“Momentum is always conserved, except when it isn’t.”
In high school physics, we learn all manner of conservation laws, one at a time, when they accidentally happen to pop up, without so much as a word of explanation for WHY nature seems to care so much about these quantities. We’ve asked, of course, only to have our knuckles rapped for impertinence or, in our less corporal age, we’ve been referred to Google to figure it out as best we can for ourselves.
More than a hundred years ago, a woman who was only begrudgingly allowed a university education gave us that very WHY and with it one of the most powerful tools in all of mathematical physics. Her name was Emmy Noether and she was born in Erlangen, Germany in 1882. Her father was a mathematician and she too had a marked preference for math that only grew stronger as she delved further into its open mysteries.
In nineteenth century Germany, a woman could only attend classes at a university with the express permission of each teacher. Every course that she wanted to take, she had to set aside time with the instructor and plead her case for being allowed to sit in the same classroom with the men, promising not to be a distraction and silently swallowing their regular advice to turn to more womanly subjects (Max Planck famously rejected all women applicants to his lectures out of hand … until he met Lise Meitner).
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Noether ran the gauntlet, however, with a steadfastness in the face of rank unfairness that would mark her entire career. She received her bachelor’s degree equivalent in 1903 and wrote her doctoral dissertation (on bilinear invariant theory) in 1907 at the University of Göttingen.
It was the place to be for mathematics. David Hilbert was there. Hermann Minkowski was there. Felix Klein was there. Titanic minds who remain popularly unknown because they did their work in mathematics rather than the sexier fields of physics or chemistry, they would also be Noether’s friends and champions in her battle for recognition from the University.
Noether was not only in the right place, but also studying the right field for her moment in history. She was an expert in invariant theory and group transformations, which govern how quantities change when you transform the coordinate system where they live. Newton had some assumptions about how such coordinate shifts altered measured values, assumptions which were blown apart in 1905 with Einstein’s Theory of Special Relativity. In the fallout of that titanic event, mathematicians and physicists were looking for something that would link classical Newtonian conceptions of conservation with the new and strange world of relativity, and eventually with the even stranger world of quantum physics. Without such a unifying theory of conservation, physics threatened to fly apart into a chaos of special cases.
In 1915, Emmy Noether produced just such a theory, and published it in 1918 (a REAL math nerd, when asked about 1918, will get super excited and start talking about Noether’s theorem and then, perhaps, as an afterthought, recall something about World War I ending that year too). And now, with your tender indulgence, I want to put on my math teacher hat for a bit and talk about that very theory, because it is truly lovely and powerful, and once you wrap your head around it, the universe just shines with snazziness.
Noether’s Theorem invokes a bit of specialized vocabulary. In particular, it tells us what quantities are preserved (momentum, Energy, charge, etc) for a particular physical situation whose coordinate system undergoes a particular transformation. So, for example, if you have a falling rock, and you spin the x and y axes 90 degrees around the z axis, what measured quantities come out just the same as when you measured them in the original, unspun system? Emmy Noether’s answer encompasses every conservation law that went before, and anticipated all of the ones discovered since, even those in areas of physics she couldn’t have begun to imagine from the vantage point of 1915.
After publishing her ground-breaking work, the only material improvement Noether saw was that the university, under pressure from Einstein, Hilbert, and Klein, allowed her finally to lecture to students under her own name.
Consider two points in space, or two events in space-time. There are lots of ways for an object to move from one to the other, but only one that minimizes the difference between the Kinetic and Potential Energies (called the Lagrangian) for a particle making the trip. If you take that path, your KE and PE will be as balanced as possible, and we call that path “extremal”. (‘Cause it’s EXTREEEEEEEEME… at minimizing the Lagrangian…. MAD 80s GUITAR RIFF!!)
Now, if you’re on that path, and we shift the coordinate system around you (say, by rotating the x and y axes under you a bit), and the overall difference between KE and PE doesn’t change, or changes only verrrrrrrrry slightly, then we say the motion is “invariant” under that coordinate transformation. So, if you tell me about an object undergoing a given motion, and how you want to change the coordinate system, Noether’s Theorem will tell us exactly what conservation law must hold in that situation. It’s the last line on the paper she’s holding up in the cartoon (p???– H? – F = constant).
What is phenomenal is that using this method, you can not only derive all of the conservation laws we’re used to from high school physics, but a bunch of other things that you could not know from the older Euler-Lagrange equations and Hamilton Principle techniques that Noether fused in her own theorem. They explain d’Alembert’s insights from a century and a half before just as easily as Feynman’s ideas from three decades after her death. It was one of those grand moments in intellectual history when a shifting mass of unfathomable complexity solidified in three slick lines of text into a single over-arching theory about invariance and conservation and their role in shaping the development of the universe.
After publishing her ground-breaking work, the only material improvement Noether saw was that the university, under pressure from Einstein, Hilbert, and Klein, allowed her finally to lecture to students under her own name (until then, her classes had to be done in Hilbert’s name, as women weren’t allowed to lead classes). Of course, she still wasn’t paid or officially recognized as a professor. In 1922, the best she got was “unofficial associate professor” and a small stipend for teaching abstract algebra, a field that she was making regular and foundational contributions to since publishing her theory.
Her life continued in this fashion, recognized by the greatest minds in physics and mathematics for her piercing insights into the theory of Lie groups and noncommutative algebras (the significance of which we are only just starting to unwrap now), but without an official position proportionate to her skills or renown. And so she marched on for eleven years, writing the laws that every abstract algebra student knows by heart, until 1933 when the Nazis came to power and she, of Jewish origin, was forced from her position. Unlike Lise Meitner, who fought to retain her place in spite of unceasing harassment at the hand of Nazi officials, Noether saw the direction of the wind and fled the country for a position at Bryn Mawr College, one that she occupied for less than two years before a failed surgery to remove an ovarian cyst ended her life at the age of 53 in 1935.
FURTHER READING: The beauty of Noether’s Theorem is almost impossible to fully appreciate until you see it at work, churning through problems in wildly separate fields of physics with the same elegant ease. My own appreciation of the Theorem’s far-reaching applicability was fostered by Dwight Neuenschwander’s delightful book, Emmy Noether’s Wonderful Theorem. His buildup to the theorem itself requires really only a first year college calculus level of math fluency – if you’re cool with the chain rule for partial derivatives, you’re probably ready. After that, things get turned up a notch as he applies the Theorem to different fields, but he is very good at walking you through the thinking, and his insights into the historical development of invariance theory and the calculus of variations are clear and invaluable.