Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following:

"In this chapter we sketch a specific method for constructing scaling (continuous) limits $G^{*}(x_{1},...,x_{n})$ of reescaled correlations $G_{\theta}(x_{1},...,x_{n})$ as $\theta \to \infty$, namely the Kadanoff block spin transformations. They serve as a typical example of "renormalization group transformations". Of course there are many other incarnations of the renormalization group strategy [...]."

Well, as the name suggests, the aim of the book is to address QFT but some of these techniques are also applicable and useful to rigorous statistical mechanics. For instance, I think more useful to statistical mechanics is the infinite volume limit than the continuum limit.

What are the most important "incarnations" of the renormalization group strategy in statistical mechanics? To what types of problems they fit and what are their limitations? Are they all related?

  • $\begingroup$ Block spin transformations are really very much a statistical mechanics tools (as the name suggests) and can be used to extract e.g. critical exponents. $\endgroup$ – gmvh Apr 19 '20 at 15:52
  • $\begingroup$ @gmvh I'm a student in statistical mechanics but know hardly anything about QFT. I knew that block spins transformations were developed for statistical mechanics models by Kadanoff and, then, Wilson, but I didn't know if QFT used it for some reason. Thanks for the comment! $\endgroup$ – IamWill Apr 19 '20 at 16:17
  • $\begingroup$ In lattice-regularized QFT, real-space renormalization group techniques that are somewhat similar to block spin transformations can be used to derive "improved" actions, i.e. actions that approach the continuum limit faster (in some suitable sense). $\endgroup$ – gmvh Apr 19 '20 at 17:01
  • $\begingroup$ Thak makes sense. But is it just a discretization or one also decomposes fields? $\endgroup$ – IamWill Apr 19 '20 at 17:09

In statistical mechanics one is mostly interested in some fixed probability measure for some spin configurations on the infinite volume lattice $\mathbb{Z}^d$. The two main problems related to such a measure are P1) the construction of the infinite volume limit and P2) the study of the long distance behavior of correlations for this measure.

In QFT one looks at continuum limits. There are two types of continuum limits L1) limits $\theta\rightarrow\infty$ where the unit lattice measure is fixed and L2) limits where the unit lattice measure varies with the UV cutoff/inverse lattice spacing $\theta$. Massive QFTs are typically obtained via L2). CFTs are obtained via L1) or what probabilists mean by a "scaling limit" in a rather strict sense. Some non massive continuum limits also involve L1), e.g., the RG trajectory going from a Gaussian fixed point to a nontrivial infrared fixed point/CFT. For an example of rigorous result on the latter, see my article "A Complete Renormalization Group Trajectory Between Two Fixed Points".

RG methods are useful for all of the above problems P1), P2), L1) and L2). From a statistical mechanics point of view, L2) is not that interesting. However P2) essentially is the same as L1).

  • $\begingroup$ Awesome answer! Thanks!! $\endgroup$ – IamWill Apr 20 '20 at 20:05

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