From Schrödinger’s cat to quantum computers (II)

From analog computers to quantum computers

In this second part of the talk, we move on to discuss quantum computers. They are fashionable, they’re cute. You can boast in bars that you work in quantum computers and you’ll get free beers. But… are they, really, more powerful than their classical counterparts? I have devised a way to explain what is the main difference, and why they might well be more powerful. And it uses an old and forgotten concept: that of analog computer.

What is an analog computer? It is a machine designed to solve a certain computational problem, using physics. For example, the Antikythera mechanism. It is a device which was found in a sunk Greek ship from the Hellenistic times. It has some gears which, when spinning, represented the motion of planets, and helped predict their positions in the sky. In a certain way, the astrolabe is also an analog computer.

The Antikythera mechanism, an early analogue computer.

But let us give a simpler example. Imagine that you must order a large set of numbers, from lowest to highest. You can design an “ordering computer”, in the following way: get some spaghetti and cut each of them to a length corresponding to one of the numbers. Then, you take the full bunch and hit the table with it, flattening its bottom. Now, you only have to pick the spaghetti in order: first, second, third…

Another cute example is the problem of finding the most distant cities in a roadmap. I give to you a list of cities, and the distances those which are linked by a road. An analog computer can be built with a long thread. We cut it to pieces representing the roads joining
each pair of cities, and them we tie them in a way resembling the full roadmap. The knots, of course, represent the cities. Now we pick the roadmap by one of the knots, and let the rest hang freely. We look at the knot which lies the lowest. Now we pick it, and let the rest hang freely. In a few iterations we converge to a cycle between to knots. Those are the most distant cities in the roadmap.

And another one! Consider a 2D map on which we are given the positions of a few cities. We are asked to design the road network of minimal length which joins them all. Sometimes it will be convenient to create crossroads in the middle of nowhere. This is called the Steiner tree problem, and it is considered a “hard” problem. There is a way to solve it fast with an analog computer: we get a board and put nails or long pins on it representing the cities. Now we immerse the full board into soapy water and take it out very slowly. A soap film will have developed between the nails, with some “crossroads”, joining all pins. If we do it slowly enough, the film will have the minimum energy, which means that the total roadmap will have the minimal length. This idea is important: we have converted our computational problem into one that Nature can solve by “minimizing the energy”.

Experiments by Dutta and coworkers,

Experiments by Dutta and coworkers,

Our paradigm problem to solve will be the spin glass problem. It sounds like a technical problem, but I have a very nice way to explain it: how to combine your goals in life and be happy. Well, we all want things which simply do not go together easily: work success, health, true love, going out with the buddies, children, etc. Let us represent each of those goals as a small circle, and give a weight to each of them. Now I draw “connection” lines among the goals, which can be positive, if they reinforce each other, or negative, if they are opposite. Each line has a “strength”. So, for example, “having children” is highly incompatible with “going out with the buddies”, but it is strongly compatible with “finding true love”. Which is the subset of those goals I should focus on in order to maximize my happiness?

Each node is a "target", blue links mean "compatible", and red means "incompatible".

Each node is a “target”, blue links mean “compatible”, and red means “incompatible”.

I will tell you how to solve this problem, which is truly hard, using an analog computer. We put atoms in each node, and we make “arrow up” mean “focus on this goal”, while “arrow down” will mean “leave it”. Now, experimental physicists, which are very clever guys, they know how to “lay the cables” so that the energy is minimized when the system represents the maximum happiness. Cool.

No, not so cool. The problem is the “false minima”. Imagine that you’re exploring the bottom of the ocean and you reach a very deep trench. How can we know that it is, indeed, the deepest one? Most of the time, the analog computer I just described will get stuck in the first trench it finds. And, believe me, there are many. It’s just the story of my life: I know I can do much better, but… I would have to change so many things, and intermediate situations are terrible. But today I feel brave, I really want to know how to be happy.

