Matthias Dellago's Shortform

The act of coarse-graining/scaling up (RG transformation) changes the theory that describes the system, specifically the theories parameters. If you consider in the space of all theories and iterate the coarse-graining, this induces a flow where each theory is mapped to a coarse-grained version. This flow may posess attractors, that is stable fixed points x*, meaning that when you apply the coarse-graining you get the same theory back.

And if f(x*)=x* then obviously f(f(x*))=x*, i.e. any repeated application will still yield the fixed point.

So you can scale up as much as you want - entering a fixed point really is a one way street, you can can check out any time you like but you can never leave!

You're making an interesting connection to symmetry! But scale invariance as discussed here is actually emergent - it arises when theories reach fixed points under coarse-graining, rather than being a fundamental symmetry of space. This is why quantities like electric charge can change with scale, despite spacetime symmetries remaining intact.

And while spacetime symmetries still seem scale invariant, considering the above argument they might also break down at small scales. It seems exceedingly unlikely that they would not! The initial parameters of the theory would have to be chosen just so as to be a fixed point. It seems much more likely that these symmetries emerged through RG flow rather than being fundamental.