Noether’s theorem

Question

Noether’s theorem

In: Physics

Answer 1

Put crudely, Noether’s (first) theorem states:

> If a system has a continuous symmetry property, then there is a corresponding quantity whose value is conserved.

So if you have a physical system that follows certain physics models, and if in that system there is some “symmetry”, (meaning that it doesn’t change when you shift it in a particular way), then there is some quantity linked with that symmetry that doesn’t change.

For example, if you have a system where the maths works just the same forwards and backwards in time you get conservation of energy. If you have a system where it works the same if you shift it a bit in space you get conservation of linear momentum, and if you have a system where rotating it a bit gives you the same equations you get conservation of angular momentum.

Emmy Noether was an early 20th century German mathematician. She proved this theorem in 1915, when she’d just moved to the University of Göttingen to work with David Hilbert and Felix Klein – who were working on some fancy maths around geometry and mathematical physics (her paper on the theorem wasn’t presented until 1918, and then it was presented by Klein, because she was a woman and so not allowed to teach officially, wasn’t getting paid, and wasn’t a member of the Royal Society of Sciences).

Noether, Hilbert and Klein, among other things, were working with a young(ish) German physicist called Albert Einstein. Einstein had come up with some new ways of looking at physics (and in particular, space and time) and needed help with the maths (not because he was bad at maths, but because this area of maths was new, fairly niche, and incredibly difficult). Noether and her colleagues were playing around with the maths of General Relativity – in particular, looking to see how conservation laws work (or don’t work) – when she figured out what we now call Noether’s Theorem.

The version Noether came up with is quite a bit more complicated than the one usually given, as she was working in field theory, with four-dimensional spacetime, rather than in classical physics, with Lagrangians and Hamiltonians. Essentially she was showing that conservation laws didn’t really work or make sense in General Relativity, and as a consequence proved a really neat result about how they work in classical physics.

Noether went on to develop a whole load more maths throughout the 1920s, mostly in abstract areas of algebra, and eventually got official status (and pay) at Göttingen until 1933 when she was kicked out of office for being Jewish. She managed to get a job at Bryn Mawr College in Pennsylvania, but died a couple of years later, aged 53.