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.