Quantum mechanics comes to our help. Remember that atoms are quantum, and that their arrows can point in some other directions, not just up or down. So I put my analog computer, made of real (quantum) atoms inside a giant magnet, which forces all spins to point rightwards. Remember: \left|\rightarrow\right> = \left|\uparrow\right> + \left|\downarrow\right>, so now each one has 50% probabilities of pointing up and pointing down, like Schrödinger’s cat. Maximal uncertainty. I know nothing about what to do. Let us lower slowly the power of the magnet. Nature always wants to minimize the energy, so we pass through some complex intermediate states, highly “entangled”, in which some atoms decide early which position to choose. They are the “easy atoms”, for which no competition with other atoms exist. The “difficult atoms” are the ones in which you have more doubts, and they stay in a “catty” state for longer time. When, finally, the power of the magnet has come to zero, all atoms must have made up their minds. We only have to read the solution, which is the optimal happiness one.

Sure? It all depends on the speed at which we have turned the magnet off. If I am greedy and do it fast, I will ruin the experiment, and reach any “false minimum” of the energy. So, a false maximum of the happiness. All this scheme is called “adiabatic quantum computation”. For physicists, “adiabatic” means “very slow”.

How slow should I go? Well, this is funny. Apparently, most of the time it’s not very important. But there is a critical moment, a certain “phase transition” point, when the entanglement between the atoms is maximal. Then, it is crucial to advance really slowly. As an analogy, think that you have to take a sleeping baby from the living room to his cradle. There is always a fatidic moment, when you have to open the damned doorknob. If you are unlucky, you may have more than one. But, for sure, at least you’ll have one.

And, what if we’re so clever that we have left all doors open? That’s what worries us physicists most. There is a conjecture, that there is some kind of “cosmic censorship”, that will impose a closed door in the path of every difficult problem. Nature might be evil, and has put obstacles to the possibility of solving difficult problems too fast. It would be a new limit: the unsurmountable speed of light, the unstoppable increment of entropy… and now that? It is worth to pay attention: the next years will be full of surprises.

This is the second part of a lecture I originally delivered in the streets of Madrid for the “Uni en la calle”, to protest the budget cuts in education and science in Spain, in March 9, 2013. And, later at a nice high school in Móstoles.


From Schrödinger’s cat to quantum computers (I)

From Schrödinger’s cat to entangled cats

If you are also fans of The Big Bang Theory, you will be aware of Penny and Sheldon’s discussions about Schrödinger’s cat. Penny wants to know whether she should hook up with Leonard, and Sheldon tells her that, in 1935, Erwin Schrödinger designed a mental experiment in which a cat was put inside a closed box with a vial of poison which can be opened at random times. You may not know whether the cat is alive or dead until you open it. Penny thinks that the lesson is that she should try, and only then she will know. But, really, Sheldon only wanted to get rid of her. The question, as all human interaction, was completely irrelevant to him.

Penny & Sheldon

I have a board. If you like boards, this is my board.

Although I love the series, the explanation about Schrödinger’s cat is lame. You put a cat in a box and a vial of poison. There’s 50% chance that the vial opens and the cat dies. According to our intuition, the real state of the system is one of them: alive-cat or dead-cat. Since we don’t know which one it is, we represent our knowledge with probabilities:

50% alive & 50% dead

But that’s not quantum mechanics! Quantum mechanics is far more weird, and tell us that the cat may be alive and dead at the same time. We represent it this way:

(That notation, \left|X\right>, is called a “ket”… yes, we physicists are very fond of funny notations.) If, while in that state, you open the box, the cat is forced to choose. With 50% probabilities, it becomes an alive-cat, and with 50% a dead-cat. But then, how is it different from before!?  Because the “alive-and-dead-cat” is a new “catty state” that we may represent this way:

cats3and which has different properties.

Well, with cats this doesn’t really work. We tried, but they move a lot, and miaow, and scratch. We better try with atoms, which are far more peaceful. Most atoms behave like small magnets, and their magnetization can be thought of as a small arrow, called “spin”, pointing in any direction, something like this:

spinningatomWe have our (cat-like) atom in a closed box, with its little arrow pointing in any direction. With cats, you may ask: “are you alive or dead?”, and it gives you an answer. With atoms you may ask, for example: “is your little-arrow (spin) pointing up or down?” Of course, not only for the vertical direction. You just pick up any direction and ask, but let’s say that the vertical direction is clear enough. So, you may have the atom in state \left|\uparrow\right>, and the answer will be “up”, or \left|\downarrow\right> and the answer will be “down”. But what happens if you mix them? You can have the state \left|\uparrow\right> + \left|\downarrow\right>. Then when you ask “Is your little-arrow pointing up or down?”, the atom chooses \left|\uparrow\right> with 50% probability and \left|\downarrow\right> with 50%.

