Talk:Conformal field theory

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:46, 10 November 2007 (UTC)

WikiProject importance rating
How on earth is conformal field theory of low importance? It's one of the active research areas of quantum field theory, because it's needed to state the AdS/CFT correspondence, which, in the words of the article, is "the most highly cited article in the field of high energy physics."

Warning: Once this gets a higher importance, I'll start griping about it's class rating! Adam1729 (talk) 02:32, 8 October 2015 (UTC)

Actually this article is High importance in my opinion - adjusted, and is in dire need of rewriting since it's a total mess.PhysicsAboveAll (talk) 12:55, 9 October 2015 (UTC)

Are conformal field theories really always quantum ?
The examples of the Ising, Potts, and random cluster models at criticality used in most articles (i've read) are classical -i might be mistaken but it seems Ising and Lenz thought in classical terms when defining the Ising model. That is, we have a measure on classical arrangements of the classical spins (without arguing whether spin is intrinsically quantum mechanical) and not a measure on a magnitization field, or on a lattice-indexed family of quantum spin states (qubits, elements of a 2D Hilbert space). And those theories are considered conformal field theories, the field being then a random classical field. Shouldn't the notion of conformal field theory be wider than just requiring it to be quantum ? Of course most of the formalism (computations of correlations functions, OPEs) apply regardless of the "microscopic" definition of the theory, and many results are exactly the same when we can consider a classical and quantum version of a system, like for Ising models, but a classical statistical model is just not quantum. So i think we should just focus on conformal invariance and not require quantumness in general.

Also, i am not an expert but it seems to me that conformal invariance also requires the notion of boundaries, to be defined rigorously, otherwise the conformal transformations are not well-defined, at least in 2D, unless we restrict just to global conformal (Möbius) transformations. In probability they define conformal invariance in terms of the image of domains of the complex plane and curves inside those regions; they have a measure on such curves (for instance the 2D Wiener measure), which is invariant by composition with conformal transformations -the term "covariant" can also be used instead, as the measure is strictly-speaking not invariant, unless the conformal transformation is the identity. See for instance Gregory Lawler's https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/098/38098204.pdf

So i would say that because conformal field theory is getting alot of attention from mathematicians it would be interesting to cater to publics that likes more precision and generality, in this article, by defining/stating things a little more rigorously. Plm203 (talk) 07:04, 14 June 2023 (UTC)


 * About classical vs quantum, we could indeed mention classical CFT and give a few references, if there are enough prominent texts on the subject. But most of the literature uses the expression "conformal field theory" to mean "quantum conformal field theory". Similarly, in the 1990s di Francesco, Mathieu and Sénéchal wrote a book called "Conformal field theory" which was almost only about 2d. Nowadays "CFT" no longer means "2d CFT" by default, not because anything changed in principle, but because many works in other dimensions were done since then.


 * Your other remark would rather apply to the article Conformal map. Could anything be clarified in that article? Sylvain Ribault (talk) 07:29, 14 June 2023 (UTC)
 * Thank you very much Sylvain. Regarding texts: as i was saying most of the articles i read about critical statistical mechanics models are classical. Also when dealing with properties of 2D CFTs, and computational methods like the bootstrap, representation theory, etc. (not that i understand them) it seems that the quantum nature of the underlying fields is not used and those developments (the book by DFMS) seem to apply as much to classical systems -when they exist with those characteristics; actually can one realize all (2D) CFTs as classical lattice models ? Of course renormalization methods and universality contribute to explaining why classical and quantum systems have the same statistical properties. I have not read a great deal but the work of Cardy and collaborators seems to be mostly on classical statistical models. All the works in mathematics which have been prominent: by Schramm, Lawler, Werner, Smirnov, Duminil-Copin and their numerous collaborators -and they have proved important results relevant to physics in the past 2 decades. Most or all of their publications are available on the arXiv. See for instance the review https://arxiv.org/pdf/1109.1549.pdf.
 * I remember Itzykson-Drouffe and Le Bellac have published books using the term "statistical field theory". And some may argue on the basis of a hidden variable theory that quantum mechanics are but a large-scale view of random objects. So perhaps quantum field theories are just examples of statistical field theories... but i'm kidding. :)
 * Anyway it is not very important. Perhaps it is best to just call everything "quantum" and let people figure out the details -im not ironic here. I just hope that newcomers will not be afraid by the "quantum" and will  realize that if they only care about statistics for their applications they don't have to deal with anything quantum, objects which are often not yet proved to exist. Plm203 (talk) 10:11, 14 June 2023 (UTC)


 * The quantum nature of CFT is probably clearer from a high-energy point of view, if you think of applications to gravity or string theory. For example, Liouville field theory is defined by a functional integral. Solving the corresponding classical equations of motion amounts to focussing on saddle points, and has an interpretation in terms of classical geometry.


 * Things are a bit different in the stat-phys point of view that you are thinking about, and maybe that point of view could be emphasized more in the article. Beware however that we have to reflect explicit statements from the cited works, as opposed to your interpretations and inferences from these works. Ideally, you would find a version of the considerations that you wrote above in the literature, and use that as a source. Sylvain Ribault (talk) 11:48, 14 June 2023 (UTC)
 * Thank you again Sylvain. I don't feel confident enough about possible improvements to influence the actual wiki page. I am happy we've written here a few comments. Have a nice day. Plm203 (talk) 13:02, 14 June 2023 (UTC)
 * Let me add that we could think of the statistical sum over a lattice as a discrete version of the path integral. See this recent preprint for the case of Liouville theory. Sylvain Ribault (talk) 14:21, 14 June 2023 (UTC)

The discussion on classical vs quantum above is confusing. It seems the author of the question means rather Euclidean vs Lorentzian. It's true that Euclidean CFTs are used for describing critical phenomena in classical stat-mech models, while Lorentzian CFTs arise in describing (quantum) phase transitions in quantum cond-mat models. But both Euclidean CFTs and Lorentzian CFTs are quantum, in the sense that there is behind a Hilbert space with operators acting on it. PhysicsAboveAll (talk) 19:01, 16 June 2023 (UTC)