We’ve all seen how a certain section of the populace of our universities decry and dismiss the contributions of “dead white men”. These miserable party-poopers tend to be studying useful things like The Relationship Between Microwave Ovens, Systemic Racism, and Afro-Ethnic Consciousness1.
Today, I want to talk a little bit about someone for whom we can say “well, 2 out of 3 ain’t bad”.
Emmy Noether, who was white and who died in 1935, happened to be a woman, of all things.
And what a remarkable woman she was. She is, without question, one of my scientific heroes - despite being a mathematician.
I’m not a mathematician and so I can’t properly assess the significance of her work for mathematics, but I understand it to have been profound. From the wiki entry on Noether :
“The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries” (Nathan Jacobson - a distinguished American mathematician)
In 1918 she published a paper that has had the profoundest effect on physics. In it she proved a theorem that has become known as Noether’s Theorem. It is hard to overstate the significance of this theorem. It’s difficult to say which paper has been the most “important” work in physics of the 20th century, but Noether’s paper would easily make the shortlist.
In this paper she discovered a deep connection between symmetry and the conservation laws of physics.
Most of us, particularly if you’ve been on the blob/diet merry-go-round, are aware of conservation laws; eat less and move more being the kind of unthinking slogan that results. Whilst this is actually very good advice, it is part of a disastrous oversimplification of how we get into trouble with weight and find it so difficult to shift.
This slogan expresses the idea of energy conservation as it applies to food. There’s energy in food and if you don’t use it all up you’ll end up with thunder thighs, goes the simple picture. OK - we get it - stop shovelling the donuts in and walk a bit more - but not many people are aware that energy conservation arises because of a time symmetry.
Most of us have an intuitive grasp of symmetry; we’re aware that a square has a rotational symmetry, for example. If you rotate a square through 90 degrees (about its centre) you end up with something that looks identical to what you started with. A square also has a reflection symmetry; bisect it and the two pieces look identical, you can “fold” them onto one another.
But when physicists talk about “symmetry” they often mean something a bit more than this, and this takes a bit of explaining.
We first need to understand a bit about what a coordinate system is. Whether or not we realize it, or use such fancy technical terms, we all use these things in our daily lives without thinking about it. You’re driving somewhere and you see the sign telling you there’s 10km to go to your destination. That’s making use of a coordinate system.
These things can be thought of as a way of “labelling” our world so that we can communicate important information - like how far we still have to travel, how long it will take us, etc.
There’s no really fundamental importance to a kilometre, for example. It’s just a length we’ve all agreed to use. Provided we all use the same notion of what a kilometre is, we can use it to convey useful information.
So, we imagine Alice creating a labelling system so that she can talk about things like distances and times. We could imagine a whole network of rulers and clocks extending from where Alice is. She lays everything out and fixes her “start” points - which are often taken to be the zero of distance (where Alice is) and the zero of time (she presses the stopwatch). This is Alice’s coordinate system.
If “something” happens 10 metres away from Alice, 3 seconds after she started her stopwatch, then she will assign the coordinates (10,3) to this.
But Bob can do the same thing - and he might be in a different location. Naturally, he’s going to want to fix his coordinate system with him sitting at his zero point. He might even press his stopwatch at a different time.
He might give the same event that Alice saw the coordinates (15, 4).
The idea here is that when Alice and Bob write everything down in mathematical terms they’re going to have different coordinates - and we might write (x,t) for Alice’s version and (X,T) for Bob’s version.
Alice and Bob have both been to university and, wisely, avoided studying about the relationship between microwave ovens and systemic racism. They’ve learned a bit about the laws of physics instead.
Alice is going to use these physics laws to describe what she observes - and she’s going to write everything in terms of her x and t. Bob is going to do the same, but he’s going to be using his choice of X and T.
The thing is, is that there’s a relationship between Alice’s (x,t) and Bob’s (X,T).
In the above example we can infer that Bob is sitting 5m away from Alice and so the relationship between their distance coordinates is going to be
X = x + 5
We call this kind of ‘rule’ a coordinate transformation.
Where does the “symmetry” come in?