(By the way, if someone is thinking of erotic analogies, flash news: we physicists have already thought of all of them.)

But I told you that \left|\uparrow\right> + \left|\downarrow\right> is more than just 50% up and 50% down. Let’s change direction. Now, instead of asking about up or down, we ask “is your little-arrow (spin) pointing rightwards or leftwards?” The atom answers “rightwards” with certainty. 100% probability!! So… that was the point!! It answered randomly when asked about up or down, because it was pointing to the right!! Is there any way to prepare the atom so it points always leftwards? Yeah, we write \left|\uparrow\right> - \left|\downarrow\right>. And the same happens if you ask the other way round: if you have \left|\uparrow\right> and ask “are you pointing left or right?”, it will answer randomly. Wrong question, random answer.

But let’s come back to cats. We can go beyond Schrödinger and put two killer cats in the same box. They hate each other, and only one will survive. The quantum state can be written as

cats4but… it could be the other way round! It might be

cats5Classically, we would have 50% of each, But, in quantum mechanics, we can have the state

cats6I put both signs because both are possible. It depends on the cat breed, I think.

But, I insist, that’s hard to do with cats. Do it at your own risk. With atoms, it’s a whole different story. We may prepare atoms such that their little-arrows (spins) point, for sure, in opposite directions. Let’s say that they are in the state

\left| \uparrow\downarrow \right> - \left | \downarrow\uparrow \right>

This state suffers from what we call entanglement. And very weird things happen to it. That was studied by Einstein and some of his buddies, called Podolsky and Rosen, in 1935 (also) (yeah, good year), when they showed that we could do the following. Take the box containing both atoms and split it in half, making sure that a single atom stays in each half-box. Now, take one of the boxes very far away. When you ask one of the atoms “are you pointing up or down?”, you don’t know what the answer is going to be, because it chooses randomly. If we do it with cats, you don’t know if the box you’ve kept contains the dead or the alive cat. But let’s assume that you get the answer “up”. Then we know what will the other atom reply when asked whether its arrow  points up or down. It will say “down”.

The surprise comes when you ask the atom: “is your arrow pointing left or right?” Its answer will also come randomly, 50% right and 50% left. I am not going to justify that, just believe me. But if you ask the same question to the other atom, no matter how far away it is, its answer will be the opposite!!! You may ask about any direction, and both atoms will give you opposite answers. The question that we may ponder is, of course… how does the second atom know what was measured on the first? Apparently, entangled atoms hold a bond that, like good loves and good hates, survives distance.

This is the first part of a lecture I delivered in the street in Madrid, as a part of the “Uni en la calle” program to protest the budget cuts in education and science in Spain, on March 9, 2013. I have delivered it also at Manuela Malasaña high school. Thanks to you all, guys!

Feeling acceleration (Rindler spacetime)

This is the first article of a series on the Unruh effect. The final aim is to discuss a new paper on which I am working with the ICFO guys, about a proposal for a quantum simulator to demonstrate how those things work. We are going to discuss some rather tough stuff: Rindler spacetime, quantum field theory in curved spacetime, Hawking radiation, inversion of statistics… and it gets mixed with all the funny stories of cold atoms in optical lattices. I’ll do my best to focus on the conceptual issues, leaving all the technicalities behind.

Our journey starts with special relativity. Remember Minkowski spacetime diagrams? The horizontal axis is space, the vertical one is time. The next figure depicts a particle undergoing constant acceleration rightwards. As time goes to infinity, the velocity approaches c, which is the diagonal line. But also, as time goes to minus infinity, the velocity approaches -c. We’ve arranged things so that, at time t=0, the particle is at x=1.

Minkowski diagram of an accelerated particle.

Minkowski diagram of an accelerated particle.

