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The Noether's theorem that I want to mention is the following: Noether's theorem.

I know the importance of Noether's contribution to modern algebra. Can anyone write about Noether's theorem in physics, and its role in modern physics?

Please write it as simple as possible.

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migrated from math.stackexchange.com Feb 28 '12 at 7:30

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I think this is better suited for physics.SE –  Alex Becker Feb 28 '12 at 7:21
    
Good question for theoreticalphysic.stackexchange –  Riccardo.Alestra Feb 28 '12 at 7:27
    

2 Answers 2

up vote 5 down vote accepted

(I only know the importance, not the mathematical treatment &c, but I doubt you want that)

Significance

Noether's theorem let's us obtain conservation laws. Conservation laws are pretty much the life of physics. If you want to calculate the outcome of any process, you have to see what is conserved in the process. Without these laws, you'd be left with an incomplete system. Mathematically, you'd have more variables than equations.

Noether's theorem let's us obtain conservation laws in a beautiful way: Any symmetry in the system gives rise to a conservation law/invariant. This is somewhat semi-intuitive, so I'll give an example (lifted from The Theory of Almost Everything by Robert Oerter):

Example

Imagine a skateboarder in a half-pipe. He can skateboard in two ways: along the direction of the pipe, and perpendicular to it (in an oscillatory fashion). Now, let us look at what happens when we shift the pipe. If nothing apparently changes on such a shift ("space shift"), then we have a conservation law. So lets say we shift it along the direction of the pipe. Obviously, the skateboarder will not feel a thing. But, if we shift the pipe in the direction perpendicular to its length, the skateboarder will be floating in the air, and he definitely notices a change. This is shown in the below diagram. The black arrow indicates how I shifted the pipe. The red arrow shows the (relative) shift of the skateboarder, and the red stickfigure is the final (relative) position of the skateboarder.

enter image description here

So, what is the conserved quantity? It should be conserved along the pipe but not perpendicular to it. Over here, it is the velocity (actually momentum). The skateboarder will have constant velocity along the pipe, but his velocity can change in the perpendicular direction.

It seems more obvious how the theorem works now; if there is a type of symmetry, then doing a "shift" according to the symmetry does not change anything, and thus something is invariant or conserved.

More about it

In modern physics, you see symmetries left right and center. A lot of focus is put on symmetries, as they let you obtain conservation laws without the need of specifying the law as an axiom.

A few more symmetries and their conservation laws:

  • Time-invariance: Energy
  • rotational symmetry: Angular momentum
  • phase shifts (change the phase of the electron field): Conservation of charge
  • Color gauge invariance (change the colored fields by a phase rotation): Conservation of (QCD) color

Note that QCD color is conserved in a trivial way in scattering--- everything that comes in comes out neutral. This conservation law is only a short-distance thing.

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They don't have any issues--- you are just wrong. Charge conjugation is wrong, the right symmetry is global phase invariance (global gauge invariance), and parity conjugation is wrong, there is no conservation of spin separate from conservation of angular momentum. The right conserved quantities corresponding to discrete symmetries are a consequence of the (much more obvious) quantum Noether theorem. "Color symmetry" is not a standard name. You can say "Color gauge transformations" if you want to be accurate. –  Ron Maimon Feb 28 '12 at 14:28
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Instead of downvoting, I fixed it. The early parts are good, but please try to be more careful and not put things of which you are unsure in the answers. –  Ron Maimon Feb 28 '12 at 14:31
    
@RonMaimon thanks for that! I'd read this from a probably outdated source. Next time ill do a bit of Googling before I write stuff I'm not sure about :) –  Manishearth Feb 28 '12 at 14:44

Noether theorem is one of the critical parts of the modern physics which puts a foundation to the connection between symmetry and conservation laws.

I would say that it is extremely important, but its importance is not too well-known by the students. It was even missing from some old theoretical physics courses. Probably, due to the fact that in most simple cases the statement of this theorem is obvious.

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