Emmy Noether | Mathematician who proved Noether’s theorem

Emmy Noether was a mathematician who discovered perhaps the most profound idea in contemporary physics. Noether’s theorem, which she formulated in 1915, says that symmetries in the universe give rise to mathematical conservation laws. This statement is a crucial underpinning of physical laws, from those that govern the rotation of a wheel or the orbits of planets around stars, to the intricate mathematical frameworks of general relativity, quantum physics and particle physics.

Noether was born in the small German town of Erlangen, near Nuremberg, in 1882. Despite the fact that her father, Max Noether, was a professor at the University of Erlangen, she was initially forbidden from enrolling there because of her gender.

Such discrimination dogged Noether’s career. Although she eventually gained both an undergraduate degree and a PhD, no university would hire her for a permanent faculty position. She eventually became one of the world’s foremost experts in the fields of abstract algebra, algebraic topology and the mathematics of symmetry, working at the University of Erlangen and subsequently the University of Göttingen.

But for over a decade, she was without appointment, pay or formal title, despite the championing of her work by many of the most prominent mathematicians of the age, chief among them David Hilbert and Felix Klein. That only changed in 1919, when the end of the first world war and the replacement of the German Reich by the liberal Weimar Republic brought a sea change in attitudes towards women’s education.

Noether’s theorem

Noether’s eponymous theorem was inspired by Albert Einstein’s work on relativity in the early years of the 20th century, culminating in his general theory of relativity in 1915. It formalised an idea that was implicit but unstated in the general theory of relativity and many other theories of physics: that symmetries hold the key to new theories that describe the workings of nature.

Mathematician Hermann Weyl, a contemporary of Noether who was greatly influenced by her work, once described a very simple way of thinking about symmetry. “A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before,” he wrote. Noether’s central insight was that every symmetry you can observe is connected with a mathematical conservation law.

Translational symmetry, for example – the idea that physics remains broadly the same if you move a little to the left or right, or backwards or forwards – is nothing other than the law of conservation of momentum. The symmetry of moving around in a circle amounts to the law of conservation of angular momentum. Symmetry in time – that is, physics remaining the same when translated forwards or backwards in time – amounts to the conservation of energy.

Noether’s theorem adds up to a practical prescription for making progress in physics: identify a symmetry in the world’s workings, and the associated conservation law will allow you to start making a meaningful calculation. Much of physics since its discovery has been a search for these symmetries – or, in the case of the development of the standard model of particle physics, broken symmetries in how quantum fields work that point to where symmetries existed at higher energies when the universe was young.

A broken symmetry in the first split second of the universe, for example, allowed matter to win out against its symmetrical twin, antimatter, creating the matter-dominated universe we live in today. Another broken symmetry, associated with the existence of the Higgs boson, caused the electromagnetic and weak nuclear forces to have the very different strengths that they now possess.

So, as ideas in physics go, they don’t come any more fundamental than Noether’s theorem. Sadly, Noether’s life after discovering the theorem wasn’t a happy one. She came from a Jewish family, and on the accession of the Nazis to power in Germany in 1933, her hard-won right to teach at the University of Göttingen was revoked. She emigrated to the US and taught at Bryn Mawr College in Pennsylvania, but died of complications from cancer surgery two years later.

The reverence that many of Noether’s colleagues felt for her was only increased by her calm spirit and support for others in the face of the Nazis’ oppression. Weyl, whose wife was Jewish and who also emigrated to the US, later wrote that “Emmy Noether – her courage, her frankness, her unconcern about her own fate, her conciliatory spirit – was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace”.

But it is her seminal work that is most celebrated, although perhaps not as much as it should be – as is the case with many pioneering female mathematicians and scientists. As Albert Einstein wrote in The New York Times, “Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began”. Others might suggest that the last seven words of that sentence are superfluous.