Now we are told that the particle is, really, a vehicle carrying our friend Alice inside. Since the real acceleration points rightwards, she feels a leftwards uniform gravity field. Her floor, therefore, is the left wall.


Alice in her left. Acceleration points rightwards, “gravity” points leftwards.

Are you ready for a nice paradox? This one is called Bell’s spaceship paradox. Now, imagine that Bob is also travelling with the same acceleration as Alice, but starting a bit behind her. Their trajectories can be seen in the figure


Alice and Bob travel with the same acceleration. Their distance, from our point of view, is constant.

From our point of view, they travel in parallel, their distance stays constant through time. So, we could have joined them with a rigid bar from the beginning. Wait, something weird happens now. As they gain speed, the rod shrinks for you… This is one of those typical paradoxes from special relativity, which only appear to be so because we don’t take into account that space and time measures depend on the point of view. This paradox is readily solved when we realize that, from Alice’s point of view, Bob lags behind! So, in order to keep up with her, and keep the distance constant, Bob should accelerate faster than her!

So, let us now shift to Alice’s point of view. Objects at a fixed location at her left move with higher acceleration than she does, and objects at her right move with lower acceleration. Her world must be pretty strange. How does physics look to her?

One of the fascinating things about general relativity is how it can be brought smoothly from special relativity when considering accelerating observers. In order to describe gravity, general relativity uses the concept of curved spacetime. In order to describe how Alice feels the world around her we can also use the concept of curved spacetime. It’s only logical, Mr Spock, since the principle of equivalence states that you can not distinguish acceleration from a (local) gravity field.

Fermi and Walker explained how to find the curved spacetime which describes how any accelerated observer feels space around her, no matter how complicated her trajectory is. The case of Alice is specially simple, but will serve as an illustration.

The basic idea is that of tetrad, the set of four vectors which, at each point, define the local reference frame. In German, they call them “vier-bein”, four-legs, which sounds nerdier. Look at the next figure. At any moment, Alice’s trajectory is described by a velocity 4-vector v. Any particle, it its own reference frame, has a velocity 4-vector (1,0,0,0). Therefore, we define Alice’s time-vector as v. What happens with space-vectors? They must be rotated so that the speed of light at her point is preserved. So, if the time-vector rotates a given angle, the space-vector rotates the same vector in the opposite direction, so the bisector stays fixed.


The local frames of reference for Alice, at two different times.

Now, each point can be given a different set of “Alice coordinates”, according to local time and local space from Alice point of view. But this change of coordinates is non-linear, and does funny things. The first problem appears when we realize that the space-like lines cross at a certain point! What can this mean? That it makes no sense to use this system of coordinates beyond that point. That point must be, somehow, special.

In fact, events at the left of the intersection point can not affect Alice in any way! In order to see why, just consider that, from our point of view, a light-ray emmited there will not intersect Alice’s trajectory. Everything at the left of the critical point is lost forever to her. Does this sound familiar? It should be: it is similar to the event horizon of a black hole.


Red: what Alice can’t see. Green: where Alice can’t be seen.

Let us assume that you did all the math in order to find out how does spacetime look to Alice. The result is called Rindler spacetime, described by the so-called Rindler metric. In case you see it around, it looks like this

ds^2=(ax)^2 dt^2 - dx^2 - dy^2 - dz^2

Don’t worry if you don’t really know what that means. Long story short: when Alice looks at points at her left (remember, gravity points leftwards), she sees a lower speed of light. Is that even possible? That is against the principle of relativity, isn’t it? No! The principle of relativity talks about inertial observers. Alice is not.

So, again: points at her left have lower speeds of light. Therefore, relativistic effects are “more notorious”. Even worse: as you move leftwards, this “local speed of light” decreases more and more… until it reaches zero! Exactly at the “special point”, where Alice coordinates behaved badly. What happens there? It’s an horizon! Where time stood still.


The world for Alice, Rindler spacetime: speed of light depends on position, and becomes zero at the horizon.

