What if everything in the universe was actually a bit to the right of where it is now?
Or 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 think,
is the most profound and far-reaching idea in physics.
Knowing what happens to a system under these imaginary transformations, gives us insight
into the systems real behaviour.
The usual summary is: symmetries imply conservation laws.
In this video, I’ll explain what that means.
We’ll start with symmetry.
Normally we use the word symmetry to mean that if we took the mirror image along some
line, a symmetric 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 just the same as before, or another example is translational symmetry.
In fact mathemations took the idea of symmetry and generalised it completely.
a symmetry is anything where you have some sort of object and apply some sort of transformation
to it, and you can’t tell the difference- in some sense.
This might seem like they’ve taken a good descriptive word and then generalized it till
it’s meaningless.
But actually this idea is very useful.
These abstract symmetries are a constantly reoccurring theme in mathematics - in fact,
the study symmetry helped motivate a one of the most important fields of modern mathematics
called abstract algebra.
Emmy Noether was an expert in symmetry, developing foundational concepts in abstract algebra.
It was during a small pause from 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 lead
to her beautiful theorem.
This is the symmetry that she considered.
The object is some system, a part of the universe.
It could be a thing someone is throwing.
Or a particle in a void.
Or maybe some binary stars.
Or if you want, the whole universe.
Then you transform it.
For example, you could rotate it by some angle lambda.
Or shift it up or down by lambda, or stretch all the distances by lambda.
Now we’re interested in if the system is ‘the same’ in some sense.
Noether decided the interesting thing to check is if the total energy of the objects would
be the same.
So we say that a system has a symmetry under a transform if the total energy of the objects
didn’t change.
For example, if I had a particle all by itself and then compared it to a shifted version,
clearly the energy is the same.
So this system is translationally symmetric.
On the other hand, say there was a big planet near by.
A particle that is closer has got more gravitational potential energy, so this isn’t translationally
symmetric.
Or consider this object orbiting in a circle, and compare it to a rotated version.
Both objects are an equal distance from the planet and so both ways, they have the same
gravitational potential energy.
So this system is rotationally symmetric.
So that’s the symmetry part of Noether’s theorem.
Now let’s look at conservations.
If you’ve ever studied physics, for example at school, you’ll know how important these
things called conservation laws are.
It means that if you have a bunch of things and you counted up their total momentum let’s
say, then you let them go for any amount of time and counted the momentum again, it would
be the same number.
Technically, you can do physics without ever needing to use these conservation laws.
But.
Often they’ll give you some insane problem that looks like you shouldn’t be able to
solve- at least not easily...
But if you invoke the magical conservation laws your answer just falls out.
Conservations laws aren’t just useful for classical physics either, they help out in
quantum mechanics and really all of modern physics.
I used to not like using conservation laws because they can make it seem too easy.
As in, I’d 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 is conserved or why momentum is conserved,
so if I used one of those to solve a problem then clearly I 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 that momentum is conserved.
But this, is kind of not true not always true.
If I choose my system to be a ball rolling on the ground, we all know that eventually
it stops.
Or if I dropped something, it gets faster and faster.
Sure, if you take everything as your system momentum is always conserved, but how can
I know whether a particular system’s momentum won’t change.
Noether’s theorem gives us a simple way to know, regardless of whether the system
is one particle or the whole universe.
She proved that you only get conservations if the system has the right 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 that’s on its own has this symmetry, so it’s momentum is conserved.
That’s true, it will continue on at the same speed in the same direction forever.
If we instead had a bunch of particles by themselves as our system, this system is also
translationally symmetric-if they all over there instead, that doesn’t change their
energy.
So again, Noether tells us we have conservation of their total momentum, which wouldn’t
be that obvious otherwise.
In fact, if we consider a shifted version of the universe, no one would be able to tell
the difference and so there’s no difference in the energy.
Hence the momentum of the universe is conserved.
When isn’t momentum conserved for a system?
What about this object that gains speed as it falls?
Noether’s theorem says that this system can’t have translational symmetry, so let’s
check.
What if this object was nearer to the ground?
It would have had less gravitational potential energy- Good!
It isn’t symmetric.
How about rotational symmetry?
Like 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 it has angular momentum in this direction, and that it goes at the same speed
the whole way, so its angular momentum is conserved.
And this is what noether’s theorem predicts, if you have rotational symmetry around one
axis, then the angular momentum in that direction is conserved.
One last example, this one is a weird one.
We’ve talked about translating in space and in angle, but what about translating in
time?
In otherwords, you have a system doing something at the 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 conserved then?
It’s energy.
I know, that’s a bit circular here, but it is more important when we come to quantum
mechanics- so I had to mention it.
Noether didn’t just come up with these three examples.
Instead, she gave us a mathematical way to turn any symmetry into a conservation and
vis versa.
See these conserved quantities are called the generators of these transformations and
you can calculate what the generator is for any transformation you come up with.
If I encountered some exotic system and noticed it is symmetric under a transformation, there
is a mathematical way for me to calculate what’s conserved.
There’s also the converse.
Say I notice noticed that some mysterious new quantity
is conserved.
Noether’s theorem says that conservation is from some symmetry, and the conserved quantity
is the generator of the transformation, so I can calculate which transformation it is.
That’s very powerful, but the theorem is amazing because it is just as beautiful and
it is useful.
Symmetries appeal to us, and seem natural.
We think it makes sense that if the universe was shifted, or rotated that nothing should
change, there’s no difference between here and there.
So showing that symmetry and conservation laws are equivalent shows that conservation
laws must be just as natural.
Homework Let me know what you think of this idea.
Have you heard of it before?
Maybe you’ve heard about things like super symmetry in physics- try find out how that’s
related.
The version of Noether’s theorem I talked about here is the one for classical physics
(including GR), only its much less powerful version of the theorem than she created (but
I don’t understand that one so...).
If you know some calculus and classical physics, try and find a proof of this theorem.
And this is a fun activity, try come up with strange systems with strange symmetries- then
see if you can figure out what’s conserved.