What is the connection between Conformal Field Theory and Renormalization group in QFT?

Question

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?

This post imported from StackExchange Physics at 2014-04-23 15:13 (UCT), posted by SE-user xxxxx

Comments

Well it seems that you have all pretty much figured out: conformal field theories are a subset of quantum field theories corresponding to the point(s) at which the beta function vanishes. It may seem not much interesting to look at subset of 'vanishing measure' in the space of quantum field theories but actually: conformal symmetry is a really strong constraint and is enough to solve exactly some theories in 2d, and you can derive results near critical points from the so called conformal perturbation theory. The main references are Ginsparg lecture notes (arXiv) and thee book by DiFran & al.

This post imported from StackExchange Physics at 2014-04-23 15:13 (UCT), posted by SE-user Learning is a mess

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@Learningisamess: There's a lot of good writing about CFT which doesn't stick to the comforts of 2 dimensions. See, e.g., sites.google.com/site/slavarychkov or physics.ipm.ac.ir/phd-courses/semester7/CFT-course-2013.pdf

This post imported from StackExchange Physics at 2014-04-23 15:13 (UCT), posted by SE-user user1504

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Morally speaking, CFTs are the "fixed" points of RG flow, in the space of theories. Assuming that RG flows behave "nicely" (a subject being researched currently, but generally believed to hold), you can think of any QFT as _relevant_ deformations (operators that break conformal symmetry) on top of a particular CFT. (As an aside, I wonder if one can then use the power of spurion-like techniques.)

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Huh, do all fixed points correspond to CFTs which are invariant under the full conformal group? I always thought that fixed points just imply scale invariance. So what is the difference between a fully conformal invariant fixed point and an only scale invariant fixed point?

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That is precisely the question being hotly debated and there's been a lot of back-and-forth in the last few years. For the ideas involved and references, have a look at this fairly comprehensive note by Yu Nakayama: http://arxiv.org/abs/1302.0884