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…

spaghetti
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.

citynetwork
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, http://arxiv.org/abs/0806.1340

Experiments by Dutta and coworkers, http://arxiv.org/abs/0806.1340

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.

landscape
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!

All paths to happiness

(Invited post by NP-complete)

fig11.- The discovery of the machine

My first gut feeling was that that chap in the hardware store was teasing me. In my fifteen years as an electronic engineer I had never seen such stuff. That product simply couldn’t be real. But ok, for just one euro, I could well afford the risk.

When I reached home I couldn’t stop thinking of all the things I could have bought with that euro. An electronic device that splits the universe into different universes? Well, that’s what the manufacturer’s instructions claimed. “Fork Industries Ltd.” I had never heard about a manufacturer by that name. The device was a simple black panel with a few buttons and a numeric display. If I got swindled, at least I could take it apart and reuse the components to recover my investment.

There was a “reset” button, to start the system, and a “bifurcate universe” button, according to the instructions. A presumptuous name for a button, right? According to those instructions, when pressing that button, the universe would split into two almost identical parallel universes, no less than that! The only difference between the two universes would be that, in one of them, the panel display would show a 0, and in the other universe it would show a 1. In both universes, all the rest would be exactly the same, no more differences. I thought that, even assuming that the machine was not a scam and that it really did that thing, the difference was really stupid. What use is that, a 0 in one universe and a 1 in another?

The text explained also that you could do consecutive bifurcations. If the user pressed that button a single time, then two universes would be created: the one showing a 0 in the panel display and the one showing a 1. But if you pressed twice, then four universes would be created: in one of them you would see first a 0, and then another 0; there would be another one where you would see a 0 and then a 1; another with 1 and 0, and yet another with 1 and 1. If you pressed three times, then eight universes would open up: one with 000, another with 001, other one with 010… and so forth until you get all combinations, ending with 111.

I am sorry but, even assuming that that story of the bifurcated universes were true, it would still be useless to do all such bifurcations: what’s the point in being in a universe where the machine has showed, after five consecutive bifurcations, 00101 or 11101? If the rest of the universe is identical, if the only difference is made by a few little digits in a screen, what’s the big deal?

Then I saw that the instructions went beyond that. They explained that there was a way, a single way, to communicate all those universes among themselves. The numerical keyboard in the panel allowed users to type any number whatsoever. At its side, there was an enter button. If the user types a number in the keyboard and presses enter, then the machine will show in the panel the sequence of 0s and 1s of bifurcations performed in one of the bifurcated universes. Concretely, that sequence would correspond to the universe where the user had typed the highest value among all those submitted by the users in all the universes. For example, if after five bifurcations the user who types the highest number (e.g. 543) is the one living in the universe 10011, then, when users write their respective numbers (all smaller than 543), they will see the sequence 10011 appear in their panel displays of their respective machines.

When such a communication is performed (or when you press reset), the possibility of further communication among the previously bifurcated universes is cancelled forever. From that time on, you will only be allowed to communicate in such a way with the new universes you bifurcate with the machine in the future.

And that’s all. The machine didn’t allow anything else.

What a scam.

But, ok, there was only one way to be sure. I decided to try. I would press the button eight times, and that way I should be creating 256 parallel universes (with eight 0s or 1s there are 256 possible combinations). And my parents always said I would never reach anywhere! They would see me multiplying by 256 the divine creation! I decided that, after I was happily done with my bifurcations, I would convert to decimal the sequence of eight 0s and 1s which had appeared in my machine’s display, and then I would type such a number in the keyboard.

And that’s what I did. After pressing eight times the button, that is, after bifurcating the universe eight times, the sequence of values I observed was 00100101. That is, number 37 in decimal basis. Then I wrote 37 in the panel keyboard and pressed enter. Immediately, the sequence 11111111 appeared in the panel.

It made sense, of course. Out of the 256 universes which I had created, the universe that had witnessed the highest number must have been 11111111, which is 255 in decimal. So 255 had beaten all the others (including, of course, the 37 of my own universe). Thus, all the “myselves” of the 256 universes I had created would be watching at that very moment the very same sequence of values which gave rise to the winning universe: 11111111.

Wonderful, but that didn’t prove anything. That bifurcation story could be a lie. The machine might be programmed to show only the highest possible number.

Then I fancied a better experiment. Now I would press the bifurcation button eight times, but this time I would type the number of values which change from 0 to 1, or 1 to 0, when reading the sequence left to right. For example, if I saw 111111111, then I would type 0, since there is no change in the sequence. If I saw 11100110, I would type 3, because there is one change between the third and fourth symbol (1 to 0), another between the fifth and sixth symbols (0 to 1) and another between the seventh and eighth (1 to 0).

I pressed eight times and I got 10101111. Four changes. I wrote 4 and pressed enter.

Then the machine showed 10101010.

Dammit! There were 7 changes! And that amount of changes was unbeatable, there was no way to get more changes with eight values. It’s tied with 01010101, but there is no other sequence which can win. In the 256 universes that I had opened up, it was really the maximal amount of changes that one could get.

