This sounds like a fascinating insight, but I think I may be missing some physics context to fully understand.
Why is it that the derived laws approximating a true underlying physical law are expected to stay scale invariant over increasing scale after being scale invariant for two steps?
Is there a reason that there can’t be a scale invariant region that goes back to being scale variant at large enough scales just like it does at small enough scales?
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!
This sounds like a fascinating insight, but I think I may be missing some physics context to fully understand.
Why is it that the derived laws approximating a true underlying physical law are expected to stay scale invariant over increasing scale after being scale invariant for two steps? Is there a reason that there can’t be a scale invariant region that goes back to being scale variant at large enough scales just like it does at small enough scales?
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!