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Not a Tomato!

Here is my creation for the Not-A-Tomato … initiative? game? puzzle? – well, for the Not-A-Tomato-whatever-it-is, anyway !

It is a photograph of Einstein playing the violin which I gimp-shopped a little. I found the image using the Creative Commons search tool plugin for firefox. The original photo is hosted at the wikimedia commons at http://search.creativecommons.org/?q=einstein+violin&sourceid=Mozilla-search

Einstein playing the tomato!

For more about Not-a-Tomato, see notatomato.com

Filed in Uncategorized and tagged notatomato 5 months, 1 week ago

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Emmy Noether – Ada Lovelace Day!

This is began just a really quick posting, as I had hoped to be the first to post about Emmy Noether as part of Ada Lovelace day, but I didn’t have time to properly fill this out until I got home!

However, as it turns out, another blogger had already created a great post about Noether, which you can read here.

In it the author discusses a little about Noether’s background, and has a link to the wikipedia article about her, which is a great place to start if you want to know more.

Since the other blogger has already introduced Emmy Noether much better than I could have, I have decided to discuss her life only very briefly, and focus more on her work.

In particular, I’m going to talk about a theorem she proved which links the worlds of physics and abstract mathematics in a very deep way. This theorem is called, naturally enough, “Noether’s Theorem” (*) , and to this day it remains one of my favouritest theorems of all time It is definitely one of the most important and beautiful theorems in mathematics.

Noether was a mathematician early last century. Her work ocurred around the time that quantum mechanics and the theory of relativity were shaking the foundations of our understanding of the physical world. She was a contemporary, peer, and friend of men such as Einstein, Hilbert, and Klein amongst others.

Mathematicians often know her best for her main body of work, which was in the area of modern abstract algebra. Physicists, on the other hand, are more likely to come across her work by way of the theorem I am about to describe.

When I first read about this theorem, during my university traning in physics, I thought it was one of the most wonderful things I had ever learnt. It was at one and the same time singularly beautiful and yet extremely useful. You can imagine how excited I was when I found out the the Noether of ‘Noether’s Theorem’ was a woman!

Most people will have heard of things like ‘the conservation of energy’, and those who studied high school or college physics will have also heard of the ‘conservation of momentum’, and even the ‘conservation of angular momentum’.

We all have an intuitive notion that, during physical interactions, there is a certain amount of book-keeping going on. If something is moving very fast and is suddenly stopped, it tends to make a lot of noise and generate heat. So we might say that the ‘movement energy’ has been transformed into heat and sound energy. Or we could say that there is a balance of energy which needs to be restored. Whatever is lost from one column of the accountant’s ledger needs to be regained in another. The trick is discovering what all the columns are!

If an ice-skater is spinning slowly with their arms spread out, and then they pull their arms in towards them, they will begin to spin much faster… this time, informally, we could say that slower ‘wide spinning’ has been transformed into faster ‘narrow spinning’. Again, there appears to be some kind of balance, a set of quantities which can be traded for one another.

In all these cases, a physicist would say that there exists a ‘conserved quantity’. In the first case above, the conserved quantity is energy. In the case of the ice-skater, the conserved quantity is something called ‘angular momentum’.

Finding out ways to describe and to quantify which physical attributes of bodies in interaction can be ’swapped’ with which others has been a main-stay of physics for a long time. Ever since the incredibly fertile linking of empirical methods with abstract mathematical analysis around the time of Kepler, you could probably say that once you have found out exactly what is conserved, under which conditions, you know everything that can be known about a physical system!

So on the one hand we have the bread-and-butter work of physicists – measuring quantities before and after interaction. Drawing up equations to see how these quantities change, relating these equations to one another and to theories about ‘how the world works’. [ Of course, there's actually a lot more to it than this, but I am trying to give the simplest description I can! ]

This is all wonderful stuff, and we could say a lot about physical systems using the techniques, but we wouldn’t expect to say much of interest about topics such as one-dimensional Lie groups of transformations.

On the other hand, we have a purely mathematical world, the abstract theory of symmetries and group theory. In this world we can describe the ’symmetries’ of ’sets of permutations’, we can wonder about the possible roots of polynomials, and much more. But we wouldn’t particularly expect to have anything very interesting to say about, say, a spinning ice-skater.

Noether’s theorem bridged these two worlds, producing an understanding in which statements made in seemingly seperate fields could be related. Naturally, this new way of understanding things had a profound influence on the development of physics throughout the rest of the century and beyond.

The theorem states, roughly, that conserved quantities in physical systems (energy, angular momentum and so on) are in reality the reflections of symmetries in certain deeper mathematical structures underlying them (**).

Thus, the theorem makes a connection between these two apparently disparate areas, and shows that not only that they are connected, but also describes in precisely what way they are connected.

The lovely thing is that we discover that talking about the symmetries of a physical system and talking about what kind of ‘book-keeping’ that system uses are in an important sense both just different ways of saying the same thing.

That’s all for now, I hope to refine this post a little more and perhaps add a couuple of images at a later date

(*) Also sometimes known as “Noether’s First Theorem”.

(**) A reasonably exact wording, as I recall it from my physics training, is something like this: “To every continuous symmetry in the Hamiltonian of a Dynamical System there corresponds a conserved quantity”. Wikipedia gives the following informal wording: “To every differentiable symmetry generated by local actions, there corresponds a conserved current.”