This year is the centennial of Noether’s theorem, which is often called the most beautiful result in mathematical physics. Developed by Amalie Emmy Noether (1882-1935), the theorem resolves questions raised by the general theory of relativity and continues to be an essential theoretical tool.
A consensus first-ballot Hall of Famer—Einstein called her “the most significant creative mathematical genius thus far produced since the higher education of women began”—Noether is famous among mathematicians primarily as one of the architects of modern abstract algebra. Her eponymous theorem, which seems almost a sideline in a very productive career, establishes a fundamental connection between conservation laws, such as the law of conservation of energy, and “symmetries,” a term that needs some explanation.
In ordinary usage we say that an object is symmetrical if it looks the same when reflected about an axis, as when we talk about the symmetry of faces. We might also, in a more technical sense, say that an object is symmetrical with respect to some change if the change leaves it looking the same. A perfect sphere looks unchanged after it’s rotated, so it is “symmetric” with respect to any amount of rotation. Wallpaper may look identical after it’s slid a certain distance right or left, or if it’s reflected in a mirror, etc. It may have a variety of such symmetries. (A nontrivial theorem says that so far as symmetry is concerned, there are precisely 17 different kinds of wallpaper.)
Noether’s theorem concerns the symmetries not of objects but of physical laws, transformations that leave the laws themselves unchanged. For example, laws are symmetric under translation if they don’t distinguish one point in space from another. Roughly speaking, such laws predict that whether I do an experiment here or there—that is, after translating the experimental apparatus—I expect the same results. (This is speaking “roughly” because it requires some commonsense qualifications: Experiments in the room down the hall might differ because the room is overheated, bombarded by radiation, shaken by passing trains, and so on.) Laws are time-symmetric if they apply in the same way at all points in time—so that, again roughly, it doesn’t matter whether I do an experiment today or tomorrow.
Galileo proposed that experiments performed inside the cabin of a ship, so long as they don’t refer to anything outside the cabin, could never distinguish between a ship resting at anchor or one moving smoothly straight across the sea. For example, something dropped will be seen to fall straight down. A ball rolled across the floor will travel in a straight line along the direction in which it was released.
The right way to express this kind of symmetry mathematically depends on our assumptions about space and time. Under the commonsense belief that time and distance mean the same on a smoothly sailing ship as they do on an anchored one, the math is simple and defines what has come to be called symmetry under “Galilean transformation.” Einstein pondered deeply the fact that while Newton’s laws have this symmetry, the laws of electromagnetism do not. The result was the special theory of relativity. It preserves both Galileo’s proposal and the laws of electromagnetism by adopting a different understanding of distance and time, which leads to a different mathematical formulation of Galileo’s idea (symmetry under “Lorentz transformation”). The laws of electromagnetism and a modified version of Newton’s laws are both symmetric under that.
Noether’s theorem applies to laws that are formulated in a particular way and therefore to a particular way of thinking about nature. In Newtonian physics, bodies interact by exerting on one another forces that compel their motions. It’s a bottom-up view that is easy to imagine simulating on a computer: Based on where the bodies are and how they’re moving now compute where they’ll be and how they’ll be moving a fraction of a second later; repeat ad infinitum.
An alternative is to describe physical processes not as if they are pushed from the past but as if pulled from the future, as if acting in order to produce a certain outcome. In the first century a.d., for example, Hero of Alexandria noted that some properties of light—the way it is reflected by a mirror, the fact that it travels through air in straight lines—can be explained by supposing that light proceeds from place to place by taking the shortest possible path. That principle can’t explain refraction, the fact that a straight stick partly submerged in water appears to bend at the water line. In the 17th century, Pierre de Fermat managed to incorporate refraction by generalizing Hero’s idea, proposing that light follows a path that takes the shortest possible time. If light travels more slowly in water than in air, it saves time by making a bit more of its journey through the air and a bit less in water than if it were to proceed directly—hence the stick’s apparent bend.
Further generalizations were investigated intensely during the 18th century and reached their modern form in the 19th with Hamilton’s principle (named for William Rowan Hamilton, an Irish mathematician). It tells you how to express the dynamics of a classical physical system by defining a quantity called action and postulating that the system will evolve in such a way as to minimize it. Pierre Louis Maupertuis, generally credited with originating this line of inquiry, regarded it as testimony to God’s wisdom in achieving His effects by the most economical means.
Noether’s theorem applies to systems described using Hamilton’s principle. It defines precisely what constitutes a “continuous symmetry” of laws that are defined by an action and, importantly, from each such symmetry derives a conservation law. So translational symmetry implies that momentum is conserved. Time symmetry implies that energy is conserved. Rotational symmetry—all directions are the same; it is irrelevant which way the experimental apparatus faces—implies that angular momentum is conserved.
Hamilton’s principle was devised for classical mechanics, the physics of planets and pendulums and springs and billiard balls. There it gives the same results as the bottom-up application of Newton’s laws (and in many circumstances, much more easily). But it has also provided an immensely helpful roadmap for discovering and developing theories in other domains—for example, the so-called Standard Model of particle physics that gives an account of the known elementary particles and their interactions via three of the four known fundamental forces (gravity being omitted). Much of its development proceeded by proposing symmetries and investigating the corresponding conservation laws, and Noether’s theorem has been an essential part of the toolkit. Many of the furthest-reaching stretches of cosmology and theoretical physics today, from speculation about string theory to the hunt for dark matter, depend in some way on Noether’s theorem.
So who was Emmy Noether? There seems to have been only one short scholarly biography so far. Her father was a distinguished mathematician at the University of Erlangen in Bavaria, which didn’t admit women; she did, however, audit classes; after the rules changed she enrolled and, in 1907, obtained a Ph.D.; she worked without pay at the Erlangen Mathematical Institute until 1915, when she moved to Göttingen, an important center for mathematics. David Hilbert, then arguably the world’s greatest mathematician, asked for her help investigating conservation laws in general relativity. This is what led to her formulating the theorem that bears her name, first described in an article in August 1918.
Hilbert wanted to keep her at Göttingen with an academic appointment, an attempt that led to a famous exchange in the faculty senate. A (nonmathematician) member objected, “What will our soldiers think when they return to the university and find that they are expected to learn at the feet of a woman?” Hilbert’s reply—which, alas, did not carry the day: “I do not see that the sex of the candidate is an argument against her admission as a privatdozent. After all, the senate is not a bathhouse.” Some years later she did get a modest appointment, with modest pay, and in the meantime gave lectures by means of a ruse: Hilbert would announce a course and she would be his permanent substitute. Hermann Weyl, a prominent Göttingen mathematician who also tried and failed to get her a position commensurate with her talent, wrote to a friend, “I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as a mathematician in many respects.” (Several web pages and at least one popular science book wrongly attribute this remark to Hilbert.)
Weyl described her personality as “warm like a loaf of bread.” She was an inspiration to her students—through passionate conversations, not her lectures, which were poor. “She had a very clear understanding of what she was saying,” said one colleague, “but she didn’t have a clear idea of what she was going to say.” Shortly after the Nazis came to power in 1933, she—a Jew and a pacifist—was dismissed from the university. In this dark time, Weyl said, “her courage, her frankness, her unconcern about her own fate” were “a moral solace.”
The Rockefeller Foundation helped fund a professorship for Noether at Bryn Mawr. (The recommendation from Einstein couldn’t have hurt.) In 1935, after surgery to remove a uterine tumor, she died, age 53, from a postoperative infection.
It was not a tragic life. Noether was not deterred or embittered by indignities from academic bureaucrats. She was, and was recognized as, a peer of towering intellects. And she did her work, by all accounts, with joy.