The fundamental idea here is that when Alice writes down the laws of physics in terms of her x and t she’s going to have a bunch of equations. Bob is going to have a bunch of equations in terms of X and T.
They have to look the same (mathematically).
If we use the rule X = x + 5 to “transform” between Alice’s coordinates and Bob’s coordinates we have to end up with the same physical laws - the idea being that the laws of physics don’t change because we’re in a different location (or have used a different set of clocks).
That’s the “symmetry” - the invariance of our mathematical description when we change between the coordinate systems.
Newton’s Laws will look the same when written in terms of x and t as they do when written in terms of X and T.
It turns out that this is much more than just mathematical or notational nicety. If we’d like our laws of physics to be the same at both ends of the lab bench then the laws themselves must have this kind of invariance property. In more practical terms, if we do an experiment at one end of the bench we do not expect to get a different result when we do the same experiment at the other end of the bench.
This places an important constraint on the kinds of laws of physics we can have.
This kind of reasoning about the invariance of physical laws is what led Einstein to his theories of relativity (both the special and general theories). There was a problem with Maxwell’s laws (for electromagnetism) back in Einstein’s day; these laws did not have this required mathematical invariance property. It was really Einstein who said - the laws aren’t the problem, it’s how we ‘transform’ between Alice’s and Bob’s sets of coordinates that is the problem.
It’s true that others (notably Lorentz) had figured out what kind of coordinate transformation was necessary before Einstein, but it was Einstein who properly explained just why it had to be so.
Then Emmy Noether took things even further.
She showed that the conservation laws of physics were a consequence of these kinds of symmetries.
In simple terms :
The fact that we get the same result when doing an experiment at either end of the lab bench implies that momentum is conserved (symmetry under linear translation in space).
The fact that we get the same result when we rotate our experiment implies that angular momentum is conserved (symmetry under rotational translation in space).
The fact that we get the same result for the same experiment done 10 minutes apart implies that energy is conserved (symmetry under translation in time).
The symmetry of the laws of physics under a linear translation in space is why momentum is conserved. For each of these kinds of symmetries there is an associated conservation law - and this is what Noether proved.
Amazing, deep, beautiful, stuff. She uncovered something so very profound about the way our universe works.
She did all of this, her profound mathematical work and her spectacular contributions to theoretical physics, whilst facing very severe discrimination because of her sex. The world’s pre-eminent mathematician, David Hilbert, wanted her to work at Göttingen, that astonishing powerhouse of mathematics that had given a home to such useless dead white men as Gauss and Riemann. The men in other departments were not at all pleased by having their superlative maleness sullied by a mere woman and Hilbert had to “game the system” by advertising her lectures as his own so that she could be paid. It took 4 years before she was finally given an official university position there.
Noether was, by all accounts, remarkable not only for her brilliant, brilliant academic contributions. She was known, for example, to let her students take credit for work that she had (mostly) done in order to help their careers. From Noether’s wiki page, a colleague described her as being,
Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all
I could compare and contrast the discrimination Emmy Noether faced with the discrimination that women in STEM today are, still, said to face, but that would be churlish of me. But I do very much regret the foolishness and short-sightedness of our past societies.
How many Noethers have been lost to time because of the shitty attitudes of men?
I just made that title up. But you can easily find real examples throughout the “grievance” and “humanities” literature of even dafter things.
Thank you sir. What a marvellous woman and a tragedy that she died so young. A cautionary tale too, in that she was discriminated against for being a woman and a Jew. The tens of thousands of imbeciles marching in the street in support of Hamas at the moment aren't worth a single toenail clipping of a woman like Dr Noether.
Your discussion of coordinate transformations also rang my bells. A radar working on a ship at sea needs to have the ships motion (pitch, roll, yaw etc) removed from the data stream so that the operators are presented with a stable view. All of this is accomplished through a series of linear and rotational coordinate transformations implemented by a bit of matrix maths. Wonderful stuff.
If someone enjoys doing something, and shows an affinity for it, and it’s something that benefits us all (mankind, that is)— I’ve never understood why we would care what that person looks like. I like to fancy that because I don’t understand it now (in terms of identity politics) I wouldn’t have then (in terms of prejudice), but who knows?