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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Life versus math

(Dedicated to Michele)

The main thesis of this article is the following: hiding from life is still living. In other words: you can choose your life (maybe), but you can not escape from life.

A good friend of mine uses to think that physics (the way he puts it, but I could also say maths, or hacking…) is opposed to life. I’ll start by supporting his argument, with a quotation from Joseph Weizenbaum (creator of ELIZA) talking about the MIT hackers:

…bright, young men of disheveled appearance, often with sunken glowing eyes, can be seen sitting at computer consoles, their arms tensed and waiting to fire their fingers, already poised to strike, at the buttons and keys on which their attention seems to be as riveted as a gambler’s on the rolling dice.  When not so transfixed, they often sit at tables strewn with computer printouts over which they pore like possessed students of a cabalistic text.  They work until they nearly drop, twenty, thirty hours at a time.  Their food, if they arrange it, is brought to them: coffee, Cokes, sandwiches.  If possible, they sleep on cots near the computer.  But only for a few hours—then back to the console or the printouts.  Their rumpled clothes, their unwashed and unshaven faces, and their uncombed hair all testify that they are oblivious to their bodies and to the world in which they move.  They exist, at least when so engaged, only through and for the computers.  These are computer bums, compulsive programmers.  They are an international phenomenon.

Joseph Weizenbaum, Computer power and human reason (1976)

There is a trend to think that hackers of any kind, let them be computer programmers, mathematicians, physicists… want to escape from reality, that’s why they create their small world in which they are almighty, their own Asgard, a world of purity. Some of them may even think that they’re escaping from life, but life can not be escaped. It is a natural trend within life!

Mathematics is not as pure as mathematicians think. Arnold used to explain that there is no such thing as good maths outside physics. Otherwise, it is just an intellectual game, and not a quite funny one. Physics is not so pure either, but that is easier to grasp after the Manhattan project. About the hackers Weizenbaum talked about… they thought they lived in a clean world there in the MIT AI lab, but it was funded by the US military…

The world is imprevisible and chaotic. Often, this is nice and fine. But sometimes, when you feel in pain, you’d like some cozy environment where you can hide from the stupid people outside. Maths, physics… can be such a place. Sometimes. Life has resources for everything…

More about Euler’s crazy sums

Recently we talked about how Euler managed, through some magic tricks, to find that the sum of the inverse of the square numbers is \pi^2/6… That was a piece of virtuosismo, but even geniuses sometimes slip up. This one is funny. You may know the sum of a geometrical series:

1+x+x^2+x^3+\cdots = {1\over 1-x}

(There are many ways to understand that formula, I’ll give you a nice one soon). Now, using the same formula, sum the inverse powers:

1+x^{-1}+x^{-2}+x^{-3}+\cdots = {1\over 1-x^{-1}} = {x\over x-1}

So far, so good. Now, imagine that we want the sum of all powers, positive and negative:

\cdots+x^{-3}+x^{-2}+x^{-1}+x^0+x^1+x^2+x^3+\cdots = {1\over 1-x} + {x\over x-1} -1 = 0

Amazing!!! The sum of all powers is zero!! Nice, ein? Euler concluded that this proved the possibility of the creation of the world from nothing, ex nihilo. So, this is easy… what is the problem with this proof?

BTW, I read this story and many others in William Dunham’s book The master of us all

I'm better than you all even when I'm wrong!

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!

 

Euler’s crazy sums

Rigour is the hygiene of the mathematician, but it is not its source of nutrients… Here you have a fantastic piece of work by Leonhard Euler, where he showed how to reason mathematically, maybe without rigour, but with a rich and healthy intuition.

