On the oddly pervasive notion of the conservation of energypic.twitter.com/j0iWw1sbTY
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one might argue that energy ought to be that which is conserved (by time evolution) by definition; you can sorta make this precise using noether's theorem, although that might be circular
Noether's theorem is a consequence of Lagrangian mechanics, so it's sorta begging the question. In some sense, though, you can get energy conservation in both GR and NM this way, which I guess is a good point!
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Somewhat embarrassingly, I don't actually know what the quantum version of N's theorem is. I presume it's likely some obvious statement about operators commuting with the Hamiltonian or something...
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Yep, by definition a symmetry (e.g. x or t translations) commutes with the evolution operator e^{-iHt}, which means the generator of the symmetry commutes with the Hamiltonian so is conserved. For time translation symmetry the Hamiltonian commutes with itself so is conserved.
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whence the lagrangian with time translation symmetry? (a fair amount of my QFT course was the lecturer writing down a lagrangian and then ¯\_(ツ)_/¯ing)
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this tweet chain just led me to this thread which is greathttps://mathoverflow.net/questions/12602/noethers-theorem-in-quantum-mechanics …
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Regardless of how one actually proves Noether’s Theorem, I think it’s correct to think of energy as something that is conserved because of time-translation invariance. Evidence: with non-time-invariant laws, energy is not conserved!
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Nice way of thinking; I like the last point. It still leaves as a puzzle why qm and classical mechanics see this. You can formulate QM as (classical) Hamiltonian evolution on a suitable phase space, but it's sorta artificial...
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There is a sort-of proto-version of Noether's theorem in the simpler Newtonian formulation as well - at least when you can write down explicit solutions: https://arxiv.org/abs/1812.10557
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