∫ Emmy Noether Was a Wonderful Woman, and Now I Have the Proof

In classical mechanics today we talked about Noether’s theorem, which is one of the coolest little things in physics. Ms. Noether, nee Noether (she never married), was an absolutely brilliant woman, often regarded as one of the greatest mathematicians of the 20th century, and certainly one of the greatest female mathematicians ever. Her work spanned a wide, wide variety of topics, partly because her gender made it incredibly difficult for her to find profitable work, and so she would up attaching herself to whichever groups would take her.

Emily Noether in 1930

To hear Dr. Van Nieuwenhuizen tell it (and I don’t know for sure if this is accurate) it seems that, after years of working without pay after finishing her thesis, she was offered a job by David Hilbert, another of the century’s greatest mathematicians and owner of a very strange hat. Unfortunately, Hilbert hadn’t exactly cleared it with the university first, and so when she showed up, they wouldn’t make her a professor, so he paid her to give lectures at courses taught in his name instead. To keep herself busy on the research end of things, she started doing work for a theoretical physics group at the school, who would come to her for her mathematical expertise to ask her to help solve problems without giving her any of the context. Fortunately, the fact that she only ever saw the mathematics and wasn’t distracted by the physical quantities involved allowed her to come up with some brilliant, overarching statements about physics from the patterns she saw popping up (which the fully-paid professors kept overlooking).

You can read all about her on her wikipedia page. It’s full of sad stories about how hard it was for her to find work and how, once she finally gained some respect in the academic community in spite  being persecuted for her gender, the Nazis gained power and started persecuting her for her Jewish heritage. But what I want to talk about at the moment is this excellent theorem of hers.

So now you have a choose-your-own adventure game to play; If you already know what Noether’s theorem is, you should skip ahead to the mathematical section, marked with the triple integral (∫∫∫). But if you don’t you should stick around for a minute and I’ll give you a rough overview.

In a nutshell, Noether’s theorem is the statement that every time there is something symmetric about the laws of physics, there is also a conserved quantity to be found somewhere, and vice-versa (this vice-versa gets overlooked a lot the first time this is taught to people, but it’s critically important).

For example, the laws of physics are symmetric under changes in time: if I do some experiment tonight, and then do it again tomorrow, well then I may be a nerd but I will be a nerd who got the same result from his experiment two nights in a row. That’s what physicists mean by a “symmetry” by the way: something you can change without affecting the physics. In every day language, “symmetry” usually means “reflection symmetry,”  meaning that two sides of something (like Pierce Brosnon’s face) look the same. But that’s just a special case of the physicists’ definition: You can look at one side of his face, and then change the side you’re looking at, and you’ll be attracted to him either way.

Back to the matter at hand: the world is symmetric in time, and Noether’s theorem tells us that, because of this symmetry, energy is conserved. You’re probably familiar with the phrase “conservation of energy” but just in case, I’ll recap: you can’t make energy, and you can’t get rid of it. You can burn gasoline to run a car’s engine, and that might make it look like you were making energy out of gasoline, but no, the energy was trapped inside the gasoline all along, it was busy holding the carbon atoms together.

This as deep philosophical consequences for the way we think about physics. For example, it means that conservation and temporal symmetry are not two different concepts. They are two manifestations of the same thing. I find this quite satisfying, because the fact that the world is symmetric in time is very natural and common sense (it would be really hard to live in a world where the laws of physics kept changing all the time!) but the fact that energy is conserved isn’t immediately obvious to me. We could live in a world where that wasn’t true. So that’s always left the nagging question of “why should energy be conserved”? And now the answer is, ” it has to be if you want the laws of physics to be the same from day today. And you do want that, don’t you?”

The other thing it does is give us a way to define energy. Believe it or not, it’s a very hard thing for physicists to define energy in a precise but uncomplicated way. It’s like asking a mathematician to define what a number is: it’s used so commonly that it’s just hard to think of energy as anything other than, well, energy.

