Here's something I've always wanted to know the answer to. There's a result in mathematical physics by Emmy Noether about symmetry and conservation: generally, if a physical system has a symmetry built into in its description, there's a corresponding conservation law. Physical law looks the same today as it will tomorrow - it's isotropic in time. Physical law is the same if you move up or down a few miles, or left or right a few feet - it's isotropic in space. Where there's isotropy, there's a conservation law - in the case of time, conservation of energy, and for space, conservation of momentum.

In geometry, symmetries preserve volumes. Let's throw that out and talk about scaling maps. And lets suppose that we don't already know that physics looks different on the small scale (I'm talking very loosely here - I mean, let us consider classical mechanics.) If materials are homogenous at all scales, so there are no atomic effects and no limit to divisibility, we can take any mechanical system and shrink or magnify it arbitrarily - and it ought to work the same way, intuitively. Does it or doesn't it? Possibly not, because features like mass, and surface tension would scale differently. So that would be odd, right? You could detect that change experimentally. But if the physics is identical at all length scales, then that shouldn't happen - you should be able to magnify every distance in the universe uniformly and not detect it experimentally.

Let's pretend that's true, although I think the scaling issue kills it. If so (here's the real question), can we weaken Noether's theorem to get a new conservation law? What would that conservation law (conservation because of scale invariance?) look like?

- posted by Scott @ 5:57 PM