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  Dilation operator in CFT viewed as 'hamiltonian'?

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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,Pμ]=iPμ and [D,Kμ]=iKμ, so that Pμ and Kμ are raising and lowering operators, respectively, for the operator D. This is analogous to the operators ˆa and ˆa being creation and annihilation operators for ˆH when discussing the energy spectra of the n dimensional harmonic oscillator.

My question is, while ˆa and ˆa raise and lower the energy by one unit (±ω) for each application of the operator onto eigenstates of ˆH, what is being raised and lowered when we apply Pμ and Kμ 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(Φ(x))=λΔΦ(x), where λ is the scale of the coordinates and Δ is the scaling dimension of the fields. Can I write F(Φ(x))=DΦ(x)=λΔΦ(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

asked Jul 26, 2014 in Theoretical Physics by CAF (100 points) [ revision history ]
edited Jul 27, 2014 by Arnold Neumaier
2D CFT or general CFT? Also, are Pμ,D and Kμ what one would usually call L1,L0,L1 in the Virasoro algebra?

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

1 Answer

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The global conformal group in 2D is SL(2,C) and is generated by the subset of Virasoro generators (L0,L±1,ˉL0,ˉL±1). In particular, the generator D=L0+ˉL0. Using the operator-state correspondence, one identifies the eigenvalue of D on a state with the scaling dimension of the corresponding operator.

answered Jul 29, 2014 by suresh (1,545 points) [ revision history ]

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