I was casually reading about Noether’s theorem when I had a Keanu Reeves moment. Conservation of energy comes, by Noether’s theorem, from independence of time translation in the theory of physics (i.e. experiments’ results are not dependent on starting time). Conservation of momentum comes from independence of space translation (i.e. experiments’ results are not dependent on the place they are performed, assuming the conditions are the same). And then I realized that position and momentum are a fourier pair (you know, that thing in quantum mechanics that says you can’t know both at once), and so are time and energy. But Noether’s theorem does not talk about quantum mechanics; it talks about classical mechanics.
Hmm, now I realize a paragraph in the Noether’s theorem article that I missed:
Noether’s Theorem is deeply tied to quantum mechanics as it identifies physical variables that are related by the Heisenberg uncertainty principle (such as position and momentum) using only the principles of classical mechanics.
Well, that’s… cool. The fact that that article points it out explicitly takes away some of its punch. Still, I’d like to investigate this correlation. Unfortunately I don’t understand the math behind Noether’s theorem yet.