From the commutation relations for the conformal lie algebra, we may infer that the dilation operator plays the same role in CFTs as the Hamiltonian in quantum mechanics. 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.

This post imported from StackExchange Physics at 2014-07-26 20:08 (UCT), posted by SE-user CAF