The most beautiful idea in physics - Noether's Theorem

What if everything in the universe was actually a little to the right of where it is now? Or what if this orbiting planet was actually half a rotation ahead? What changes? More importantly, what stays the same? These seem like fun but useless thought experiments until Emmy Noether discovered what I believe is the deepest and

most

far-reaching

idea

in

physics

. Knowing what happens to a system under these imaginary transformations gives us an

idea

of ​​the real behavior of the system. The usual summary is: symmetries imply conservation laws. In this video, I'll explain what that means. We'll start with symmetry.

We normally use the word symmetry to mean that if we take the mirror image along some line, a symmetrical object looks the same. Mirror symmetries are pretty, but we can make the word symmetry mean so much more. For example, rotational symmetry: when you can rotate an object a certain amount and it looks the same as before, or another example is translational symmetry. In fact, mathematics took the idea of ​​symmetry and completely generalized it. A symmetry is anything where you have some kind of object and you apply some kind of transformation to it, and you can't tell the difference, in some sense.

It might seem like they took a good descriptive word and then generalized it until it made no sense. But actually this idea is very useful. These abstract symmetries are a constantly recurring theme in mathematics; In fact, the study of symmetry helped motivate one of the

most

important fields of modern mathematics called abstract algebra. Emmy Noether was an expert in symmetry and developed fundamental concepts in abstract algebra. It was during a short lull in her extremely influential mathematical career that she thought about

physics

. She wondered if she could apply the idea of ​​symmetry to the world, and that's what led her to her

beautiful

theorem

.

This is the symmetry she considered. The object is some system, a part of the universe. It could be something someone is throwing away. Or a particle in a vacuum. Or maybe some binary stars. Or if you want, the entire universe. Then you transform it. For example, you could rotate it some lambda angle. Either move it up or down using lambda, or stretch all distances using lambda. Now we are interested in knowing if the system is "the same" in some sense. Noether decided that the interesting thing to check is whether the total energy of the objects would be the same.

So we say that a system has symmetry under a transformation if the total energy of the objects did not change. For example, if you had a single particle and then compared it to a displaced version, clearly the energy is the same. So this system is translationally symmetric. On the other hand, let's say there is a large planet nearby. A particle that is closer has more gravitational potential energy, so this is not translationally symmetric. Or consider this object orbiting in a circle and compare it to a rotated version. Both objects are at the same distance from the planet and therefore have the same gravitational potential energy in both directions.

So this system is rotationally symmetric. That's the symmetry part of Noether's

theorem

. Now let's look at the conservations. If you have ever studied physics, for example in school, you will know how important these things called conservation laws are. It means that if you have a bunch of things and you count their total momentum, say, then you let them go for any amount of time and count the momentum again, it would be the same number. Technically, you can do physics without using these conservation laws. But. Often they will give you some crazy problem that it seems like you shouldn't be able to solve, at least not easily...

But if you invoke the magical laws of conservation, your answer simply fails. Conservation laws are also not useful only for classical physics, but they also help in quantum mechanics and, indeed, all of modern physics. I didn't like using conservation laws because they can make it look too easy. I mean, I would get the solution with so little work that it really feels like magic and so I didn't feel like I understood why it worked. After all, I didn't understand why energy or momentum is conserved, so if I used one of those to solve a problem, then I clearly didn't understand the solution.

Noether's theorem is powerful because it explains where conservations come from. Let me go back to an example. I said momentum is conserved. But this is not always true. If I choose to have my system be a ball rolling on the ground, we all know it will eventually stop. Or if I drop something, it gets faster and faster. Sure, if we take everything as if the momentum of your system is always conserved, but how do I know if the momentum of a particular system won't change? Noether's theorem gives us a simple way to tell, regardless of whether the system is a particle or the entire universe.

She showed that conservations are only obtained if the system has the correct symmetries. Again, let's look at examples. If you have translational symmetry, the theorem says you have conservation of momentum. We know that a particle standing alone has this symmetry, so its momentum is conserved. That's true, it will continue at the same speed in the same direction forever. If instead we had a bunch of single particles as our system, this system is also translationally symmetric; If, however, they are all there, that does not change their energy. Again, Noether tells us that we have conservation of its total momentum, which otherwise would not be so obvious.

In fact, if we consider a displaced version of the universe, no one would be able to tell the difference and therefore there is no difference in energy. Therefore, the momentum of the universe is conserved. When is the momentum of a system not conserved? What happens to this object that gains speed as it falls? Noether's theorem says that this system cannot have translational symmetry, so let's check. What would happen if this object was closer to the ground? It would have had less gravitational potential energy. Good! It is not symmetrical. How about rotational symmetry? As we said, this object could have been rotated here and the energy wouldn't change, so it has rotational symmetry around this axis.

We also know that it has angular momentum in this direction and that it is going with the same speed all the way, so its angular momentum is conserved. And this is what Noether's theorem predicts: if you have rotational symmetry about an axis, then angular momentum in that direction is conserved. One last example, this one is strange. We've talked about translating in space and angle, but what about translating in time? In other words, you have a system doing something at that moment and you compare it to the same system some time later. If it has the same energy, then it is time translation symmetric.

What does Noether say is preserved then? It is energy. I know, this is a bit circular here, but it's more important when we get to quantum mechanics, so I had to mention it. Noether didn't simply come up with these three examples. Instead, she gave us a mathematical way to convert any symmetry into a conservation and vice versa. See, these conserved quantities are called the generators of these transformations, and you can calculate what the generator is for any transformation you can think of. If I find some exotic system and notice that it is symmetric under a transformation, there is a mathematical way to calculate what is conserved.

There is also the opposite. Let's say I notice a new and mysterious amount being conserved. Noether's theorem says that conservation comes from some symmetry and the conserved quantity is the generator of the transformation, so I can calculate what transformation it is. That's very powerful, but the theorem is amazing because it is just as

beautiful

and useful. Symmetries attract us and seem natural. We think it makes sense that if the universe shifted or rotated so that nothing changed, there would be no difference between here and there. So proving that the symmetry and conservation laws are equivalent shows that the conservation laws must be equally natural.

Task Let me know what you think of this idea. Have you heard of him before? Maybe you've heard of things like supersymmetry in physics; Try to figure out how it's related. The version of Noether's theorem I talked about here is the one in classical physics (including GR), it's just a much less powerful version of the theorem than the one she created (but I don't understand it, so...). If you know some calculus and classical physics, try to find a proof of this theorem. And this is a fun activity, try creating strange systems with strange symmetries and then see if you can figure out what is conserved.