Amalie Emmy Noether, or Emmy, as she preferred
to be called, was born on March 23rd in 1882.
Her dad was a professor of mathematics at
the University of Erlangen in Bavaria and,
as a young girl, she trained in languages.
By age 18, she had mastered French and English
well enough to be certified to teach them.
But rather than pursuing an instructional
position, she decided to enter the university
of Erlangen, where she studied mathematics,
and it’s because of this choice that we
now know of Noether’s Theorem, which is
perhaps one of the most interesting insights
discovered at the intersection of mathematics
and physics of the last century or so.
That’s a huge claim and others might argue,
but there is no question that the impact has
been huge.
So, what is Noether’s theorem?
It’s an insight that connects mathematical
symmetries and conservation laws in physics.
And, to understand the connection, you need
to understand the meaning of those two phrases.
Conservation laws are something that you encounter
a lot in physics classes.
You may have heard about conservation of energy,
momentum, or charge.
There are others too.
Something is conserved when it doesn’t change.
Probably the simplest illustration of what
I mean is conservation of charge.
If you have a particle with zero charge, then
it decays into a particle with a charge of
plus one, it must also decay into a particle
of minus one.
Plus one added to minus one equals zero.
That way you have the same amount of charge
before and after the decay.
Conservation of energy is also commonly encountered
in physics class.
If you take a ball and pick it up, the ball
has potential energy.
If you hold it still, it has zero kinetic
energy, which is the energy of motion.
But if you drop it, the ball moves quicker
and quicker, losing potential energy and gaining
kinetic energy.
But the total amount of energy doesn’t change.
It just changes forms.
Students can use the principle of conservation
of energy to quickly calculate the relationship
between the height that the ball was lifted,
to its velocity when it hits the ground.
So, those are conservation laws.
They are quantities that don’t change.
So, what are symmetries?
Symmetries have a colloquial meaning.
Faces are symmetric because if you flip them
around a vertical axis and they look pretty
much the same.
Or if you rotate a triangle by 120 degrees
it looks the same.
That meaning is related to the mathematical
sense encoded in Noether’s Theorem.
In Noether’s Theorem, symmetries are more
of an explicitly math thing.
It means if you make a change in the equation,
you can’t tell the difference.
For instance, suppose in an equation you change
every instance of the term x with minus x.
What happens?
Well, of course, it depends on the equation.
Suppose you have the equation y equals x.
Replace every x with a minus x and you get
y equals minus x.
Those two equations are different.
On the other hand, what happens to the equation
y equals x squared?
If you replace x with minus x, this time you
square the minus x and you get x squared again.
This particular equation doesn’t change
under the replacement of x with minus x.
We then say that the equation is symmetric
under changing x with minus x.
When we say the phrase replacing x with minus
x, what does that really mean?
It means that you’re swapping the direction
of the x-axis.
That’s because what used to be positive
is negative and vice versa.
Since the arrow of the axis points in the
positive direction, that’s like swapping
left and right.
If we swap the direction of the x-axis and
then turn it around so it looks like we expect,
we see that this is equivalent to swapping
left and right on the graph.
For the equation y equals x, replacing x with
minus x, which is the same as reflecting the
line around the y axis, we see that before
and after the flip, the graphs look different.
On the other hand, for the y equals x squared
equation, flipping right and left makes no
difference at all.
In this case, we say that the equation y equals
x squared is symmetric under flipping left
and right.
So, this brings us to Noether’s theorem.
What she proved was that in a physical system,
certain symmetries always mean a conservation
law and conservation laws always mean a corresponding
symmetry.
For instance, if the equations describing
the laws of nature don’t care if you are
here or here.
They’re the same.
Motion side to side, front and back, and up
and down are called translations.
The fact that the laws of physics are unchanged-
that is to say symmetric to translations through
space implies that momentum is conserved.
Another thing one can change without changing
the laws of nature is the exact number for
the time you do the experiment.
I mean, suppose I drop this ball.
If I started a stopwatch last night at midnight,
or the moment I was born, the ball would fall
the same way.
The amount of seconds that have elapsed since
I started the stopwatch will be very different,
but the ball doesn’t know or care.
This indifference to when you set your clock
to zero is called time invariance and the
fact that the laws governing the behavior
of matter are symmetric under these sorts
of changes in time lead to the law of conservation
of energy.
A third symmetry is interesting because of
the fact that the laws of the universe don’t
care which direction you call zero.
You can rotate in any direction and the laws
of the universe are unchanged.
This symmetry results in the law of the conservation
of angular momentum.
These three familiar conservation laws arise
from symmetries in the spatial translation,
time translation and rotational translation
properties of our equations.
Others arise from less known properties, for
instance- the law of conservation of charge
arises from some symmetries in the equations
of quantum mechanics.
So why is Noether’s theorem such a big deal?
It’s because she showed where conservation
laws come from.
They come from symmetries.
When you think about it, there is no real
reason the laws of the universe should be
mathematically symmetric.
But the observation of conservation laws suggests
that these symmetries are important.
It might even give us some direction for future
study.
Emmy Noether was an incredibly intelligent
person, unfortunately born in a world that
didn’t fully appreciate her.
She was, perhaps obviously, female at a time
where women’s opportunities were limited.
She was not allowed to enter some universities
and, even when she proved herself to be a
first-rate mind, was not paid when she taught
others.
She was never promoted to the rank of full
professor, in spite of the recognition and
support of some of the leading luminaries
in the world of physics and mathematics.
Einstein wrote “In the judgment of the most
competent living mathematicians, Fraulein
Noether was the most significant mathematical
genius thus far produced since the higher
education of women began.”
David Hilbert, who was a famous mathematician
at the time, wanted Noether to be given the
rank of Privatdozent, which would entitle
her to be paid to teach at the university
level.
When some of his colleagues objected on the
basis of her gender, he responded by saying
roughly “We should invite her to join.
This is a faculty, not a locker room.”
Noether finally was accorded some status at
the University at Gottingen, where she encountered
further obstacles.
It was just in time for the rise of Nazism
in Germany.
As a person of Jewish heritage, she was expelled
from the university and left the country.
She found a position at Bryn Mawr University
just outside of Philadelphia and she occasionally
lectured at Princeton.
Unfortunately, her life was cut short soon
after she arrived in the U.S., when she died
from a fever that struck her after surgery
to remove a tumor.
Emmy Noether’s contribution to humanity’s
knowledge was much broader than her eponymous
theorem.
She really was a genius and she made many
contributions to mathematical theory.
But, in physics, it is Noether’s Theorem
for which she is known.
For any of a particular class of symmetries,
there is a conservation law.
Her theorem is relevant to modern day searches
for new physics.
One of the most popular ideas for extending
our current understanding of the laws of physics
involves an idea called supersymmetry.
And supersymmetry, if it’s real, of course,
has a conserved quantity.
Finding that conserved quantity is one of
the ways we’ll know we’ve found supersymmetry
when we see it.
Well, if we see it.
Emmy Noether really is an unsung historical
hero of math and science, deserving of more
recognition even today.
So, let’s do it.
Get a book on her life and accomplishments
and read up on them.
She really was pretty awesome.
If you liked hearing about her story and accomplishments,
or if you simply think that Noether’s Theorem
is one of the coolest things ever, be sure
to like, subscribe and share.
Others should hear about her.
She was super smart, so I’m sure she’d
agree with me when I say- remember that physics
is everything.
