By definition a system will exhibit scale invariance at low energies if it has an IR fixed point.

I am having some doubts on how to interpret this fact in terms of quantum field theory and to understand how it affects nature.

A first statement I would like to consider is the following: if the theories we use to describe nature had all an IR fixed point our world should be scale invariant.

I am not convinced about this. Take for instance electromagnetism: is it scale invariant at low energies? On the one hand it has an IR fixed point so the answer should be yes, but on the other hand if we consider problems of electrostatics in which we insert scales by hands (e.g. place some charged objects of fixed sizes around space and consider the resulting field), we will have that the field configuration is not scale invariant.

A possible explanation could be that the configurations in which scales are relevant in electrostatics are not IR enough, for instance because we have scale invariance when we can approximate any object of finite size with a point. But this seems to weaken the initial statement because it should be possible to regard electrostatics at finite scales as low energy enough, at least for small enough charges.

A different explanation could be that inserting scales by hands spoils scale invariance from the beginning because you are assuming that there is something else that is not scale invariant which gives birth to your finite scale objects.

For instance one could say that QCD is ultimately responsible for bringing scales around in our world, since it does not have an IR fixed point: conductors have a size because of QCD and that makes possible states that despite being IR have scales.

Summing up there are two main points that I would like to understand:

  • how would the IR scale invariance appear if all the sectors of nature had a fixed point at low energies? Would there be "macroscopic" objects of finite size or everything with size larger than a certain UV scale would be scale invariant?

  • In the cases where is possible to neglect UV scales for the various theories having an IR fixed point, what else other than QCD determines the scales around? Is it important to have at least a theory not having an IR fixed point in order to have scales at low energies?

  • 1
    $\begingroup$ The world doesn’t look scale-invariant to me. Protons, atoms, stars, and galaxies all have their own sizes. There are no galactic- sized protons or proton-sized galaxies. So why should physical laws be scale-invariant at all? $\endgroup$ – G. Smith May 14 '19 at 16:37
  • $\begingroup$ yeah sure, maybe the title is misleading. The goal of the question is to understand how the scale invariance of some classes of phenomena gives way to this scale-full world of ours $\endgroup$ – AoZora May 14 '19 at 19:24
  • $\begingroup$ The masses are introducing scales into nature: the field inertia (or field propagation/diffusion) scales, defined by lenght $\ell = \hbar / m c$. $\endgroup$ – Cham May 14 '19 at 21:28

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