Is there a difference between renormalization and renormalization group?

Question

Is there a difference between renormalization and renormalization group? In his book 'Scaling and Renormalization in Statistical Mechanics', John Cardy states the following about the term Renormalization Group:

"This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not of a group. Neither is renormalization in quantum field theory an essential element, although is has an intimate connection with some formulations of the renormalization group."

Well, my academic research is concerned to the study of critical phenomena in statistical mechanics using RG techniques. I have to admit that I have hardly any knowledge on quantum field theory, although I know both areas share some common ground when we talk about the study of RG. I'd like to clarify John Cardy's statement. Is there a difference between renormalization and renormalization group? If so, what is it? How are these two techniques related?

Answer 1

There is definitely a difference between renormalization and the renormalization group. Renormalization is a broader concept of getting finite answers out of theories that involve diverging quantities. For example naively one might think that a certain coupling strength appearing in your theory (in the lagrangian) is observable. It is then surprising to find that the value diverges. The solution can be that the coupling strength as you defined it is not an observable at all (hence it is allowed to diverge). One then needs to find a related observable quantity that is finite. Redefining quantities so that infinities cancel is called renormalization. (For example it might turn out that the size of the coupling appearing in your theory is not an observable but that ratios of couplings as measured at a certain energy are.)

The renormalization group is (mainly) a way to effectively deal with theories where there is some large separation between scales. Often the UV (high energy) degrees of freedom are irrelevant for low energy physics and an effective description can be found by imposing that that the system is invariant under coarse-graining.

The renormalization group is an effective tool for renormalization hence the two are closely connected. To find and compute the appropriate finite quantities in a diverging QFT it can for example be useful to introduce a high energy cut-off. The dependence on the cut-off should then disappear from any true physical observable. In fact, one can demand that the exact value of the energy cut-off does not effect the physics. This is of course exactly the kind of condition that gives you a renormalization group equation. This can then be used to find the appropriate renormalization needed to compute finite quantities in your theory.