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Noethers Theorem is Awesome!

A very lofty title, I know, but hear me out!

As far back as physics was still "natural philosophy" (which it still is, if we're honest), as far back as the ancient greeks (and innumerable others, if we're still being honest), we have observed that things do not simply come and go out of existence --- the water does not vanish as you boil it, but goes into the air, and comes back as rain. In short, we would be nowhere without conservation laws. I dare you to do a book of physics problems without them!

For most people, then, one takes the existence of conservation laws as a base fact: just how the world is, no particular reason as much as why anything else is put here in this world. In the more common applications of these laws, this is perfectly sufficient to assume and forget (sure, you can point to inelastic v.s. elastic collisions, but energy is still conserved, just not within the system). But, as I'm very obviously foreshadowing, this is not the case !

We now approach Noether's theorem as a simple statement, that conservation laws arise not from nothing, but as an unavoidable, mathematical consequence of symmetries in our universe!

Why do I call this the most beautiful idea in all of physics? (I mean, i haven't here yet, but i regularly do!)

  1. We have an objective, mathematical reason why the world is the way it is --- not a description of reality, a perscription of the truth!
  2. Symmetry is a very artistic, aesthetic sort of thing, so it makes sense to associate this symmetry-based theorem with beauty
  3. We have the fascinating consequence of when we don't observe conservation laws, such as CP violation, which we can assert to a lack of symmetry! Or, maybe if you're more interested in the grandiose, the non-energy conservation of the Big Bang? All of these amazing things can be described by Noether's Theorem!
  4. It's obscure enough to make me sound smart when I namedrop it
  5. Other people have already called it that and I'm shamelessly copying them

... ignore those last two

That's nice, but how do we know its true? And how do we actually use it?

Math. A lot of math, that wouldn't make for a nice poetic waffling introduction like I just did. I take these notes I wrote down from Peskin and Schröder's QFT textbook, and hope my handwriting isn't TOO shameful

Of course, the above explanation requires an understanding of Lagranian Mechanics and the principle of least action, which given my upload schedule will only be mentioned by me in seven years time, but nevertheless, this is a shorthand form of the proof anyway (and FAR from the only one!)