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Scale invariance is itself an emergent phenomenon.
Imagine scaling something (say a physical law) up - if it changes, it is obviously not scale invariant as it will continue changing with each scale up. If it does not change it has reached a fixed point and will not change in the next scale up either!
Scale invariances are just fixed points of coarse-graining.
Therefore, we should expect anything we think of as scale invariant to break down at small scales. For instance, electric charge is not scale invariant at small scales!
In the opposite direction: We should expect our physical laws to continue holding for the macro scale, if they are fixed points of scaling. This also explains the ubiquity of power laws in the natural sciences; power laws are the only relations that are scale invariant and thus preserved!
All of this may seem tautological but is actually truly strange. To me this indicates that we should expect to be very, very far from the actual substrate of the universe.
Now go forth and study renormalisation group flow! ;)
Epistemic status: Just riffing!
2This 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!
1The main source of scale-invariance itself probably would have to do with symmetry meaning that an object has a particular property that is preserved across scales.
Space symmetry is an example, where the basic physical laws are preserved across all scales of spacetime, and in particular means that scaling a system down doesn't mean different laws of physics apply at different scales, there is only 1 physical law, which produces varied consequences at all scales.
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.
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.
While this is an interesting idea, I do still think space symmetries are likely to remain fundamental features of physics, rather than being emergent out of some other process.
I'll bet you! ;)
Sadly my claim is somewhat unfalsifiable because the emergence might always be hiding at some smaller scale, but I would be surprised if we find the theory that the standard model emerges from and it's contains classical spacetime.
I did a little search, and if it's worth anything Witten and Wheeler agree: https://www.quantamagazine.org/edward-witten-ponders-the-nature-of-reality-20171128/ (just search for 'emergent' in the article)
Can you have emergent spacetime while space symmetry remains a bedrock fundamental principle, and not emergent of something else?
I don't know if that is a meaningful question.
Consider this: a cube is something that is symmetric under the octahedral group - that's what *makes* it a cube. If it wasn't symmetric under these transformations, it wouldn't be a cube. So also with spacetime - it's something that transforms according to the Poincaré group (plus some other mathematical properties, metric etc.). That's what makes it spacetime.
So space symmetry is always assumed when we talk about spacetime, and if space symmetry didn't hold, spacetime as we know it would not work/exist?
As a corollary: Maybe power laws for AI should not surprise us, they are simply the default outcome of scaling.