Finally, a decent(-ish) anniversary...

Tomorrow sees the 130th anniversary of Emmy Noether's birth.

Emmy Noether is a mathematician noteworthy for two reasons:

1. She was pretty good.

2. She was a woman - one of very, very few women who've ever worked in mathematics (at least to the point of being famous) - but that's another story...

Perhaps it's not surprising that Emmy Noether became a mathematician, since her father Max was also a thumping good one. Emmy Noether's remarkable career stretched over about 30 years at one of the top mathematics centres in Europe, Gottingen, working with other giants (at least to maths students) as David Hilbert and Felix Klein.

Much of Emmy Noether's work lies in highly technical subjects such as hypercomplex number theory, non-commutative algebras and representation theory, and probably will always be beyond me. But in 1915 she came up with a remarkable, powerful and hugely applicable result, now known as Noether's Theorem.

Technically this says that "for every differentiable symmetry of the action of a physical system the associated Noether current is conserved". I'll try to explain what this means, as briefly as I can.

In High school physics you're introduced to Newton's Law, F=ma and all that, that basically describes motion in terms of forces. This is all well and good, but in the full-scale world of Forces you would have to work with vectors pointing every which way and things just get tedious.

There's an alternative approach, that involves working with *energy*. Write down a term for how much energy a moving object has, and take potential energy terms (due to gravity, springs, etc.). This sets up the approach known as Lagrangian mechanics, and its cousin Hamiltonian mechanics.

To say this approach is simpler would probably be a downright lie, but there it has a few strengths. No vectors, for one. Secondly it boils down to maths that has been around for ages so there are a lot of tricks of the trade developed over the years. Thirdly, you can jump pretty much straight from this to quantum mechanics, so it's much more general. But anyway...

Again in high school physics, you might consider what happens when two balls hit each other and use "conservation of momentum" to help you solve the problem. That's cool but why is momentum conserved? It's sort of assumed that it is without really being justified.

Here's where Noether's theorem comes in. Instead of assuming that momentum, say, is conserved, you set up the equations, and find that for the two-balls problem it turns out that the balls could be anywhere in space and would behave the same. This is the *symmetry* of the system in the theorem, where a symmetry is just any way of transforming a system that doesn't actually change it. Now, from Noether's theorem, as a *direct consequence* of this symmetry there must be something that is conserved, and it turns out to be momentum.

In the same way, if you could start an experiment tomorrow instead of today without making any difference to the way it behaves, then that too corresponds to a symmetry, and from Noether's theorem we find that Energy* is conserved.

Still not excited? Ever heard of a "Theory of Everything" that would try to explain all of physics? How might physicists go about trying to construct one?

The answer, again, is due to Noether's theorem. Conservation of Energy and momentum are old-hat and were known about long before Noether came on to the scene. But this goes the other way too. Suppose you find that something is conserved. Then the equation describing the experiment *must* have the right symmetry. No ifs, no buts. So you can develop your theory to account for all the observed conservation laws, and discard any ideas that fail to do so.

Anyone who has looked at particle physics at high school will have had a swathe of conservation laws thrown at them - "strangeness", lepton number, flavour, blah blah blah. They come from Noether's theorem. It's that important - giving physicists a whole new way to get a handle on the world.

Mind, you probably still aren't that excited. Oh well.

*Strictly speaking it isn't energy that is conserved, but something called the "Hamiltonian". They're usually the same thing, though - but not always. Keep that one in mind in case it should ever come up...