Last week, I took a little poll here, on my twitter feed, on facebook, and among my students. I wanted to know if anyone had heard of Emmy Noether or of Noether’s Theorem. I don’t want to put my disappointed face on. After all, I hadn’t heard of her until well into grad school, and couldn’t have written down or derived Noether’s Theorem until well after that.
And that is a real shame, because almost nobody in the 20th century did more to explain how physics – and by extension the universe – ultimately works. Noether was a mathematician of the highest order, and makes her so important to the world of physics is that Noether’s Theorem finally gives us a real insight into why symmetry is so important.
Beyond just her work, I want to say a few words on Emmy Noether herself. In many respects, Noether’s story parallels that of Einstein’s, and the two of them intersected on a number of occasions. She was to born to a Jewish family in Erlangen, Bavaria in the late 19th century. Her father was eminent mathematician at the University of Erlangen, and Noether decided to follow in his footsteps. However, German Universities in 1900 generally did not allow women either to enroll in classes or to sit for examination. In 1898, the faculty senate of Erlangen went so far as to claim that admitting women would, “overthrow all academic order.” Noether essentially had to pursue her entire undergraduate education in mathematics by auditing classes, but was still able to pass the graduation exams at a nearby university in 1903.
In 1904, Noether began her doctoral studies at Erlangen, after the ban on women had finally been lifted. Her thesis advisor was Paul Gordan, a very close collaborator of her father, Max Noether’s. Like many other ostensibly pure mathematicians of the era, Gordan’s work found it’s way into the newly developing quantum, in this case into the so-called “Clebsch-Gordan Coefficients” which are used to describe the spin and angular momentum of electrons.
She completed her Ph.D. in 1908, and had an extremely tough time finding an official academic appointment, despite her obvious gifts. You may recall that Einstein, famously, faced similar troubles, and was exiled to a Swiss patent office until after he became mega-Einstein and discovered relativity, the photoelectric effect, and explained Brownian motion all in the same year. Meanwhile, Noether spent the next eight years as unpaid researcher at the University of Erlangen, occasionally substituting for Max Noether in his lectures.
In both her thesis and during her remaining time in Erlangen, Noether became an expert in mathematical invariants. These are absolutely crucial to our understanding of symmetry, so I should say a few words about them.
Invariants are the counterpoint to symmetries. While a symmetry describes the sort of transformations that you can apply to a system without changing it, an invariant is the thing itself that is unaltered.
To make matters concrete, think about the force of gravitational attraction between two stars. In this case, there are a number of symmetries: It doesn’t matter where you do the calculation, or when, or how the stars are oriented with respect to one another. The Invariant, however, is the strength of the gravitational attraction between the two stars. No matter how you move the system around, the magnitude of the force remains the same.
If you’re starting to get a picture of why Noether might be just the person to understand how symmetry really works in physical laws, you’re not alone.
In 1915, Einstein published his theory of general relativity. There was something incredibly elegant, and deeply symmetric about the theory, but nobody really understood how it all fit together. The eminent mathematicians David Hilbert and Felix Klein (inventor of the Klein Bottle) invited Noether to the University of Gottingen in 1915 was to help in understanding some mysteries introduced by relativity.
As Herman Weyl described the situation:
Hilbert at that time was over head and ear in the general theory of relativity…Emmy was welcome as she was able to help them her invariant-theoretic knowledge. For two of the most significant sides of the general theory of relativity she gave at the time the genuine and universal mathematical formulation.
Under normal circumstances her work would have undoubtedly allowed her to start work as a professor. But just as at Erlangen, biases against her gender interfered. Hilbert was outraged. At a faculty meeting, he argued:
I do not see that the sex of the candidate is an argument against her admission as a Privatdozent (roughly equivalent to Associate Professor in the U.S.). After all, we are a university, not a bathhouse.
Hilbert and Noether bent the rules by listing Hilbert as a course instructor, and then having Noether as the perennial “guest lecturer.” I should note that all of this was without pay. It wasn’t until 1922 that the Prussian Minister for Science, Art and Public Education gave her any sort of official title at all, and even then only a small stipend.
Almost immediately upon her arrival at Gottingen, Noether derived what’s become known as “Noether’s Theorem,” and by 1918 had cleaned it up enough for public consumption. Simply put, her theorem states:
Noether’s Theorem: Every symmetry corresponds to a conserved quantity.
If you are a bit underwhelmed after such a long build-up, you shouldn’t be. For one thing, this is more than just a blithe statement of fact; there’s a lot of mathematics under the hood. Noether’s Theorem gives a prescription for determining the conserved quantity for any system, including quantum fields. It provided the inspiration and simple interpretation for much of Quantum Electrodynamics and Yang-Mills Theories
Conservation laws are the bread and butter of physics. In the early universe, for instance, the positive charges exactly canceled the negative charges, and charge seems to be a conserved quantity, that means that the total electrical charge in the universe must still be zero today.
What Noether proposed sounds quite simple, almost content-free, unless you look at the implications. To give some simple ones:
- Time Invariance -> Conservation of Energy
- Spatial Invariance -> Conservation of Mometnum
- Rotation Invariance -> Conservation of Angular Momentum
But it goes even further than that. The last 50 or 60 years have all been about understanding quantum fields. Noether’s Theorem explains what happens when a system is symmetric upon a change of quantum mechanical phase. Answer: you get conservation of electric charge.
Likewise, her work describes and explains conservation of spin, of “color” (the equivalent of charge in the strong force) and on and on, ultimately providing the mathematical foundation for much of the standard model (despite the scant amount of attention that she gets in wikipedia).
Her story parallels Einstein’s in other, sadder, ways as well. Like Einstein, she fled to the United States in 1933. Einstein settled in Princeton, at the newly built Institute for Advanced Study. Noether went to nearby Bryn Mawr College. Her story has a rather sad ending, however. Only two years after coming to America, Emmy Noether was diagnosed with a cancerous tumor, and in the aftermath of a surgery, she died from infection. She was only 53. In Einstein’s words:
In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.
At some point in the future — if there’s interest — I may write a technical post on Noether’s Theorem and explain how it actually works in classical particle systems. But so much talk is given over to explaining how the study of physics is really the study of symmetry. I thought it would be nice to give a little credit to the work that explains what this symmetry actually means.