But I had not said anyone that this time I would type the number of changes instead of the decimal number of the sequence, as I did previously. The machine didn’t know that! Only I knew it!

Dammit! The machine worked! It worked!

Immediately I started to think about how to take profit of such a prodigious device.

I might press the button a number of times large enough so that the resulting sequence of 0s and 1s could represent a lottery combination (I may transform the 0s and 1s into a sequence of decimal numbers, and those would be the number on which I would bet). I would bet on the numbers displayed by the machine, and wait for the lottery outcome. Then I would type in the panel keyboard the number of millions of euros I had won with that combination. In most of those universes that number would just be zero, of course. Nonetheless, in one of those universes I would be a millionaire since, having created a universe for each possible combination, there must be one of them in which I had found the winning combination. In that winning universe, I would type in the machine the high number of gained millions. Then, all myselves from all the universes would receive the sequence of 0s and 1s which was used in that winning universe. This way we would find the winning combination, and use it to win the lottery.

Just a second… If I did that way, I would only know the winning combination after the lottery outcome was made public, which is when the winner would communicate it from his parallel universe. No, that would not do. By doing that I would only ensure that one of the myselves would be rich, but not the rest of us. That is useless for me. The probabilities that I would become rich are exactly the same as if I played in the traditional way, without using the machine. So, practically zero.

There must be a way to take profit of this machine…

Then I figured it out. I might use the machine to break all kinds of codes. If I pressed the bifurcation button a sufficiently high number of times, so that I would get enough 0s and 1s to codify a word of many letters (in fact, for every eight 0 or 1 I get a computer character, which can be a letter or other things), then I might find any password with the resulting letters. If every possible password which one can use in a computer appears in one of the displayed universes, then in one of the universes I would have the right password.

I connected to my computer and entered the mail server of my ex. I wrote her login. Then I pressed the bifurcation button 160 times, enough for the 0s and 1s to codify a word of 20 letters (more precisely, characters). In each of the many universes which I had just deployed, the result of converting all those 160 0s and 1s into a single word would be different. Moreover, each possible 20-letter word appears in one of the possible universes. So, if in all the universes I typed the resulting word, in one of them I would succeed and I would enter the email account of my ex. I decided that, in the universe in which I succeeded, I would type 1 into the machine keyboard and press enter, and in all others I would type 0 and press enter.

The word which was brought about by the 160 0s and 1s in my own universe was not the right password, as one might guess. So, I typed 0 in the machine, and immediately the display showed another sequence of 160 values. That sequence should come from the universe in which another version of myself had typed the biggest number into the machine keyboard, following the rules I had forced myself to follow: 1, which meant correct answer. I converted that other sequence into a word, tried again to enter the account with this new password and bingo! I was inside!

Beyond the sick interest I had in the account, obviously I thought of the more lucrative possibilities which opened up in front of me: I might enter bank accounts all over the world and order money transfers to my own account. Or, better, to some others, so that nobody could catch me. In different countries. With different names.

So my economic future was solved thanks to the little machine. Not bad for an euro!

A nice future opened up in front of me.

fig2

2.- The creative explosion

For the following years, I enjoyed all kinds of luxuries from my unlimited economic capacity.

Nonetheless, after thousands of travels, luxuries and rave-ups, a moment came when I felt empty.

Then I decided I wanted to develop my artistic vein. I dismounted the bifurcation button, opening the case, and observed that every press of the button released a five volt signal in a machine cable. So that’s how the machine perceived each bifurcation button press.

I connected that cable to my computer, and prepared by computer so that it would send 5 volt impulses whenever the computer wanted. Since my computer might send those impulses at the same speed as any other data plug (such as, e.g., the USB port), from that time on my computer would be able to “press” the machine’s bifurcation button at a speed much higher than my finger could never reach with real pressing.

Then, I dismounted the machine display and realized that the leds which made up the numbers (where the machine wrote 0 or 1), received also 5 volt impulses from the machine through certain cables.

I connected those cables to my computer, in such a way that the impulses that the machine sent to lighten the display would be immediately detected by my computer. From that time on, my computer would create the button keystrokes of that wonderful machine, and would read the sequences of 0s and 1s that it would output.

Then I programmed my computer to send eighty thousand million signals to the bifurcation button of the machine. It took barely a few minutes.

Oh my god, now I had created a huge number of parallel universes! Actually, many more than eighty thousand million, since each keystroke multiplied the number of universes by two… Do the math, and you’ll see that the number of parallel universes I had just created had more than twenty thousand million digits. Crazy!

My computer had captured the sequence of eighty thousand million 0s and 1s with which the machine had answered to those keystrokes, and stored it in a file in my hard drive… which took slightly less than ten gigabytes (that’s less impressive, isn’t it?)

Then I put an “.avi” extension to the resulting file and tried to open it in the video player of my operating system.

“The file is corrupt”, my computer replied.

That’s only logical, since most random sequences of 0s and 1s do not form a valid video file.

But a tiny fraction of those sequences do form a valid video file.

And a very tiny fraction of those make up a video that does not consist of mere snow.

And an even tinier fraction of those make up images that might correspond to a movie.

