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…

Three-body tango

Let’s start a new section in physics napkins, called scitoys, for scientific toys. The idea goes as follows: I write down code to illustrate something in physics or maths in an animation. The animation is displayed here, with some explanations and some ideas about further development… So, if you’re a jedi master, you can enjoy the challenges…

This first scitoy is just a running test… the ones I’m preparing are more spectacular. So, in three-body tango, we show three particles interacting through gravitational attraction. I’m pretty sure you all know by now that the two-body gravitational interaction (in Newtonian mechanics) is as regular as a cuckoo clock, while the intrusion of a third body makes it chaotic. You can see in the video how the planets change couple in a chaotic way…

Some technicalities… the program is in C++ for linux with X11. I will send the code to anyone interested and, in due time, I will publish it for all the scitoys. The video capture was done with xvidcap.

A technicality related to this particular program: the gravitational interaction has a short-distance cutoff. I mean: if the planets come closer than a certain minimal distance, I don’t allow the force to accumulate anymore. This avoids some instabilities…

And the challenge: how many perfectly periodic orbits can you find? Are they stable?

With a wink for Miguel Ibáñez Berganza & Daniel Gómez Lendínez

Time travel from classical to quantum mechanics

I would like to return to the time travel questions we posed on this entry. Basically, we want to understand Polchinski’s paradox, which we show in this pic So, imagine that you have a time machine. You launch a ball into it in such a way that it will come out of it one second before. And you are so evil that you prepare things so that the outcoming ball will collide with the incoming one, preventing it from entering the machine. The advantage of this paradox is that it does not involve free will, or people killing gradpas (the GPA, grandfathers protection association, has filed a complaint on the theoretical physics community, and for good reason).

No grandpas are killed, sure, but maybe the full idea of time-travel is killed by this paradox. Why should we worry? Because general relativity predicts the possibility of time-travel, and general relativity is a beautiful and well-tested physical theory. We’re worried that it might not be consistent…

There is a seminal paper by Kip Thorne and coworkers (PRD 44, 1077) which you can find here, which advances the possibility that there are no paradoxes at all… how come? In the machine described above we have focused on a trajectory which gives an inconsistent history. But there might be other similar trajectories which give consistent histories. In fact, there are infinite of them, so our problem is now which one to choose! But let us not go too fast, let us describe how would the “nice” trajectories come.

A possible alternate history: the ball travels towards the machine with speed v, but out of it comes, one second before the collision, a copy of itself with speed v’>v, in such a way that the collision does not change the direction of the initial ball (a glancing collision), but it also accelerates it… up to v’, thus closing the circle! There are no problems with conservation of energy and momentum, since the final result is a ball with speed v…

Thorne et al. described, for a case that was similar to our own, infinitely many consistent trajectories… And the question is left open: is there any configuration which gives no consistent trajectories at all? So far, none has been found, but also there is no proof for this.

And what happens when we have more than one possible consistent trajectory? My feeling is that we’re forced to go quantum! Classical physics is just an approximation. Nature, really, follows all paths, and make them interfere. But if there is a minimum action path, then it, under some conditions, may be the most important one. Quantum mechanics is happy with lots of consistent histories: they would just interfere… And a lot of funny things happen then, but let us leave that for another post…

So, what do you think? It will always be possible to find a consistent history, or not? Are there true paradoxes in time travel?

Why is g so close to π squared?

The hard facts: (a) The acceleration of gravity on Earth is g ~ 9.8 m/s2; (b) π2 ~ 9.87.

The question: Is that pure chance?

The naive answer: Sure. Just change the units, the similarity is gone. Just change the planet, the similarity is gone.

Yet… a little bit of historical research tells us that it is not pure chance. How come?

Of course, if there is a connection between the two values, it must be historical, not physical. The similarity between the two values is just on Earth, and with our units. But how is the meter defined? The definition has evolved with time (and in the US they still use units related to the lengths of their extremities… ains…). For a long time, it was one ten-millionth of the length of the Earth’s meridian. So the relation to the Earth is ensured in the definition, no doubt.

No magic involved, just history. It was the French National Assembly, during the Revolution, defining the meter. They wanted a universal definition, and they came up with that one. But it was not the first one… Before, there were others.

As far as we know, it was the marvellous mind of John Wilkins the first to conceive the idea of meter. And what was his definition? No wonder, the length of a seconds pendulum. That means: a pendulum whose period is two seconds. Now, for a bit of physics, remember that, within the small angles approximation, the period of a pendulum is

T=2\pi \sqrt{L\over g}

Now, imagine that we were using Wilkins’ meter. Then with a pendulum of length 1 length-units, we would have a period 2 time-units. Just solve for g and… hey! You get… π2.

Wilkins’ idea went all the way down to Huygens, and to Talleyrand, who proposed it to the French revolutionaries. Technical difficulties, mostly the fluctuations of length with temperature, made them change the choice, but nonetheless picking up a close value.

Le jour du mètre est arrivé!