I know that Wilson's renormalization group is not a theory per se and that there are many ways to implement it in a given system. Also, renormalization group techniques are applied in a large number of problems in e.g. QFT and Statistical mechanics.
However, I'd like to know if there is any general/abstract theory which rigorously formalizes all these incarnations or implementations in some unified way, so to give precise meaning to objects like renormalization group transformations, renormalization group flows etc, in which case these explicit implementations I mentioned become particular realizations of such a general theory. For instance, I know that we can define renormalization group maps on the space of measures and also on the space of Lagrangians. Thus, maybe these maps can be defined in abstract measure spaces or abstract functional spaces and some possible relations with, say, phase transitions and critica phenomena follow from theoretical considerations of this theory.
NOTE: I think "a general theory" accounting for all explicit models might be too much too ask, but I'd be happy with at least some general theory to some extent. It might be worth to mention that I'm mainly insterested in statistical mechanics.
EDIT 1: As pointed out in the comments, I should specify that the notion of renormalization I'm using here is Wilson's, as is now reflected in the title.
EDIT 2: I came across this beautiful article by Aernout van Enter, Roberto Fernández and Alan Sokal in which some abstract notion of renormalization group map on the space of measures and on space of Lagrangians is introduced. It seems to be possible to put Wilsonian renormalization group in a more abstract context, at least to some extent. This is the first paper (that I know) which treats renormalization in such general context (if there are other papers with similar approach, please let me know). However, I still have many questions to be answered. Some of them are listed below.
(1) The paper is pretty much self-contained but the section about renormalization group on Lagrangians is not very detailed in the sense that there are not many examples of explicit realizations of their definition of RG. Does the so-called renormalization group transformation or the renormalization group flow has anything to do with their definition, at least for finite volume systems?
(2) The authors seem to suggest that this general picture has, indeed, natural connections with the theory of Gibbs measures and, as a consequence, also with phase transitions. However, not much is said in the paper, so I'd like to know more about this. What kind of analyses can be made using this approach? What can physical phenomena can be explained by these general abstract approach?
(3) Is this approach just a particular point of view of the authors? Are there other aternative ways to define these objects? Is this the more general picture we have?
(4) Is it possible to relate this approach with dynamical systems? Is it possible to justify 'relevant directions' and fixed points and its consquences to physical theories?
NOTE 2: Although I posted the article and addressed some questions about it, it is not necessary to address the original question only in terms of this approach. My comments and questions on the paper only reflect the fact that I think this seems to be pretty much the most general picture one can come with, but I don't know much on the subject and I can be pretty much wrong, so any other approach is worthy to me.
This post imported from StackExchange MathOverflow at 2023-09-02 11:15 (UTC), posted by SE-user IamWill
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