And an even tinier fraction of those make up an excellent movie.

I had decided that, in case the resulting video made absolutely any sense, I would take the file thus generated and submit it to various film studios in order to try to get the movie in the theaters.

Since my file did not even open, that was not my case, clearly.

I waited a few months. Then, the day after the Oscar ceremony of that year, I typed a 0 into the machine keyboard. My plan was that it would mean that my (no-)movie had won 0 Oscars. Obvious, it had not even entered the competition.

Then my computer started to register the sequence of 0s and 1s that had been produced in the universe in which the value introduced by my other I was the highest. Taking into account that, among all the displayed universes, all myselves would have viewed all the movies that can be stored in a 10 gigabyte file (btw, enough to obtain a high image and sound quality), the thing was promising.

The complete sequence received from that universe was finally stored in my computer. I added the “.avi” extension to the file and tried to open it with my video player.

This time, it opened.

How I laughed. How I cried. How it made me think. I still get shivers when I remember some scenes. Moreover, I still get shivers when I remember many scenes. What the heck, I still get shivers when I remember any single snapshot of the movie! It was simply perfect!

I had my movie premiered.

A few months after that, in the next Oscar ceremony, my movie won in all categories. All 24!

For it to win in all existing categories, you may have guessed that it was not in English (impossible winning also the foreign language category otherwise). Also, it was an animated feature, but also included real actors… which, by the way, made memorable performances. With those costumes, that makeup, those visual effects… wonderful!

A few months before, when I first showed the movie around, I inscribed myself as responsible for all technical categories. But I needed other people to play as actors. I did not look like any of the movie characters at all, so I needed different people to help me. You cannot get the Oscar if there is nobody real behind, whom do you give it to? Simply, I could not have won those Oscars without real people which could be assigned the roles for the performances. But that posed no real problem. The fact that I had won all the Oscars in other universe guaranteed that, in fact, I would manage to find people which did really look like the movie characters. Those people happened to be my cousin, my sister-in-law, my mother, etc., none of them had ever acted before. All of them received, very proudly, their Oscars.

It occurred to me that I might repeat the feat on the next year, just trying to maximize the number of Oscars won by myself, personally, not by the movie. But I discarded the idea. It was complicated to win, at the same time, the Oscar to best leading actor and best supporting actor. No matter how many times I appeared in the movie, I would only be considered leading actor. And, foremost, the Oscars to best leading and supporting actress seemed hopelessly difficult to me.

I decided that I should move and explore new fields. Using the same technique, in the following years I “wrote” the best-selling novels, I “programmed” the best video-games, I “recorded” the best songs, and made the most important scientific discoveries. I won all the research prizes that could be achieved with a merely theoretical setup (since my works were, really, computer files: books, articles, essays, plots, videos, etc.) For example, I solved a tough problem in mathematics which, apparently, had remained unsolved for long (consisting in comparing two thing called “P” and “NP”). Really I didn’t understand a single word of the result, despite I had proved it myself, but what others said in the following years made me think that, perhaps, that result bore some relation to how my wonderful (and secret) bifurcating machine operated.

Once here, you may come to think that I always submitted to publishers, contests or symposia my art or scientific works twice: one with the file generated in my universe and the other with the file I received from the universe where I was most successful. In that case, you may think I would be known as the guy that submits stupid things the first time and wonders on the second. But that was not true. I can say that all the files I generated in my own universe never made any sense: only a couple of times those files opened correctly with the corresponding application, and both times all I got was noise. Take into account that the vast majority of the sequences of possible symbols simply do not mean anything. So it was very hard to obtain a file which was even remotely worth sending to anybody. But that was irrelevant, since all possible files were generated in one of the displayed universes, so in some of them the good one would lurk. For the rest of the world, I was not the guy sending stupid things on the first time and wonders on the second. I was simply, the guy that always created wonders.

fig3

3.- Obsolete instructions to be happy

Full of money and prizes, recognition and glory, the time came again when I felt empty.

Then I decided I would use my machine to be happy. In fact, the machine itself would tell me how to do it.

I used the same technique I had used the previous times to generate, with the bifurcation machine, a file of 0s and 1s, which this time I interpreted as a text file. I decided that this file would tell me what to do during the next year to be happy: throughout the year I would follow all the advices included in that file which made some sense (there is no way to follow fd%s$$gf#dfg0d78sfg) and were also reasonable. I would not jump out of a cliff, no matter how clear the text told me so, or smash the bifurcation machine with a hammer, although I decided I would follow some strange instructions, such as, for example, never saying hi, bathing every day in a tub filled with raw eggs, start all my sentences with the word krupuk or become a Methodist missionary. I decided that, when the year was over, I would type into the machine a number that would reflect the grade I gave that year in terms of happiness. Then I would receive in the machine the sequence of 0s and 1s corresponding to the advice text which had been followed by the myself which had typed the highest grade (that is, the one which had been happier by following those advices). During the next year, I would follow exactly those advices which had made my other self so happy. All of us would enjoy, with one year of delay, the same happiness our luckiest myself had received.

