The first in my Women in Science series has to be Emmy Noether, described by Einstein himself as “the most significant creative mathematical genius thus far produced since the higher education of women began”. She’s easily one of my favourite physicists of all time, even though she only has one major contribution to physics: Noether’s Theorem.
Of course, when Einstein produced that quote, the higher education of women had not been around for very long, so it’s not a very crowded competition for “most significant creative mathematical genius”. I’m also not qualified to judge how good most of her work is, since it exists well outside my area of expertise, in abstract, pure maths. Noether’s Theorem, however, deserves explaining in detail, along with a brief discussion of her life and times.
Born in 1882 in the German town of Erlangen, her father, Max, was also a mathematician. This proved to be invaluable in getting her career started, as she was able to work as an academic under her father’s name, thereby avoiding restrictions on women holding academic positions. She received her PhD in 1907 and worked thereafter in academia until her death, but wouldn’t be paid for her academic work until 1923.
She faced constant institutional sexism in every institution she worked at for decades. When two of the great mathematicians of their day, David Hilbert and Felix Klein (of the Hilbert Hotel and Klein Bottle respectively), tried to recruit her to teach at the University of Göttingen in 1915, they faced opposition from other faculty members. Hilbert, an interesting character in and of himself, responded with a quote I rather enjoy, “I do not see that the sex of the candidate is an argument against her admission as privatdozent [lecturer]. After all, we are a university, not a bath house.” Again, when at Göttingen, she lectured under Hilbert’s name.
Noether did research at Göttingen until 1933. This was, of course, the year that Hitler became Chancellor, and Noether was (you guessed it) Jewish. The expulsion of Jewish academics from university positions was among the first of Hitler’s antisemitic policies to be enacted in Germany. She was by no means the only one affected by this — a decent fraction of the Manhattan Project was staffed by German Jewish exiles. Noether was noted by many of her colleagues for being particularly courageous at this time, and for providing comfort for them. Like many of her colleagues, she moved to America. Specifically, she accepted a post at Bryn Mawr College in Pennsylvania. It was her last position — she died in 1935. She was 53.
That’s her story. I want to talk a bit about her Theorem, because it is truly remarkable. It came out of the study of General Relativity, but applies to all theoretical physics. The reason it’s so revered is because it provides a deep, profound connection between two of the most important concepts in physics — conservation laws and symmetries.
A conservation law is a law of physics which states that, for a given system, something is conserved. Energy is the most famous example — it can be neither created nor destroyed, only converted from one form to another. If one system loses energy, that must mean another system has gained the equivalent amount. There are, of course, many other conservation laws. Momentum is one, angular momentum is another, as are more exotic ones like charge, lepton number, parity, and more.
Conservation laws are loved by physicists because they impose constraints on what physical systems can do, making them easier to predict. Particle physics, for example, uses conservation of momentum to tell how many particles are involved in a given interaction. This is how the neutrino was discovered: they noticed certain decays weren’t conserving momentum, and so concluded that an undetected particle was carrying some away. You can use conservation of energy to do this:
Neil deGrasse Tyson didn’t flinch because he knew energy would be conserved — the pendulum couldn’t swing further than his nose because it didn’t have enough energy to do so, and had no way to gain it.
Symmetries are another concept beloved by physicists. We love to study symmetric systems because they’re a lot easier to study! If a system looks exactly the same after you transform it in some way, then if you can describe it in one state, you can also describe it in the transformed state equally well. For a concrete example: if I tell you that a ball is perfectly symmetric, and that one square inch on it is green, then the rest of it must be green, too, otherwise it wouldn’t be perfectly symmetric! For many non-symmetric systems, we often describe them as being symmetric, plus a correction term for any asymmetry. Asymmetries are aberrations we need to handle delicately, most of the time.
The green ball is an example of rotational symmetry. If you rotate that ball through any angle, about any axis, it will look the same. There are other symmetries associated with other transformations. The laws of physics are the same at any point in space (we assume), so you can move your experiment anywhere in space and not be able to tell — this is translational symmetry. The same applies to time — we assume physics has been and will be the same at all points in time, giving us time translational symmetry. Symmetries are also of great use to particle physicists, who use symmetry considerations to derive theoretical results. The mathematical study of symmetries is called group theory, and the theoretical underpinning of particle physics is built on that.
Now we finally get to Noether’s Theorem. According to the theorem, every conservation law is associated with a symmetry, and vice versa. It’s a theorem, meaning it’s derived from mathematical results, and is provably true. The mathematics is very involved, but essentially, if there is a system you can transform without changing it, it must have an associated conserved quantity. The best example is energy, which is associated with time invariance. If a system doesn’t change over time, then its energy must be conserved. At a fundamental level, the reason for conservation of energy is time invariance, under Noether’s Theorem.
Similarly, conservation of momentum is associated with translational invariance. If you can take a system and move it anywhere without it changing, then momentum must be conserved. (This makes sense if you think about the opposite — when moving a system does change it, it must mean something’s in the way in its new position. Whatever that is, it’ll affect the system in some way, changing its momentum.) Also, conservation of angular momentum associates with rotational invariance. Those are the simplest examples, but there are countless others.
Noether’s theorem allows you to describe conservation laws, but it also allows you to derive new ones. Just observe any symmetries a system has, and then give a name to whatever the conserved quantity associated with it is. Once you know that quantity is conserved, you can use it as an experimental tool to make powerful predictions about the system’s properties. Similarly, if you see a quantity not being conserved, that gives you information about asymmetries inherent in the system.
As I hope you can see, we like conservation laws, and we like symmetries, so to have a fundamental connection between them, as outlined in Noether’s Theorem, is quite amazing. It’s also a really powerful theoretical tool that can be used to derive all sorts of amazing results. It’s easily one of the most important theorems in theoretical physics, as it’s one of those rare results that applies to all areas of physics. I’ve even heard it compared to Pythagoras’ Theorem in terms of importance. Noether was an unparalleled genius, and I hope an appreciation of the beauty and elegance of her most famous theorem can help you appreciate that.