The connection between symmetries and conservation laws
is one of the great discoveries of twentieth century physics .
But I think very few non-experts will have heard either of it or its maker[:]
Emily Noether, a great German mathematician.
But it is as essential to twentieth century physics
as famous ideas like the impossibility of exceeding the speed of light.

It is not difficult to teach Noether’s theorem, as it is called;
there is a beautiful and intuitive idea behind it.
I’ve explained it every time I’ve taught introductory physics.
But no textbook at this level mentions it.
And without it one does not really understand why the world is such that
riding a bicycle is safe.

~Lee Smolin

 

At first, I had planned to talk about Rayleigh and Raman scattering today. However, in honor of International Women’s Day, I changed my mind. I wanted to write about the woman that Albert Einstein called the greatest female mathematician of all time, Emmy Noether.

One of the Greatest Mathematicians of All Time

Noether’s accomplishments are as incredible as they are varied. She made many seminal contributions to the field of abstract algebra and one incredible contribution to the field of physics. Her first published work helped solve the “finite basis problem,” a major open problem in mathematics at the time–even if she later called the work “crap.” (She had high standards, to say the least.) And in her later work, she:

• Made significant progress on the inverse Galois problem, which is still unsolved.
• Definitively demonstrated the role of symmetry in physics.
• Helped develop elimination theory, which is now used to solve equations on computers.
• Helped invent or completely invented commutative ring theory, module theory, algebraic topology, and representation theory, all of which are important tools in modern mathematics and theoretical physics.

In short, whenever Emmy Noether approached a mathematical problem, she invented a whole new field of study. Almost no one in history has been so successful. As mathematician Nathan Jacobson said,

The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.

And she was as brave and audacious as she was brilliant. When Noether first attended the University of Erlangen, women were forbidden from taking courses. Instead, Noether had to ask each individual professor for permission to audit his class. She was one of only two women who attempted to study there in this way.

Later, with the help of mathematical giants David Hilbert and Felix Klein, Noether became the first woman lecturer (and later professor) at the University of Gottingen, much to the distaste of several other faculty members. But at first, Noether worked for no pay and her lectures were advertised under Hilbert’s name; officially, she was his “assistant.”

I can’t possibly describe everything that Emmy Noether accomplished. So instead, I will devote the remainder of this post to describing the Noetherian idea I understand best, one of the most important ideas in modern theoretical physics: Noether’s theorem.

Noether’s Theorem

Informally, W.J. Thompson writes Noether’s theorem as:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Right now, this statement doesn’t make any sense. We need to dissect it and understand what “a continuous symmetry property” is and what it means to be “conserved in time.” We’ll discuss each of these ideas in turn. Let’s start with symmetry.

Symmetry: The Science of Sameness

To learn about symmetry, we have to go all the way back to elementary school. Bear with me; the end result may be simple, but it is completely unintuitive. First, picture a simple square in your head. Now imagine rotating it by forty-five degrees counter-clockwise, as shown below. The square looks different now. Suddenly it’s a diamond.

This is pretty unsurprising. We’re all familiar with this. (Indeed, this is an example where symmetry is not preserved.) But now let’s imagine the same square and rotate it by ninety degrees instead of forty-five. The behavior is qualitatively different. The square looks exactly the same… as if we hadn’t rotated it at all.

Incidentally this would remain true even if we rotated the square by degrees or by degrees or by degrees. Rotating by ninety degrees always returns the square to itself. Thus we say that a rotation by ninety degrees is a “symmetry” of the square.

Much in the same way that we quantify the amount of things by the counting numbers (e.g., I have two pencils or four apples), mathematicians quantify the amount of symmetry something has using “symmetry groups.” The symmetry group of an object is the collection of operations under which that object remains the same. For example, the symmetry group of the square contains every rotation that is a multiple of ninety degrees.

A symmetry group has many special properties, ones that probably seem familiar and intuitive from the world of numbers. In particular, they behave very similarly to the addition of integers we’re all familiar with–except that they may not commute. . However, if I rotate a cube by ninety degrees clockwise and then ninety degrees towards me, the composite operation is different from the operation where I rotated by ninety degrees towards me first and then ninety degrees counterclockwise.

But I digress. Let’s return to just two dimensions. Instead of a square, let’s imagine a pentagon, which is symmetric under rotation by degrees, degrees, degrees, degrees, and degrees.

Let’s add another side. A hexagon is symmetric under rotation by degrees, degrees, degrees, degrees, degrees, and degrees.

Two things are happening each time we add a side: The smallest rotation operation under which the shape is symmetric shrinks–each time we add a side, the angle is smaller–and the number of rotation operations under which the shape is symmetric grows. Indeed, the number of rotation operations under which the shape is symmetric is equal to the number of sides.

What happens if we add infinite sides? A polygon with infinite sides is a circle. In this case, our shape is symmetric under a rotation by any angle at all! This is what we call a continuous symmetry, and this is what Noether means in her theorem when she says a system has a continuous symmetry property.

Just to drive the point home, let’s look at some examples in three dimensions. Like the circle, a parabola is symmetric under any rotation around its central axis. A sphere is even more symmetric–it is preserved by any rotation at all around its center. But something doesn’t have to be round to be preserved by a symmetry. An infinitely large, flat sheet of paper is symmetric too. The operation that preserves it is motion in a straight line. If I am an ant on this infinitely large sheet of paper, and I walked twenty meters in any direction. I might never now exactly how far I walked…because everything looks the same. That’s a symmetry, too!