I took the steps. As it was likely, the instructions I received in my own universe did not make any sense. I found no single meaningful word of more than two letters in that absurd three pages sequence of symbols. So, in my universe, I had simply no instructions to follow.

By the end of the year, I received the instructions from the myself which had been the happiest in all the universes during that year, as I had foreseen. The text file instructed me to try to meet a certain lady, become her couple and live with her. It sounded good. In theory, those steps would take me to an optimal happiness, unsurpassed in any universe.

I reached the lady’s home. As it was expected, she immediately recognized me, because she was a fan of some of my works, as most of the world population was. So it was easy to start the conversation.

Nonetheless, after a few minutes I discovered she had started a relationship a couple of months before. The conversation ended cordially, but the situation was different from what I imagined.

I visited her again during the following days and weeks. I persevered, that should not stop me. I should achieve the happiness those instructions promised. I thought my immense popularity as the biggest creator in History would work its magic.

Inexplicably, it didn’t. Maybe I relied too much on that strategy based on my popularity, so she thought I was a cocky jerk that thought he would get anything he desired.

What had worked for one myself on the previous year, when maybe the lady was single, would not work for me, because everything had changed in a single year. The opportunities were different. It was like the wrong strategy to win the lottery I mentioned earlier: sometimes, the successful solution a posteriori is good for nothing.

In fact, all that year was a complete mess. Following the rest of the advices in those instructions, which were so helpful to my other self from another universe to enjoy a perfect life with that woman, were useless because it all depended on the first step, which implied the successful establishment of a relationship with that lady.

I thought for some time about that difficulty that blocked my use of the machine to get the perfect advice to happiness. The problem was that all the advices would reach delayed. If other self from another universe advised to do a certain thing, perhaps the possibility of profiting from certain action would be gone in one year, a day, or even a few minutes. The validity of the advices would always be ephemeral, and its utility uncertain.

I didn’t find a solution to that problem, so I decided to use an intelligence more powerful than mine to solve it. I used the machine to generate all possible texts in a single sheet, in search for one which might solve my problem. In each universe I would read the resulting text in that universe, and I would score it according to its potential utility to solve my problem. Thus, when in all universes we received the best scored text, we would have the best solution to the problem, if it existed.

When I read that best solution coming from one of those universes, I was quite intrigued. According to that text, the machine instructions never specified that the time in all displayed universes should advance simultaneously, so there was the chance that, in fact, each universe had its own time, whose existence was independent from all the others. Therefore, I didn’t have to assume that, in order to receive the sequence of 0s and 1s of the universe where the maximum value was entered, I had to wait in my own universe for the same amount of time that it took in that universe for the number to be entered. I should consider the possibility that, when typing a value and pressing enter in my own universe, I would receive immediately the sequence of 0s and 1s from the universe where the entered value was the highest, independently of how long it took for that value to be entered in that universe. Perhaps its universe had its own independent time, so waiting for one day, one month or one year in your own universe might have nothing to do with the same lapse of time passing by in other universe. The communication between universes through typed values in the machine might be independent from each particular universe’s time.

Well, that text was only one idea that another myself, from another universe, valued enormously for how much he felt intrigued by it, nothing else. This myself had no chance to test it before scoring it, because in order to do so he should have created his own universes in order to establish a new communication, which would have prevented him from communicating with me: every time you write a score, you break the possibility of getting in touch again with formerly deployed universes. That was explicitly said in the instructions indeed.

Nonetheless, that idea was truly intriguing. Nothing in the instructions was against that possibility.

I decided to check it out. I would perform the following experiment: I would press the bifurcation button once. If I got a 0, then I would type immediately the value 0 in the machine and press enter. On the other hand, if I got 1, then I would wait a minute and then I would press again. If this time I obtained a 0, then I would enter immediately the value 1 in the machine and press enter. But if I got 1 again, then I would wait another minute. Then I would press again and, if this time I obtained 0, I would type 2, otherwise I would wait another minute. I would follow the same procedure for a maximum of 9 times: if then I obtained a 0, I would type 9, and if that time I obtained again a 1, then I would wait for one more minute and then I would press 10 without bifurcating again. You may have realized that, in any case, I would type in the machine the number of minutes that I had would have to wait until obtaining a 0, up to a maximum value of 10.

I followed that strategy and had to wait, in my case, for six minutes. Just when I typed 6 and pressed enter, I received in the display the sequence 1111111111, that is, ten 1s.

If it had taken me six minutes for me to be able to observe a sequence which, in another universe, had taken ten minutes to obtain, then what that text said was true! Each universe had, really, its own time! Communication between the machines in all universes as time independent! My assumption that the time had to flow synchronously in all deployed universes was, simply, false. In fact, the instructions didn’t state it anywhere. My mistake was due to a bad assumption.

Very excited, I decided that I had to look for a way to profit from that novelty.