I had a professor at CU who once told me a story about the time he ambushed his thesis advisor out of the blue and asked him to define energy. The poor guy thought for half an hour before answering “Energy is the scalar quantity that isn’t electric charge which is conserved.” Now that’s not a very satisfying definition, but it’s not too far from the way I think it needs to be defined. Energy is simply too basic a thing to define in the usual way; I can’t tell you what it’s made up of or where it comes from, so how am I to say what it is? Noether’s theorem gives us a way out of this problem: Energy is the thing which has to be conserved as a consequence of the fact that the laws of physics don’t change as time passes.

You’ll notice I have yet to use a single equation or invoke mathematics at all except to use the word “number.” This is part of what’s amazing about Noether’s theorem too. The first time I heard it, I was absolutely astonished by how deep the implications it had were, but just as amazed by the thought that such deep, philosophical insights could be the result of a mathematical proof. It seems like it shouldn’t be so, but it is. And this poor woman, after fleeing Nazi persecution in Germany and coming to America, had to take a job a, no, not Princeton or Harvard or something, but at Bryn Mawr College. Which I’m sure is a fine school. As they say out here in and around Lawn Guy Land, “No disrespect.”

But although Noether’s theorem and its consequences can be explained nicely without math, the thing that caused me to want to blog about it was the cool proof of it we saw in class today, so I’m going to move along to that. Which unfortunately, really can’t be done with out getting pretty darn technical.

∫∫∫

First, we’d like to welcome those of you rejoining us from earlier in the post. And now for the technical details.

The proof for the case of time-translation symmetry, as discussed above, is actually fairly trivial, since in classical mechanics time is a special coordinate distinct from space (I still don’t know how to prove Noether’s theorem in a general relativistic environment). So what remains is to show that an arbitrary spatial symmetry produces a conserved quantity. Now, Noether’s theorem is usually proven using the Lagrangian formalism of classical mechanics, and indeed, when Dr. Van Nieuwenhuizen asked us to think about how to prove it in class last time, this was exactly what I did, having seen the proof hinted at before. Basically you equate your original Lagrangian and a Lagrangian with transformed coordinates, and take some derivatives with respect to the parameter which defines the transformation. Then you use Lagrange’s equations to subistitute some terms, and sneaky derivative terms to re-write others, until you have a total time derivative of some quantity which is equal to zero, and you just have to identify that quantity with something physical. It’s not hard, it’s just a lot of clever applications of the Liebnitz rule and it’s not particularly instructive.

But then in class, Dr. Van Nieuwenhuizen showed us this gem, which you get if you want to prove Noether’s theorem from the Hamiltonian formalism. This proof does require one sneaky piece of notation, called the Poisson Bracket, an antisymmetric operator defined as:

[ ( f(p, q), g(p, q) ] = \frac{ \partial f}{\partial q} \frac{ \partial g}{\partial p} - \frac{ \partial f}{\partial p} \frac{ \partial g}{\partial q}

Now, We’ll do the proof in the opposite direction that usually happens. Consider some system with a Hamiltonian H(p,q,t) (if it is a higher dimensional space, put indicies i = 1 ... n on every p and q and then sum over them everywhere they are repeated in the derivation below.) Now consider some quantity, any quantity, that depends on the (spatial) coordinates of the system. Call it G(p, q).

Assume G is conserved, that is,

\frac{d}{dt}G(p,q) = 0

But by expanding \frac{d}{dt}G(p,q) = 0 via the chain rule we have

\frac{d}{dt}G(p,q) =\frac{ \partial G}{\partial q} \frac{ \partial q}{\partial t} + \frac{ \partial G}{\partial p} \frac{ \partial p}{\partial t} +\frac{ \partial G}{\partial t} = 0

Invoking Hamilton’s equations that \frac {\partial H}{\partial q} = -\dot{p} and \frac{\partial H}{\partial p} = \dot{q} and using our cool new notation we have:

\frac{d}{dt}G(q,p) = [G,H] + \frac{ \partial G}{\partial t}

Consequently, this quantity G(q,p) is conserved if and only if [G, H] = 0.

