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In honor of International Women’s Day yesterday, as well as Women’s History Month, I thought I’d write about mathematician Emmy Noether, whom Albert Einstein called the most important woman in the history of mathematics. Her most influential work, called Noether’s Theorem, was a foundational contribution to mathematical physics and our understanding of the universe. My main source for this diary is this recent article in the magazine Quanta.
Emmy Noether was born in 1882 in Erlangen, Bavaria, Germany, to a Jewish family. Her father, Max Noether, was a mathematics professor at Erlangen University, where Emmy herself would study. She obtained her doctorate in 1907 at Erlangen, and continued working at Erlangen for 7 years, without pay. In 1915, she was invited to join the faculty of the University of Göttingen by David Hilbert and Felix Klein, but the faculty of the mathematics department objected to a woman being admitted to the faculty. (As a response to his department’s rejection of Noether’s appointment, Hilbert is quoted as saying “We are a university, not a bath house.”) At least she got a salary as she taught classes under Hilbert’s name.
At Göttingen, Noether worked on a problem, referred to her by Hilbert, that arose from Einstein’s recently published general theory of relativity. In classical physics, space and time are simply the canvas on which the dynamics of the universe play out, but once Einstein discovered that the presence of matter distorts spacetime, that canvas became an active player in dynamics. Her task was to determine what elements of classical physics, specifically conservation laws (such as conservation of energy, or conservation of momentum), still made sense in this newly relativistic universe. It took her three years to complete her work on this subject, but the resulting theorem that bears her name is still bearing fruit.
I have tried reading Noether’s theorem, but it is embedded in too much jargon for me to make much sense of, so I’ll give the popularized version: every conservation law in physics is related to a symmetry property of the universe, and vice versa. First, a few definitions: a conservation law means that for all physical processes, a certain dynamical quantity does not change. So, for example, when two billiard balls collide on a pool table, if we measure the sums of their momenta before and after the collision, these quantities will be the same. With respect to symmetry, if some kind of change is made to the dynamical system, but that change results in no discernible change in the outcome of the dynamics, then we say that the change is a symmetry operation, or that the dynamics is symmetric with respect to the change. As an example here, imagine a black piece of paper cut into a perfect circle. If I tell you to look away for a moment and flip the circle from one side to the other, when you look back, you’ll just see a black circle, no different from before. Flipping the circle made no discernible difference, and therefore, the circle is symmetric with respect to being flipped.
Now consider the following: I perform an experiment on a particular lab bench and I get a particular result. Then I move the experiment to a different lab bench in a different room, repeat the experiment, and I get the same result. So the result of an experiment does not depend on where in the universe I perform the experiment. Put another way, the outcome of the experiment is unrelated to where you are in the universe. This is called translational symmetry, which implies the universe is homogeneous (having the same properties at all locations). Noether’s theorem determined that the translational symmetry of the universe is bound together with the conservation of momentum. If I performed the experiment on different days, again, I will get the same outcome, which means the universe is symmetric in time. According to Noether’s theorem, the time symmetry of the universe is bound together with the conservation of energy. Finally, if my experimental setup has a particular orientation, if I perform my experiment first with a north-south orientation, and then with an east-west orientation, I will get the same result. This independence in orientation is called isotropy, i. e., the universe is the same in all directions. (Note that homogeneity and isotropy are not the same. Homogeneity is about location, and isotropy is about direction.) Noether’s theorem states that the isotropy of the universe is bound to the conservation of angular momentum.
“Before Noether’s theorem, the principle of conservation of energy was shrouded in mystery,” the physicist and mathematician Feza Gürsey wrote in 1983. “… Noether’s simple and profound mathematical formulation did much to demystify physics.”
This is true for any conservation law that physicists have discovered. For example, charge conservation (where in any process, the sum of the charges of the initial particles and the sum of the charges of the final particles must be equal) reflects the symmetry of the electromagnetic field. Noether’s theorem is a guide to physicists when creating new theories that they must reflect the symmetry dictated by the conservation laws of the particles and forces involved.
Noether was fired from the University of Göttingen in 1933, when Hitler decreed that no Jews would be employed at German universities. She was subsequently hired by Bryn Mawr College in Pennsylvania, while also lecturing at the Institute for Advanced Study in nearby Princeton, where Einstein landed after leaving Germany. Sadly, she died of ovarian cancer in 1935 at the age of 53. However, her work has made her immortal.
As a final note, I will say that before the ascent of Hitler and the Nazis, Göttingen was one of the greatest centers for the study of mathematics in the world. When Jews were banished from the faculty, Göttingen lost that status, and has only partially recovered its former reputation in the 80 years since the end of World War II. Now seeing Trump and Musk in a frenzy to defund the most prestigious universities in the U. S. feels like history is repeating itself. If the U. S. loses its reputation as the leader in scientific research, it will be difficult, if not impossible to recover.
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From Denise Oliver Velez:
This comment from one of the Caribbean writers I featured in Caribbean Mattters, Mariposa (who is Dkos member lapoetamariposa), could have been a diary.
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