The first thing that occurred to me was that the lottery problem I mentioned some pages ago (the problem that I would receive the winning combination only after the lottery outcome was made public) might be solved. I would do the following: bifurcate the universe enough times in order to be able, with the resulting 0s and 1s, to fill up a lottery ticket. Then I would press the button one more time. If this time I got a 1, then I would buy the ticket, fill it up with my sequence of 0s and 1s converted into the corresponding bet, wait for the lottery and then type into the machine the number of millions that I had won. Otherwise, if the last number was a 0, then I wouldn’t play. Instead, I would type immediately the number of millions I won with my (no-)ticket, thus, 0, and I would press enter. In that last case, I would immediately know the combination that made other myself win the lottery in another universe. Since I would know that combination before the lottery draw, I could buy a ticket, fill it up as my combination said, and win the lottery.

This way I would get half of myselves (those who obtained 0 in the last pressing) win the lottery, but I would not be able to guarantee that the rest would (those obtaining a 1), which would have to play with a very low probability of winning (exactly the same as all the other morons playing legally). The myselves which had had to play normally would repeat the same operation on the following day lottery draw. Again, they would have 50% chance of not playing and knowing immediately the winning combination (before the draw), and 50% of having to play with the combination they had obtained and (very likely) not winning the lottery for now. Then, the myselves which would have had to play would repeat the same procedure one more day, and so on. The chances of a myself being forced to play at least twenty times (so, about three weeks) without getting no winning combination before his respective draw would be about one in a million. If we increased it up to forty times (a bit more than a month), then those probabilities would be of one in a million millions. In fact, what I expected was that I would have to play normally none or just once before knowing a winning combination beforehand (try and calculate it!). Certainly, the method was worth a try.

It took me more than I imagined to win the lottery: I made it at my fourth try. But it was a huge satisfaction. Don’t get me wrong, I didn’t really need that money. The robberies I had done a few years back, by trivially breaking the codes of bank accounts of thousands of people, had made me millionaire. I abandoned those robberies when I became the most creative person on Earth. After producing thousands of works and marvelous patents in all arts and sciences, my royalties came to provide me with more money than the GDP of a middle-sized European country. So, it was obvious that I didn’t need to win the lottery for the money. Nonetheless, doing it, when I had thought that I would never be able to do it, filled me with satisfaction.

fig4

4.- Towards optimal short-range happiness

My success in the lottery case made me understand that I could now solve my past problems when, not so long ago, I tried to use the machine simply to be happy (that failure with that girl).

Such attempt failed because I knew what would give my success after some delay, when the opportunities had already gone. Nonetheless, after my previous lottery experience I knew that I might use the same method in order to avoid that delay. Again, I would use the machine to generate all the texts of advices to myself in order to be happy for the next year, for example, three pages each. Then I would press the button once more. If I obtained 1 in that last pressing, and the instructions received were susceptible of being obeyed somehow (I decided that I would not try to follow instructions that didn’t contain at least a readable sequence of seven or more letters) then I would obey those instructions within reason, and by the end of the year I would type a value scoring that year in terms of happiness. On the other hand, if I obtained a 0 in that last pressing (or if I obtained a 1 but the instructions were not readable), then I would ignore the instructions and type 0 into the machine so that those ignored instructions could not win. After doing that, I would immediately receive the sequence of 0s and 1s with the instructions that were obeyed by my other self who obtained the maximum possible happiness during the next year (or should I say “will be obeyed” and “will obtain”?). Since I would receive them immediately, I might follow those instructions from the very moment when my other self obtained them, and I would get his same success.

Regarding all others who would get a 1 in their last press and had to follow their instructions, they might repeat the same process the next year, when they might obtain a 0 after receiving new instructions and might finally profit from the success found by other. The probability that he would have to wait for forty tries (forty years) would be much less than one in a million millions (which is the probability of finding forty 0s in a row), because most of the instructions cannot be obeyed and would halt the waiting even obtaining a 1. Besides, the expected wait would be, really, between zero and one year (in fact, much closer to zero than to one): zero had half the chances and, besides, I would also have to wait for zero years if I obtained a 1 but the text was simply unreadable (and most texts are).

And so I did. I obtained a text of instructions with my pressings, but I simply discarded it because, after them, the last pressing gave me a 0. Then I introduced the value 0 in the machine, and immediately I started to receive the sequence of 0s and 1s from the happiest myself during the next year.

Just as the other time, the instructions again explained that I should start a certain sentimental relationship. The difference was that this time I knew the instructions at the same time as the successful myself knew them, not one year later.

I followed his steps, and this time I found that the lady in question was not in a relationship. In fact, in a short time I became her partner. The relationship was, in fact, wonderful.

When the year was over and the instructions were finished, I decided I needed no more instructions. I was happy. I would continue living that life.

Nonetheless, three months later she left me for another guy. I was devastated, it was horrible. In fact, by seeing how I ended up, I decided that the happiness of the previous year did not pay up for that pain. I wished I had never met her.

I cannot blame the other myself for recommending me, with his high score of happiness, those instructions. When he scored his year, evidently he did it before that breakup took place to himself, when a single year had gone by. He knew nothing.

This time, my problem was to assume that short-term happiness and long-term happiness would coincide. But it was not true.

fig5

5.- Towards a life-long happiness

There should be some way to solve that problem too. A possibility was to repeat the same process, but with a forty year span, instead of only one year. After forty years, the old myselves would score their live-long happiness, and the other myselves which had stayed in the present would be able to know which were the best decisions for a life. Some would explore all paths to happiness so others would benefit.