Imagine that Alice drops a ball, just opening her hand. It “falls” leftwards with acceleration. OK, OK, it’s really Alice leaving it behind, but we’re describing things from her point of view. Now imagine that Bob is inside the ball, trying to describe his experiences to Alice. Bob just feels normal, from his point of view… he’s just an inertial observer. But Alice sees Bob talking more and more slowly, as he approaches the horizon. Then, he friezes at that point. Less and less photons arrive, and they are highly redshifted (they lose energy), because they had to climb up against the gravitational potential. Finally, he becomes too dim to be recognized, and Alice loses sight of him.

That description would go, exactly, for somebody staying fixed near a black hole dropping a ball inside it. The event horizons are really similar. In both cases, the observer is accelerated: you must feel an acceleration in order to stay fixed near a black hole! As Wheeler used to say, the problem of weight is not a problem of gravitation. Gravitation only explains free fall. The problem of weight is a problem in solid state physics!!

For more information, see Misner, Thorne and Wheeler’s Gravitation, chapter 6. It’s a classic. I wish to thank Alessio, Jarek and Silvia for suffering my process of understanding…

On physics, maths and tenure in Spain

The following dialogue, its situations, characters and institutions are completely fictional. Or almost.

Two researchers, in their late thirties, meet at the college cafeteria.

Sandy. Hey, Cris, you’re back from Spain! How did the selection process go?

Cris. Bad, the insiders got the tenure positions, but I knew it was going to be that way. You know  how my country is. My CV was much better than theirs, and my lecture was terrific… but there was nothing to do.

Sandy. Yeah, don’t think it’s much better here…

Cris. Well, I don’t know. They evaluated my CV “logarithmically”, if you catch my drift: for example, by multiplying your papers by 2 you earn one extra point. Then, the evaluation of the lecture is very subjective… and that’s what they use to select the inside candidates.

Sandy. Why would they do so?

Cris. It’s easy: because the professors in the department are completely clueless, they don’t want new professors which can overshadow them.

Sandy. Where was the position?

Cris. It was for the mathematics department of a school of building engineering. That’s one of the most stupid things we have in Spain: the hyperfragmentation of the university. You decide your major at 18, when you reach college, and you can’t change easily. In many cases, such as UPM in Madrid, there are as many mathematics departments as there are schools. Tiny departments which do almost no research, dominated by their feudal lords.

Sandy. Like a community college here.

Cris. Much worse! They can do research, just they choose not to. But you know what was the funniest part? The committee argued endlessly about my research in theoretical physics being inappropriate for an applied mathematics department.

Sandy. No way!

Cris. Yes! I challenged them to tell me what is the difference between theoretical physics and applied mathematics…

Sandy. What did they say?

Cris. What did they say!? They mumbled about the names of the journals I use to publish in! If the title contains “physics”, then it’s physics. If it contains “mathematics”, or  “geometry”, or “algebra”, then it’s mathematics.

Sandy. And what if it’s “Communications in mathematical physics”?

Cris. That’s too mind-boggling for them! One of the guys simply asked me: “can you tell me one of your papers which is really mathematical”? I answered: “what about the one I discussed about the Riemann hypothesis?”

Sandy. Hahaha! Yeah, I remember, your fighting with proving the Riemann hypothesis using quantum mechanics.

Cris. Yeah… they told me that my papers are physics-motivated, not mathematics motivated. So, that’s where I told them that that sentence made no sense. Of course, in kinder terms… I added that it was OK to talk about papers focusing on methodology and papers focusing on the solution of a certain problem.

Sandy. Yeah, that might do as a distinction between math and physics, doesn’t it?

Cris. No, it doesn’t! A lot of physics papers focus on the techniques or the formalism. But, even worse, applied mathematics should focus on problem solving, right? I mean, real life problems. Such as… physics! Then I got a bit pedantic, and told them that mathematics comes from “máthema”, which in Greek means “what we learn”, while physics comes from “physis”, which means “Nature”. And I told them: what can you learn about, other than Nature?

Sandy. Hahaha! That made them hate you, for sure.

Cris. You bet. I even cited Arnol’d, and the speech he gave in Paris in 1999, emphasizing the close nature of mathematics and physics.

