.
The notation you propose at the end of your question seems a bit dangerous. $D$ is frequently used for denoting the infinitesimal generator of dilatations, but the function $F$ gives the action of the corresponding group element. That being said, you are of course free to introduce whatever notation you find convenient, as long as you make it clear -- both to yourself and to others -- what you mean.