Then I thought that aim was far too ambitious. Those having to live forty years obeying absurd instructions would get no reward for their loyalty to the others. Then I found an alternative way to take into account the long-term happiness and combine it with the short-term one.

To start with, I would renounce trying to get detailed instructions, telling me every moment how to proceed. Circumstantial instructions, with orders which are only valid for certain moments, are only useful if you follow them at the same time, with the same opportunities, which made them quite restrictive. On the contrary, I would try to find time-independent commands, attitudes towards life, such as “be bold”, “be greedy”, “be resentful”, “be attentive”, etc. and combinations thereof. With this new target, I would only follow time-independent orders and disregard any instructions which were dependent on the current situation. I would search for the best life attitude, the best general way to proceed, instead of running after which steps to follow every moment, which in fact would make me slave of my destiny, written and known beforehand.

Once those new objectives were fixed, I would act in the following way. I would press the bifurcation button to obtain, with the resulting 0s and 1s, about three pages instructions which would show me which should be my attitudes in life. Then I would press one more time. If I obtained a 1, then I would obey those instructions during the next year. By the end of the year, I would press again. If I obtained a 1 again, then I would obey those same instructions for one more year. The process would repeat until I obtained, some year, a 0. When that happened, I would type into the machine a value that scored my happiness during all those years following those instructions. However, that score would also take into account for how many years I had followed those instructions. I worked out a way to score my happiness in the machine so that, the more years the instructions had served me, the higher the score. Let us imagine that two instructions texts made me equally happy, but one of them worked for more years. Then the latter would get a higher score. A short period of happiness would only get a larger score than a longer one if I had been much happier in the first. In that case, I might assume that that happiness in a short period might compensate for any disgrace that might come afterwards (which, in principle, would not be more likely than through any other path).

Again, the chances that one myself had to wait for 40 years without typing his score and getting to know the instructions in the best-scored universe would be, at most, one in a million millions. The procedure was definitely worth.

I put my plan into practice. After the sequence of 0s and 1s for my own instructions, I pressed the bifurcation button once more. I obtained a 1. That meant that I would have to put those instructions in practice for, at least, one year, after which I would have a new opportunity of pressing the bifurcation button to check if I could, at last, know the instructions which would take towards happiness.

The following year, I pressed again, and again I got a 1. One more year.

And the following year, the same. Another 1. And the same on the next one, and the other, and the other. It started to surprise me the unusual number of years during which I had to follow the received instructions.

After ten years getting 1s, I started to realize that I was getting old while I was playing that weird game. I wondered if, when I finally got the instructions from my happiest myself, I would be young enough for those optimal instructions to have the same effect on me.

The 1s continued repeating year after year. That was not normal. Something strange was happening. I started to distrust.

The day I reached twenty years getting 1 after 1, I meditated very seriously about my situation. The likelihood of waiting twenty years for the coveted 0 was less than one in a million.

More years followed with 1s. Less likelihood even. My mistrust had become unbearable. Of course, since all sequences were explored, someone had to receive the twenty-five 1s in a row. Notwithstanding, the likelihood that it was me was so low that I started to consider the possibility that someone was swindling me. Maybe some of the other myselves had found a way for them not to get so many 1s, which, just by elimination, forced those unending sequences of 1s on the others. Moreover, maybe almost all others had discovered that mysterious trick and I was one of the few morons which the others were using to see what happened after many years following the same instructions and take profit.

Then I reached the thirty 1s in a row. Dammit, thirty years ago I had less than one chance in a thousand million to reach this point. Definitely, the probability that someone was playing a trick on me seemed much larger than that.

I could not forget that, many years ago, I had used the machine to tell me something I didn’t know about the machine itself. It was that time when I discovered that the time in each parallel universe runs independently. Maybe I might use the machine to find a feasible way in which the other myselves would be swindling me. I might press many times to explore all possible texts which might explain those tricks to me.

But then I thought that would not work. Let’s say that there is really a way to swindle the other myselves so that one gets benefit, for example by making the others spend years and years following stupid instructions, a 1 after the next, while the swindler would receive immediately the fruits of that massive exploration by getting a 0. In that case, if I used the machine to bifurcate more the universe and find that trick (by exploring all texts explaining that trick), then the new myself who would discover the trick might save the trick for himself, and benefit by using it against me. No, I couldn’t use the machine to discover the Machiavellian plan that was for sure acting on me. I could not rely.

More years went by, and I got more 1s. Every year I got angrier when I saw that damned number. Dammit! The likelihood that someone was swindling me so I got all those 1s had to be, for sure, much larger than the probability of getting thirty-seven 1s in a row. That made no sense.

Then that year my mistrust exploded. I said enough. I had to accept that I would never be able to know how they had played with me. Notwithstanding, I might strike back. I decided I would wait only one more year. If I didn’t get a 0 next time, I would take revenge.

Then, the next year, the 0 came. It came!

That day I wept of happiness.

In fact, that day was yesterday.