In the early days of calculus, people were fascinated by power series. Euler knew very well the Taylor series of the sine around zero:

\sin(x)=x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}+\cdots

OK, so the sine function, in a sense, is a polynomial… Well… We know a few things about polynomials, don’t we? For example. If P(x) is a degree 3 polynomial and its roots are x_0, x_1 and x_2, then:

P(x) = K (x-x_0)(x-x_1)(x-x_2)

And the constant can be fixed if we know a single value, for example, P(0). So… let us do it with the sine function. We know its zeroes: n\pi for all integer n:

\sin(x) = K \cdots (x+3\pi) (x+2\pi) (x+\pi) x (x-\pi) (x-2\pi) (x-3\pi) \cdots

A little bit of regrouping:

{\sin(x)\over x} = K (x^2-\pi^2) (x^2-(2\pi)^2) (x^2 - (3\pi)^2) \cdots

Now, what can the constant be? If x=0, then \sin(x)/x is 1. A trick and we get rid of the K:

{\sin(x)\over x} = \left( 1 - {x^2\over \pi^2}\right) \left( 1 - {x^2\over (2\pi)^2} \right) \cdots

Waw, a remarkable formula on its own! You can check it numerically: it converges slowly beyond the first maximum, but it converges indeed. But we can get more from it. It should coincide with the Taylor series of the sine, right? (hm…) The constant term is easy, just 1. The quadratic term is not too hard either. Just add up all the products which consist of all ones and a single x^2 term. You get:

-{x^2\over \pi^2} - {x^2\over (2\pi)^2} - {x^2\over (3\pi)^2} - \cdots = -{x^2\over 3!}

to make it equal to the Taylor term for x^2. Now, make the coefficients equal:

-\sum_{n=1}^\infty {1\over n^2 \pi^2} = -{1\over 3!}

Or, equivalently,

\sum_{n=1}^\infty {1\over n^2} = {\pi^2\over 6}

Waw! Of course, there are rigorous ways to prove this formula. The first one I learnt was using Fourier series. But this one is funny, isn’t it? :)

Not all of Euler’s crazy sums were right. Some were amazingly wrong. But the wrongs of the genius are also funny… So I will soon discuss them here.

Happy summing!

Why can’t I kill my grandpa? (Time travel, part I)

So, this is the first post that we will dedicate to the question of time-travel in physics. We’ll start easy, but things may get pretty confusing soon, so behold!

Of course, we’re all time traveling, right now. We’re traveling towards the future, at a rate of one second per second. Strange speeds in our time travel appear as early as the Mahabharata, when king Kakudmi visits lord Brahma for some chat and, when he returns, many years have gone by. Yet, travel to the past appears later in stories, and mostly for the pleasure of anachronism. The time machine appears by the end of the XIX century in a short story from a Spanish writer, Enrique Gaspar y Rimbau, el “Anacronópete”, where the theory is exposed that it is the atmosphere causing the flow of time, as can be checked by the conservation of food in hermetic cans… His machine travels to the past much like Superman, flying against the rotation of the Earth.

The first story to deal with the problems of time travel to the past seems to be
Tourmalin’s Time Cheques, by Thomas A. Guthrie, under pseudonym in 1891, which I can’t discuss yet… (it’s in my reading list).

To the best of my knowledge, the first story which shows the problems and paradoxes of time travel to the past in its full glory is By his own bootstraps, by Robert A. Heinlein, in 1941. If you enjoy discussion about these topics, you really should read that short story.

The first and foremost paradox of time-travel is the grandfather murder case. I travel 50 years back in time and kill my grandfather before he meets my grandmother… so I can’t be born, and can’t kill my grandfather. So, if I do A, I force not-A, which forces A… what is the way out? Somehow, something should prevent you from killing your grandfather, so that history remains coherent.

We physicists love to give a name to everything, so we’ve baptized it as the Novikov principle. History should be coherent. Perhaps, after all, I do not have free will, I can’t choose to kill my grandpa… You see, the paradox with people gets somehow out of focus. Let us put it up simply will balls. This way, we call it Polchinski’s paradox:

We have a time-machine which has an input slot, an output slot and one dial. If you put something in the input slot, it will come out of the output slot some time before given by the mark in the dial. OK. Now, we put the dial to “1 second” and throw a ball towards the input slot. The same ball will come out of the output slot 1 second before the original one hits the input slot, OK? Now we can fix the geometry so that the second ball hits the first and puts it out of the way. So the output ball will prevent the input ball from entering the machine and, therefore… where did the second ball come from?

Polchinski's paradox

There are ways to overcome this paradox. Can you think of any?

It's my turn on the time machine!!!!!

It's my turn on the time machine!!!!