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As I know, the fundamental concept of QFT is Renormalization Group and RG flow. It is defined by making 2 steps:

  1. We introduce cutting-off and then integrating over "fast" fields $\widetilde{\phi}$, where $\phi=\phi_{0}+\widetilde{\phi}$.

  2. We are doing rescaling: $x\to x/L$: $\phi_{0}(x)\to Z^{-1/2}(L)\phi(x)$.

This procedure defines RG flow on the manifold of quasi local actions: $\frac{dA_{l}}{dl}=B\{A_{l}\}$.

In this approach we have such notions as crytical points $A_{*}$, relevant and irrelevant fields, Callan–Symanzik equation etc, and we can apply it, say, to phase transitions.

Also we can introduce stress-energy tensor $T^{\mu\nu}$. And, as far as i know, if we consider scale transformations $x^{\mu} \to x^{\mu}+\epsilon x^{\mu}$ , we can obtain Callan–Symanzik equation, and if the theory have a crytical point: $\beta^{k}(\lambda^{k})=0$, then trace of stress energy tensor $\Theta(x)=T^{\mu}_{\mu}=0$, so our correlation functions have symmetry at scaling transformation.

So the question is: As far as I know, at this point they somehow introduce conformal transformations and Conformal Field Theory. Can you explain, what place in Quantum Field Theory CFT takes? (I mean connection between them, sorry if the question is a little vague or stupid). How it relates to the RG approach exactly? (This point is very important for me). Maybe some good books?

Qmechanic
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xxxxx
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2 Answers2

12

I recently stumbled upon a good comment about this in Jared Kaplan's AdS/CFT notes

  • Any quantum field theory which has hope of having an UV-completion can be viewed as as effective theory at point in the RG flow from an UV complete theory.

  • Field theories at the UV fixed point are conformal.

  • Hence all 'well-defined' field theories are either CFTs or points in the RG flow from one (UV) CFT to another CFT.

So in a sense, in Kaplan's words:

studying the space of CFTs basically amounts to studying the space of all well-defined QFTs

So that's one way of seeing the place of CFTs in general QFTs.

Gabriel Ybarra Marcaida
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zzz
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zzz's answer concerns QFTs as used in particle physics. However, OP also mentions phase transitions, thus statistical field theory, where the relationship between CFT and scale-invariant RG fixed points is more subtle.

Indeed, for local QFTs where unitarity is assumed, there are theorems showing that scale invariance implies conformal invariance, see e.g. https://en.wikipedia.org/wiki/C-theorem for the theorem of A. Zamolodchikov in 2D and discussion of the situation in 4D.

However, it is known that this link does not extend to general statistical field theories: for example, [1] use elasticity as a counter-example. The link between scale invariance and conformal invariance has been reviewed in [2].

[1] V. Riva and J. Cardy. "Scale and conformal invariance in field theory: a physical counterexample." Physics Letters B 622.3-4 (2005): 339-342.

[2] Y. Nakayama. "Scale invariance vs conformal invariance." Physics Reports 569 (2015): 1-93.

eric
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