Sandy. Yeah, I remember that. Arnol’d was amazing. He hated the Bourbaki spirit in mathematics. He said they had converted mathematics into a game, a purely logical game. All branches of mathematics are inspired in physics, right?

Cris. Right! Arithmetics comes from counting. Geometry and probability are very clearly physical. And calculus is an abstraction of the theory of motion.

Sandy. Exactly. A mathematician which considers herself to be “applied” should be someone who is able to apply all kinds of mathematical tools to real-life problems. And a theoretical physicist is usually good at that.

Cris. Also, they rejected immediately a candidate with a CV which was similar to mine, because her teaching had been in physics.

Sandy. Again, the same thing applies. In fact, the distance between teaching calculus and mechanics is the same as the distance between teaching mechanics and thermodynamics.

Cris. Absolutely.

Sandy. I assume they didn’t teach much advanced mathematics to those building engineers.

Cris. Of course they didn’t. In fact, their calculus didn’t contemplate Taylor’s theorem, their vector calculus stopped at double integrals, and their algebra didn’t include complex numbers. My god! The students never reach the e^{i\pi}= -1 level!!

Sandy. Hahaha! Yeah, I know that’s your pet peeve… You think the students only get their mathematical maturity when they understand that formula.

Cris. Of course I do! That’s the moment when they stop being little padawans and they receive their jedi sword!

Sandy. Yeah, you love beauty in mathematics.

Cris. You bet I do! In gave my lecture on eigenvalues and eigenvectors, and I produced some nice animations… But even more important, I started with the Fibonacci numbers.

Sandy. You did what!?

Animation presented by C during the lecture, showing orbits obtained when acting several times with a certain matrix.

Cris. I discussed how to get a closed formula for the n-th Fibonacci number by raising a matrix to a certain power.

Sandy. How can you do that?

Cris. I wish you had seen me! It was like making magic when you do it slowly… the golden section appears as an eigenvalue of the matrix which generates the Fibonacci numbers… it’s wonderful when you see why. Grab a napkin, I will tell you.

Sandy. Sure.

Mάθησις, Mathesis

Today I would like to make a proposal: Mathesis, a dependency road map for science.

Consider that you’re a student or researcher in science trying to learn a new topic. This topic is explained in a paper, or a book, but it is not accessible to you because there are some pre-requisites that you’ve not covered. Of course, the bibliography of the paper or book can help you, but normally they are not so useful. How to trace it back to the point where you should start reading? And what if you need to take it at several different starting points, converging in the paper that you need?

Precisely because of that, we scientists write books and review articles. But textbooks are linear structures, while knowledge is not. A textbook takes you from point A to B, along a certain excursion path. But, more likely than not, only part of it is relevant to your needs. Hopefully, you can reach your desired knowledge by linking paths taken from different books or papers.

This is the very ambitious target of mathesis: a dependency tree for learning science. This means, to create a graph whose nodes are (small) pieces of knowledge, and whose links are the dependency relations among them. Thus, if you want to learn X, then you proceed to find the node for X. Its outcoming arrows denote on which pieces of knowledge it depends. Then you can trace them back, until you find which nodes correspond to your current knowledge and proceed from them backwards.

Each node need not contain a full explanation of the topic. That would imply to build a full encyclopaedia of science, which is a meta-ambitious target. No, it should  contain some good bibliography, taking into account the dependency structure. Of course, it is much better if this bibliography is free.

This idea resembles a lot the debian repository dependency network, and an attempt to implement it for knowledge has already been done.

So, this is a call for collaboration. We need:

  • Examples. You can try to create the dependency tree for your favourite result. Or the dependency tree in order to understand one of your papers.
  • A standard format for the nodes. They should contain, at least, a brief description, and a list of the nodes on which it depends. The nodes might be weighted, with a low number meaning that only the general idea is required and 1 that the topic should be mastered. And, of course, some bibliography.
  • A nice visualization tool, in order to view parts of the total tree which are relevant to you. Maybe, in java.

This stems from an idea that I had long back, in 2004. I created project Euler, in Spanish, with the full text of my classes of maths in high school, with a dependency tree associated. And I still like the logo I prepared at that time… :)

P.S.: And out of the topic… guess some nice properties of the logo figure? ;)

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…

Robots, ho!