Nevertheless, my happiness has become misgivings and suffering since last night.

As a matter of fact, I have not slept the whole night. I have thought that, very likely, I finally got the 0 simply because the other motherfuckers estimated that thirty-eight years would be the maximal amount of time anyone would stand it while being fooled. And, moreover, exactly thirty-eight tries! Of course, they avoided the 0 appearing exactly on the fortieth attempt, too round a number. That’s why they have given my 0 at the thirty-eighth attempt. Moreover, making me spend all my life following instructions is only useful if I score them before I die. Only then the instructions will enter the bag of all those evaluated and compared, out of which the perfect instructions will come and from which they all will benefit. So, in no way they would have given me 1s forever. They needed a 0 to come when I was very old. They’ve exploited me as much as they could. Fucking bastards!

I am sorry, but it’s too late for those motherfuckers. I’m a fucking old man, they cannot give me my youth back. With certainty, if now I enter my score, the new instructions I will then get, coming from the happiest myself, will be useless for me, because I am far too old to do anything worth it. Fucking bastards. I must complete my revenge against all of them in any case. It’s not important that I finally got a 0. They’ve fucked my life with their damned trick.

Now I find myself decrepit and disgusted, in front of my machine, ready to complete my revenge.

I approach the machine and I introduce the value which, according to what we all agreed four decades ago (when it was just I), would represent the maximum possible score. In fact, we had (I had) agreed that that score would be, in fact, unreachable, and that all scores would have to be beneath it. I press enter.

Fuck you, fuck you, bastards!! Now you all will receive the instructions that lead to this shitty life of mine! You all will spend your fucking life following shitty instructions, having a blind faith in the happiness that they should bring you, and waiting year after year that the disappointing time which has gone is to be compensated with the wonders that would come! Exactly what you have done to me, when I thought I would finally know the perfect instructions after receiving my 0! Fuck you, bastards!

I step away from the machine. Then I take a look at the instructions which, with iron discipline, I have stupidly followed during the last thirty-eight years.

“Mistrust always” is what those instructions say. Nothing more. The rest of the three pages is just blank symbols. That’s the end of the instructions.

It doesn’t say “Mistrust sometimes”, or “Mistrust moderately”. Simply, I should always mistrust. It was hard to do it the first years. But mistrust is an attitude to which you can get easily used, because it feeds back on itself when you think about all the screwing over that you can get. After a decade, I was not even conscious that I was applying my mistrust without thinking on any established plan. I didn’t even remember that I was compelled to be mistrust in order to obey some stupid instructions. I was mistrustful out of pure devotion.

I’m exultant. My final revenge against all the other myselves, against all those motherfuckers that no doubt have fooled me, is complete. It doesn’t matter that I won’t ever be able to prove what they did. Of course, somebody would have to receive that amount of 1s because all combinations of a number of 1s and a text of instructions are explored, as one universe is created for each. However, what’s the chance that this extremely unlucky myself would be, precisely, me? Something like one in a hundred thousand millions or so, right? The machine proved in the past that it hides unexpected surprises, so the chance that other myselves found a way to swindle me is obviously larger that the chance that the extremely unlucky myself would be, precisely, me. Obviously, they swindled! I would have to be an idiot not to see it! So they deserved a punishment.

The accomplishment of my revenge makes me happy. All those morons will now follow faithfully for years some instructions which, really, will never take them to happiness. All those years I have spent among broken hopes will be, finally, the just revenge that I desire for those who caused those years.

A moment… I realize I have to moderate my happiness. This is only a real revenge if coming to this very point where I am is not worth. If they all end up as happy as I am now, then this is not a revenge. I must not be happy!

Then I realize, relieved, that I am wrong thinking that way. The other myselves will not have, at the end of their days, the opportunity to revenge as I had. Unlike me, they will not be pressing the bifurcation button year after year waiting for the day when they would score their lives, having influence on the others. They will simply follow my instructions, expecting them to be the route to happiness.

When I realize that my revenge is really perfect, I smile again in peace.

I am really happy.

(Translated by me from the original Spanish version by Ismael Rodríguez Laguna, aka NP-complete in this blog. Thanks a lot to Tom LaGatta for his comments on this version.)

Epijournals

I hold the theory that ideas are much more social than we think. When the time is ripe, they simply flourish in many minds at the same time. For example, one year ago, in this blog we published an entry discussing the possibility of a peer-review system on the ArXiv. Some hot debate ensued, and some people used very harsh language. But time was ripe, and people more influential than I have started a project, called epijournals. So far, the only source of information is Tim Gower’s blog and the comments on it. [Note added: it's been pointed out to me that there is already an official link]

So, again, what is the problem? The problem is that basic research is funded, mainly, by public money. But the results of research are published in journals which are privately owned, and make a profit. Well, one might think that if it is a reasonable profit, and they add value to the publication, everything is fine. But the added value approaches zero asymptotically, and the profits are completely insane. Scientific journals get their stuff for free, since we scientists do not get any money for the articles. We do the research, typeset the articles, and make them camera ready. No charge. Also, we act as referees to judge the quality of other people’s papers, and very often also edit them. No charge! Yet, the journals are amazingly expensive. And sometimes they play harsh with the rights of the scientists to use their own work. E.g.: the dutch publisher Elsevier, which has become our bête noir, and against which we’re holding a boycott.