But what else can we say based on the fact that [G, H] = 0? Well this quantity G can be used to define a specific coordinate transformation, namely

\delta q = [q,G]
\delta p = [p,G]

But look, the variation of the Hamiltonian under this action is simply:

\delta H = \frac{\partial H}{\partial q} [q,G] + \frac{\partial H}{\partial p} [p,G]

Writing out the Poisson bracket in full and then simplifying, we find that

\delta H = \frac{\partial H}{\partial q} \frac{\partial G}{\partial p}  + \frac{\partial H}{\partial p} \frac{\partial G}{\partial q} = [H,G]

And we had just decided that [H,G] = 0.

Note that all these steps are reversible (iff statements) and we arrive at a wonderful state of affairs, with three statements, all of which are equivalent:

  1. \frac{d}{dt} G(p,q) = 0
  2. [G,H] = 0
  3. \delta H = 0 when \delta q = [q,G] and \delta p = [p,G]

The familiar Noether’s theorem is the result of the fact that 1 implies 3, and vice versa. If we have some conserved quantity G, we can find a coordinate transformation such that the Hamiltonian is unchanged; this is our symmetry. I haven’t shown the details here but from group theory you can show the reverse: given any transformation for which the Hamiltonian is invariant, you can construct G, the generator of the transform, and then that G turns out to be conserved. Pretty cool, eh?

Oh, and yes, in many ways that linkage step, statement number 2, is indeed the coolest part of the theorem. And yes, it should remind you very, very strongly of something you know about quantum mechanics.

And yes, I should probably go do something less dorky before I go to bed. Goodnight all.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

2 Responses to ∫ Emmy Noether Was a Wonderful Woman, and Now I Have the Proof

  1. Anna says:

    That is a marvelous proof. It’s such a great example of why I love mathematical physics: we fix some portion of reality onto some limited universe of symbols which obeys some very simply and clearly delineated rules which at first glance have absolutely nothing to do with reality. Then we shuffle the symbols around in their little universe according to the aforementioned rules and at find out something about reality which- and this is the wonderful thing- was not necessarily present in the original meaning we gave to the symbols.
    But the thing is, that process is more or less true of any remotely successful mathematical description of anything, whether it’s a probablistic description of of the stock market or a path-integral description of weak decays. So why is it that the likes of Noether’s theorem strike me as deep and wonderful in a way that other mathematical descriptions do not?
    I can shrug this off partially with a comparison of how successful the descriptions are: a particle physicist has to increase the size of their error bounds so that they are visible to the naked eye, whereas an ecologist is delighted if their model accounts for even 50% of some observed phenomenon.
    I can’t, however, shake the sense that certain corners of physics occupy (or perhaps just approach) a space where the mathematics is connected more deeply, and the seemingly artificial rules of calculus and group theory and whatnot are not merely tools or approximations.
    It is however, also possible that given that it is way past my bedtime, none of this actually makes any sense.

  2. S says:

    Damnit, Colin. That seems really cool. I wish I had a day to play around with the math to properly understand the implications.

    So because everything is conserved, you can never create or destroy spin-quanta? That’s pretty cool and directly demonstrates why particles have to be moving so fast in order to break them. They have to get close enough to interfere destructively with the smaller and stronger forces governing objects at the next level down. I’ve always wondered if the relationship between ‘fundamental forces’ was fractal: gravity ~ weak && magnetism ~ strong. That would really frustrate the math. It could also, in theory, explain quantization. Any sub-quanta energy would implicitly get absorbed by smaller and stronger particles working below the current level any time particles interacted.

    P.S. – I cheated using wikipedia a little: “Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.”

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