May I introduce you to Paquito? Less shiny than C3PO and less obnoxious than R2D2… yet a great basketball player! Poor Paquito didn’t win the Galapabot’10 competition, which took place last weekend. For some strange reason, I was appointed judge in the event. In the pic, my face is strategically covered by one of Paquito’s gears.

So the game, organized by one of my little padawans (recently upgraded to young jedi), Irene by name, was basically to clean your ground half of balls, by dumping them to your opponents’ half.

In the pic you can see BEAST III in action, aka Terminator, it was really a great piece of work! As a remarkable side issue: you can see a girl driving a robot. There are geek girls in robotics!!  BEAST III was created and handled by a Swedish-non Swedish team. OK, I guess this last sentence needs explaining. The Svartmetall team came from the Intl. school of Stockholm, yet none of the members were Swedish! Also worth mentioning the other robots: Chucky, Franky, Smurf (made on the spot!) and BEAST 3.5.

The competition had an autonomous stage, where the robots had to act on their own, following their programs only. After that, the human drivers take control. I guess it is the first stage when you have the feeling that you have created something… The soul of Isaac Asimov was looking upon us, I am pretty sure.

Superhighway (or funny minima)

I proposed this problem to my calculus students. It turned out to be more interesting than I thought (thanks, Ignacio and Noema).

The government intends to build a superhighway without speed limit so, therefore, without curves. It will start from city A, and should pass as close as possible to cities B and C.

In order to solve it you should start by stating what you mean by “as close as possible”. An option is to minimize the sum of the distances. But then, you get the following funny configuration:

Cities B and C are at the same distance from A, and make up a right angle. Intuition dicatates that the best route for the highway would be the bisector. Let d(A,C)=d(A,B)=1. Then, the sum of the distances from the cities to the highway is \sqrt{2}. But there is a better highway! You can just break the symmetry between B and C and make the road pass through C exactly. Then, the sum of the distances is just… 1.

If this was a real highway, the politician in charge would tell us that symmetry is also worth. Citizens of B may riot with un asymmetric solution… So, now let us change our target function. Let us minimize the sum of the squares of the distances. In that case, the bisector gives a square total distance of 1, same as the asymmetric road… Can you explain it?

Which is the best choice? Of course, it depends on the reason for which you’re fighting with the problem. If it is just to pass an exam, any of them will do…:) [Of course, there are many other alternatives. A student minimized the sum of the squares of the distances from the points to the straight line in the y direction. This makes sense in some cases, e.g.: when you’re fitting experimental points to a line.]

So, choosing the right function to minimize is crucial in practice… Kadanoff once explained in a talk that the government of the city of London has ordered a huge global study of traffic, taking into account both public and private transport, energetic and economic issues, taxes and prices, eeeeeverything. They made an enormous computer program that was running for days and, finally, told them the answer, how to optimize traffic in London. They had to remove all traffic lights. Why???? The program had many things into account… also the fact that in street car accidents, it’s normally old people who die. And old people do not pay taxes, they receive their pension money from the government. So it was convenient to remove the traffic lights… So, you see: garbage-in, garbage-out. Yes, maths is a nice girlfriend, she gives you more than you put in the relationship… but she can do no miracles. If you’re stupid, she can’t fix that…

Teaching fresh(wo)men calculus

I have just finished, for the second time, the calculus fall term for engineering freshwomen (and freshmen) ;) at UC3M. The classes were in English, split into two for practical sessions, around 70 students in total. It was a nice group (yes, some of my students will read this and no, I do not say this because of that… the teacher evaluation polls are already over) ;) I have been thinking about what do we teach, what is its purpose and how we should do it… and I have reached a few conclusions.

  • There are two reasons to teach maths to non-mathematicians: (a) because they will need some tools which are standard in their trade or (b) because they should learn to think, they should learn real problem solving techniques. The contents of the calculus term (derivatives and integration in one variable, basically) is already covered in high school, only a few new things are taught here (Taylor, polar coords…) So, the real reason must be the second one.
  • That’s why I have introduced two novelties: first of all, problems in “real life format”. With this I mean that they’re formulated vaguely, with no data. For example: “I want to leave my can of beer on the ground, but it is irregular and I’m afraid it might fall down. I should drink a little bit of beer so that it becomes more stable. How much?”
  • Another point that was important for me was the ability to give numerical solutions, approximations… I mean: to obtain numbers even when an analytical solution is not available. We also introduced numerical calculation techniques via octave, but it had to be out of the class hours.

