sábado, maio 12, 2018

Lagrangean Systems: "Levels of Infinity - Selected Writings on Mathematics and Philosophy" by Hermann Weyl, Peter Pesic




“It is a well-known anecdote that Hilbert supported her [Emmy Noether] application by declaring at the faculty meeting, ‘I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bathing establishment.´”


In the memorial address “Emmy Noether (1935)” delivered in Goodheart Hall, Bryn Mawr College, 26 April 1935, and included in “Levels of Infinity - Selected Writings on Mathematics and Philosophy” by Hermann Weyl, Peter Pesic

Mathematics is, in a sense, profoundly anarchistic - you can't use authority to change or control its progress, and nothing is ruled in our out without proof agreed by the collective of practitioners, and Weyl was one of our most distinguished practitioners of the art of doing beautiful mathematics and physics. Sometimes practitioners have a brave and frankly generous stab at letting the layman get a feel for some of the broader concepts, but ultimately this is an intellectual edifice that's been built by thousands of people over the last five centuries or so and there's no reason whatsoever that we should be able to understand it at all without putting in the hard yards - the problem is not with math, it's with us and our arrogance in assuming that's possible. Weyl, as this homage book testifies, was able to put math into language people could understand and it's absolutely essential for a general audience. Language needs to be a vehicle of understanding and not an obstacle to it.

What amused me as an engineer is how engineers are taught many mathematically valid shortcuts that they use to solve many problems, while mathematicians are not taught them. Then again, how engineers and mathematicians interpret the ideas expressed in the mathematics that they use is obviously different, so perhaps although I find it amusing it is not particularly important in the greater scheme of things, (if there is a scheme). Of course, we do get taught be shortcuts, but only in the context of understanding exactly where they break down. We engineers get to live in a world of 'nice' functions where we can do things like differentiate under the integral or assume sin theta equals theta without getting too antsy about it...

I'm glad both Hilbert, Einstein and Weyl made a top shout out to Emmy Noether! She proved one of the most important and foundational results in modern physics - in a just world she'd be as well-known as Einstein, but (a) she was a woman and (b) there's no easy way to explain what she did with a glib pop science metaphor...but after having read Weyl's kind of mathematical eulogy for her, and because today is woman's day (8th March), I'll just have to give my two cents... 

Noether proved it as a theorem specifically about physical systems. It only works because the physics is fully determined by a Lagrangean which is minimised. And if that Lagrangean is covariant under a continuous symmetry (e.g. spatial translation) it leads to a conserved quantity (e.g. momentum). If the system cannot be described by a Lagrangean whose action is minimised then Noether's Theorem does not necessarily hold. Noether showed that physics being the same whatever time it is leads to Conservation of Energy. Being the same regardless of your position leads to Conservation of Momentum and being the same no matter what direction you look at leads to Conservation of Angular Momentum. All of which are examples of a symmetry which results in a conserved quantity. I'm not sure it really requires the usual glib metaphors to explain, most people have heard of Conservation of Energy and Momentum. You can explain Conservation of Angular Momentum by the usual example of a skater rotating faster as they pull their arms in. And the idea that physics is the same at all times and places and whatever direction you look at should be straightforward to understand with a small amount of thought. The extraordinary thing is that it isn't a particularly complicated proof and isn't really about physics particularly. What is surprising is no one discovered it earlier. Even Newton had the mathematical tools to do so. That he and none of the succeeding two centuries of mathematicians did suggests she had a special talent. Maybe because she was really a mathematician where she is famous for solving much more difficult problems. But it is strange nevertheless that Noether's Theorem isn't more famous. Certainly up there with Einstein's Special Theory of Relativity. And of course is widely used in theoretical physics today.

It is still important today because the basis for any theory of physics such as particle physics is also a Lagrangean whose action is minimised. If that Lagrangean is covariant under a continuous group then there is an associated conserved quantity called the Noetherian current. Another conserved quantity which can be explained by Noether's theorem is conservation of electric current as a result of phase symmetry in the wave function of quantum mechanics.

As always the ghost of Emmy Noether, one of the greatest mathematical physicist of the 20th century for her work on symmetry and conservation of quantities (energy, momentum, angular momentum), presides over all. It is a pity she was never awarded a Nobel Prize of her own. I would describe Noether's work as (a) mathematical physics for her work on symmetry and conservation and (b) pure mathematics, for everything else. For her work on symmetry alone she deserves to stand in the pantheon of great mathematical physicists. Both for its insight and subsequent centrality to modern particle physics and quantum mechanics.

Thanks Hermann Weyl for doing what you did at the time.



NB: The essay on Noether, along with the essays “The Mathematical Way of Thinking” (1940), and “Why is the World Four-Dimensional?” (1955, the year Weyl died), on their own, are worth the price of admission.


2 comentários:

Luís Filipe Franco disse...

It's been so much time since I have heard talk about Weyl... I believe it was during the classes in Diferential geometry on the math graduation on Ist. Amazing memories!

Manuel Antão disse...

I'm glad you liked it. When I finished my college degree I did a (Theoretical) Physics at FCL class and I finally understood what the Noether theorem was all about...