Sameness in Time

Now we know what a continuous symmetry property is. What does Noether’s theorem say this implies? The easiest way to understand this, I think, is by example. Let’s go back to the parabola, which we know is symmetric under rotation about its axis. Now let’s imagine dropping a marble into the parabola. If we drop the marble straight down it will oscillate back and fourth across the parabola under the force of gravity, just like Bart Simpson in a half-pipe, as shown below.

But if we throw in the marble in with just a little bit of spin, so that it enters the paraboloid traveling a little bit north or south (as opposed to up or down or radially outward or inward), we get totally different behavior where the marble appears to clamshell around the paraboloid:

(Note that although the qualitative behavior of these animations is correct, it isn’t the exact solution. It wouldn’t be too difficult to solve the equations of motion–it’s an undergraduate classical mechanics problem–but I didn’t do so. Call it an exercise left to the reader. 😉 )

Now the marble’s motion is far from constant. It speeds and slows, bobs up and down. However, it turns out that there is a property of the ball’s motion that never changes. The exact product of the mass of the marble, times its distance from the origin, times how fast it rotates about the origin, will always stay constant. This product is called the angular momentum of the ball, and it never changes as the ball wobbles and bobs and accelerates and decelerates. Because it is constant in time, we say that the angular momentum of the marble is conserved.

Noether’s theorem tells us that the reason that the angular momentum of the ball is constant in time is because the paraboloid is symmetric under rotation. Every symmetry of the space our marble lives in generates a conserved quantity of the motion of our marble–something about it that is constant in time. Of course, these conserved quantities are rarely obvious. They’re usually some product of things like mass, acceleration, and even position.

But Why? The Parable of the Rockies

Now that we know what Noether’s theorem says, can we get some intuition as to why it should be? My hometown of Boulder, Colorado is nestled in the eastern foothills of a huge mountain range, the Rocky Mountains, which span the entire western horizon from north to south. To the east of Boulder, Colorado is completely flat.

When settlers first traveled west towards Boulder and saw the mountains in the distance, they rejoiced. They believed that by nightfall they would be safely nestled in the foothills. But it was not to be. When night fell, the world looked exactly the same. The settlers appeared to have made no progress towards the mountains. The settlers were discouraged, but they carried on. They were sure they would be in the foothills by nightfall the next day. But again, it was not to be. The next morning, the mountains were as eternally distant as ever. This went on for two weeks before the settlers reached the foothills.

What happened was that the mountains were so vast, one could see them from a great distance away. To a good approximation (from the settlers’ point of view), they were infinitely wide and infinitely tall. Despite this, though, the settlers still might not have been fooled if eastern Colorado weren’t so absurdly flat. With no landmarks to reference, the settlers were tricked into believing that nothing had changed as they traveled–that perhaps they hadn’t traveled at all. The only landmarks were the unreachable mountains.

Because the settlers were essentially on an infinite flat plane, the world was symmetric under motion along a straight line. As they traveled along that line, the world continued to look the same to them, so they were unable to tell that time had passed. A particle moving on an infinite plane will behave exactly as it did before it moved, since nothing in its environment has changed–thus, momentum is conserved. The same goes for a particle moving in a circle of constant radius along the surface of a paraboloid, along one of the great circles along the surface of a sphere, or in the direction of symmetry along the surface of any other corner-less, three-dimensional shape. (More precisely, the surface must be “smooth,” which is a term defined as “infinitely differentiable.”)

This isn’t so surprising. But what is surprising is that, no matter what the particle does, the component of its motion along the direction of symmetry (e.g., around the axis for a paraboloid or around the center for a sphere) remains unchanged. And this corresponds to a property of the particle’s motion that is unchanged in time.

Just to recapitulate that last bit: A symmetry in space corresponds to a symmetry in time. If nothing changes as I travel in space, then I can’t tell that time has passed. So some aspect of me ceases to change in time.

At the end of the day, Noether’s theorem is beautifully, surprisingly simple…and deeply profound.

Implications

At first glance, Noether’s theorem just seems like an esoteric quirk of geometry. But its implications are very deep and very far reaching. Let’s step back a little and look a few centuries into the past. Newton’s first law of motion is that:

An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted on by an external force.

Well, that’s a statement about a conserved quantity! Newton is telling us that there is some property of the motion, in this case momentum, that doesn’t change in time. Newton didn’t know about Noether’s theorem. But it turns out that there exists a symmetry that generates Newton’s first law: translation invariance. Empty three-dimensional space looks the same everywhere you go. If you were a bird–or a particle–and could fly in any direction in a straight line, everything would appear the same no matter how far you went. This symmetry is what generates the conservation of momentum described in Newton’s first law! (As a side note, Noether’s theorem also only holds in the absence of external forces, so Newton’s law is consistent with that.)

And replacing Newton is just the start! Since its discovery, Noether’s theorem has become an integral part of theoretical physics. In particle physics, the symmetries of a system generate particles, and Noether’s theorem has inspired the discovery of many subatomic particles. In general relativity, Einstein’s equations are so difficult that an exact solution is often unsolvable without the aid of symmetry–but Noether’s theorem allows us to find the spacetime symmetry related to a quantity we believe to be conserved.

It’s not an understatement to say that Noether’s theorem is one of the most important developments in theoretical physics in the last two hundred years. And this theorem is only one of Noether’s myriad brilliant achievements. Yet I’d be willing to bet that most of you never heard her name in school.

Further Reading

• The New York Times did an article on Emmy Noether here.
• The Examiner also did an article on Noether here.
• The University of California at Las Angeles has some records of her here.
• Theoretical physicis John Baez has an article on Noether’s theorem for the interested physics student.
• Professor Nina Byers of UCLA, details the story of Noether’s discovery of her theorem here.

Questions? Comments? Insults?

As always, if you have any questions or corrections, or if you just want to say hi, please leave a comment or shoot me an email.