Dilation operator in CFT viewed as 'hamiltonian'?

Question

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

Comments

2D CFT or general CFT? Also, are $P_\mu$,$D$ and $K_\mu$ what one would usually call $L_{-1},L_0,L_1$ in the Virasoro algebra?

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

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Hi ACuriousMind, hmm, I am yet to study 2D CFT (but I know it is a special dimension as far as CFT's go) or the Virasoro algebra. I am using $P_{\mu}, D$ and $K_{\mu}$ to mean, respectively, the translation, dilation and special conformal generators of the infinitesimal transformations.

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

Answer 1

The global conformal group in 2D is $SL(2,\mathbb{C})$ and is generated by the subset of Virasoro generators $(L_0,L_{\pm1}, \bar{L}_0, \bar{L}_{\pm1})$. In particular, the generator $D=L_0+\bar{L}_0$. Using the operator-state correspondence, one identifies the eigenvalue of $D$ on a state with the scaling dimension of the corresponding operator.