Emmy Noether

March 8th is the international working woman’s day, so I guess it’s just fair to write a blog entry about my favourite woman physicist… which happens to be Amalie (Emmy) Noether. I will not focus so much on her life, but on the most wonderful theorem on mathematical physics imagined by human minds, which was her brain-child…

About her life, I will only remind you that she was the first woman teacher at the University of Göttingen, recruited by Hilbert and Klein, in 1915. Göttingen was the most important center for theoretical physics at that time. It took a lot of arguing… One faculty member said “What will our soldiers think, when they come back home and are asked to study at the feet of women?”, and Hilbert gave his famous response: “This is a university, not a bath house”… Being a jew and socialist, she had to flee from Germany when Hitler came to power, and escaped to Russia and then to the US… You can read Wikipedia and many other sources for more info.

About her work… well, for me, the most impressive result of mathematical physics is known as Noether’s theorem, I’ll try to explain it in simple terms: if your physical system has a symmetry, then it has a conserved quantity. Conservation of energy is due to the invariance under time translation: physics is the same today or tomorrow. Conservation of momentum, due to invariance under spatial translations: physics is the same here, in Vladivostok or in alpha-Centauri. And so on. How come? I’ll try to give a derivation that makes you feel the thrill, yet does not get stuck in technical details…

Let us consider the space of all possible physical configurations of a system. In classical mechanics of point particles, a configuration is specified when you give all the positions and momenta, so a point in it will be given by x=(q_1,q_2,\cdots,p_1,p_2). Time-evolution is a flow in this configuration space. A flow is just putting a vector at each point of space, indicating the direction and speed with which you should move if you’re there. But there are many other interesting flows in configuration space, which correspond to other operations different from time evolution. You might consider the flow induced by rotating the whole system, or translating it, or stretching it…

All of those flows can be expressed in terms of generating functions. Consider any scalar function defined on the configuration space,  f(x). Its flow is defined in the following way. Get the gradient, \nabla f, which is a vector field. You might consider it to be the flow, but it is not convenient. We apply on it a certain matrix, call it ω, the symplectic matrix. This way, the flow of a function f is given by u=\omega \nabla f.  The only thing that you need to know about ω is that ωu is always perpendicular to u. If you move along a direction which is perpendicular to the gradient of a function, you keep the value of that function constant, right? So, moving along the flow \omega\nabla f preserves the value of f. The flow of f preserves f.

Now, apply this story to time evolution. Its flow is induced by the hamiltonian: u_t=\omega\nabla H. Of course, this means that time evolution will preserve the value of H. OK, we knew that! The equations of motion are

{\dot x}={\partial x\over \partial t}=\omega \nabla H(x)

What about other flows? Since I’m trying to keep things non-technical, I won’t prove the following assertions. Spatial translations are generated by the momentum f(x)=p. Rotations are generated by the angular momentum (on the z-coordinate, say): f(x)=L_z=yp_x-xp_y… What does it mean? Let’s say that you’re rotating your system by an angle α around the z-axis. You want to know the position of all the particles after such a rotation. Then, you get the “equations of motion”:

{\partial x\over \partial \alpha} = \omega \nabla L_z(x)

Let’s say that we want to know how one of these functions f evolves with time. Then, we derivate that thing with respect to time:

{\partial f\over \partial t}= {\partial f \over \partial x} {\partial x\over\partial t} = \nabla f \omega \nabla H

This object is important, so we give a name to it, the Poisson bracket, {f,H}.

So,  {f,g} means “how evolves f under the flux induced by g. Its main property is that {f,g}=-{g,f}, because of the properties of ω.

Now, Emmy Noether’s magic in action. Let us say that f is a symmetry of the system. This means that the hamiltonian does not evolve under the flux induced by f. So, {H,f}=0. But then, {f,H}=0 also! And this means that f does not change under the flux induced by H, i.e: under time evolution. So, f is a conserved quantity!

And this is Noether’s theorem: for every continuous symmetry of a system, there is a conserved quantity. It is, of course, the generator of that symmetry. If you have translation symmetry, momentum is preserved. Rotation-symmetry: angular momentum is preserved. For more intrincate symmetries, there are more abstract conserved quantities. For example, the esoteric gauge symmetry explains, via Noether’s theorem, the conservation of charge! And the conservation of energy? That’s the easiest, it’s just the symmetry under time-evolution…

For more info, besides Wikipedia (not the best site…), check John Baez’s explanation, or this page, or any good book on classical mechanics.

OK, this was a tribute to my favourite woman physicist of all times… But,  as of today, I also want to pay tribute to the ones I’ve met in my life: Silvia, Pushpa, Mar, Lourdes, Carmen, Nuria, Lola, Nina, Sagra, Elena, Vanessa, Susana, Rosa, Arantxa, Diana and all the rest…

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