My only complaint: such a course, if it has to be taught correctly, requires a rather low number of students per class. When I teach linear algebra, it’s ok for me to talk about eigenvalues to 120 students. That’s because the idea is fully different. I don’t know how to teach problem solving from the blackboard. There are always a few lucky cases in which you have to teach nothing: they already get your point, almost before you’ve finished stating it. With the rest, we normal mortals, it has to be done one by one…

Another important point. I would like to change the “blocks”. There should be four of them

  1. Visualization: sketching functions, curves in polar coordinates or parametric, surfaces… And the reverse: see data and “guess” an analytical expression. Fitting experimental data.
  2. Computing: approximation schemes, estimation skills. Tayor, mean value theorem… And numerical programming skills.
  3. Optimization: all sorts of problems where some target function has to be maximized or minimized. There are few “real life” problems which can not be re-cast in this form…
  4. Cutting into pieces and pasting back: (for want of a better name) with this I mean all kinds of problems which “reduce” to integration: areas, volumes, lengths, work of a force, average of a function, etc.

Calculus at this level can be seen by the students as a bunch of tricks. And they’re right. All of us making a life as “applied mathematicians”, we have a bag of ideas that come to our mind when we see a new problem. Applied mathematics is just that: the ability to tackle a new problem, to make the “right metaphor” with another problem that you solved years back.

Just a finishing remark: why do we have so few girls??????? I want a convincing answer, or I’ll move to nursery school next year! And I’m serious about that!

Is this the reason?

Arco capaz… is there an English word?

Hm… I should start this story from the beginning, as Alice once was told. I make a living from teaching college maths in Madrid, but half of the time I do it in English. (No, no randomness involved, though). I asked my calculus students to solve this problem:

“A soccer player runs with the ball perpendicularly to the goal line, but a little bit to the right of it. Forgetting about the opponent players, when should he shoot?”

My idea was just an optimization problem, but, alas, some of my boys and girls are really smart… and one of them asked me: “Javi, how do you say ‘arco capaz’ in English?” Oh, embarrassment! I didn’t know the word!

Ehm… let’s go by parts, as Jack the Ripper said. For those of you not fluent in the language of Cervantes, arco capaz is the geometrical locus of the points in the plane from which a given segment is seen under a certain angle. It must be an arc, as you can see from the pic:

Okey dokey, so the boy was right, you can solve it using this construction. All you need is… well,  won’t say, just give your proposals :)

Anyway, the interesting part comes now. OMG, I felt so bad that there was a technical word I couldn’t say in English… so I run to wikipedia (all praise be given to her), clicked arco capaz in Spanish, clicked the English button et… voilà? No! It took me to a concept which is completely unrelated!

Hm… how could wikipedia (APBGTH) fail me???? Then I found this link in wordreference (also all praise… whatever) in which… the hypothesis was advanced that there was no word in English for that concept! They even cited a webpage where Patrick Morandi, head of dept of maths in the New Mexico state university, used the Spanish term…

After that I found few entries with the French term “arc capable”, but the term in Spanish is really common!! All first year students in science and engineering have heard about it! So, perhaps this is the second word (first in science) that goes from Spanish to English. The first one was mosquito

But why is the concept not lexicalized in English? I can give no citations, but I heard (my grandpa, long back) that it was born in navigation science. Using it, you can find out where you are on the map very easily as soon as you have three landmarks that you can recognize… Did sailors use a different technique in other countries? Maybe they used it but they didn’t give a name to it? I can imagine: “Captain Haddock, please draw the two… ehm… yes, draw the two circles and see where do they intersect”, “Which circles?”, “Yes, the circles from which the two given segments are seen under the correct angles…” Messy.

Yes, giving names to the correct things can save a lot of work. Or give a lot of work, if you do it wrong…