What value do the publishers give, anyway? Prestige. Their stamp of approval. If you publish an article in Nature, Physical Review Letters, or Cell, it means that it has gone through a careful selection process, it must be worth reading. But this revision process is done by… us scientists! And is done for free. So… if only the stamp matters, why not using it directly on the ArXiv?

The idea is simple: epijournals will work as regular journals, with editors and referees. The only difference will be that the journal webpage will not hold the articles themselves, only links to the ArXiv. So, you upload your paper to the ArXiv. Then, you write an email to the editor of your favourite epijournal, telling her your ArXiv number. She will send it to referees, make a decision and, if finally published, you’ll get a link in the epijournal webpage. Easy peasy.

As of now, it’s only mathematicians taking the step. It’s crucial that big guys send their papers to these epijournals, only then they will gather momentum and become prestigious. We, little people, need big publications to get tenure… so, please, please, if you have an important breakthrough in maths, consider submitting to one of them (when they start, of course).

What should the next step be? As of now, the “ranking of publications” is held by Thomson Reuters, a private company with a lot of power. For example, they compute (using a secret formula) the Libor and Euribor indices. They decide whether a journal is good enough or not to enter their Journal of Citation Reports, which charges the journals their revolutionary tax in order to stay in the list. Shouldn’t UNESCO take care of this task?

Other interesting issues related to epijournals:

  • Should we allow (signed) comments on the articles? I would like to.
  • Not only comments: I would love to chat with the authors of the papers I like. And, as an author, it would be so nice to chat with the readers. It will help reduce the feeling of irrelevance of theoretical work…
  • So, this would evolve in the direction of creating an on-line social network for scientists.
  • Is blind peer-review good enough? Why not double blind? (The referees do not know the authors either)
  • Make a two-stage refereeing system: a quick and dirty one, which would correspond to the current system, and another five years after publication, to assess what was the real impact.
  • About authorship: shouldn’t we allow for “partial” authorships? Or, more concretely: specifying what has everyone done in each paper. The current strategy of listing the authors in a certain “relevance order” is clearly not enough…

I hope more questions will come up in the comments… ;)

Quantum particle near an event horizon

As the next episode in our series about the Unruh effect (it gets hot when you accelerate), here can watch a video I have prepared which depicts how a quantum particle behaves near an event horizon.

So, what are we watching? The left and right panels show the spin-down and spin-up wavefunctions for a massless Dirac particle (a massless electron), initially at rest in Rindler spacetime. Colors correspond to phase. Because of the principle of equivalence, there are two alternate physical interpretations:

  • You are moving with constant acceleration rightwards. At time t=0 you drop a Dirac particle. It seems to move leftwards, just because you leave it behind. When it gets far away, it slows down. This is due to relativistic time-dilation.
  • There is a uniform gravitational field pointing leftwards. That’s why the Dirac particle accelerates in that direction. As it falls, it slows down. This is due to gravitational redshift.

Of course, the interference pattern which develops at the center is just quantum mechanics, nothing else. But when the particle reaches the edges of the box (top, bottom and right), new interference patterns appear which are spureous to our problem. That’s just the handicap of a finite-size simulation.

Nice, ein? This was work we developed at ICFO, Barcelona, along with Maciej Lewenstein, Alessio Celi and Jarek Korbicz. I have just showed it as a premiere during the Quantum gases meeting at CSIC in Madrid.

It’s hot when I accelerate!

Unruh effect and Hawking radiation

Let us discuss one of the most intriguing predictions of theoretical physics. Picture yourself moving through empty space with fixed acceleration, carrying along a particle detector. Despite the fact that space is empty, your detector will click sometimes. The number of clicks will increase if you accelerate further, and stop completely if you bring your acceleration to zero. It is called Unruh effect, and was predicted in 1976.

That’s weird, isn’t it? Well, we have not even scratched the surface of weirdness!

So, more weirdness. The particles will be detected at random times, and will have random energies. But, if you plot how many particles you get at each energy, you’ll get a thermal plot. I mean: the same plot that you would get from a thermal bath of particles at a given temperature T. And what is that temperature?

T = \hbar a / 2\pi c

That is called the Unruh temperature. So nice! All those universal constants… and an unexpected link between acceleration and temperature. How deep is this? We will try to uncover that.

In our previous Physics Napkin we discussed the geometry of spacetime felt by an accelerated observer: Rindler geometry. Take a look at that before jumping into this new stuff.

Has this been proved in the laboratory?

No, not at all. In fact, I am working, with my ICFO friends, in a proposal for a quantum simulation. But that’s another story, I will hold it for the next post.

So, if we have not seen it (yet), how sure are we that it is real? How far-fetched is the theory behind it? Is all this quantum gravity?

Good question! No, we don’t have any good theory of quantum gravity (I’m sorry, string theoreticians, it’s true). It’s a very clear conclusion from theories which have been thoroughly checked: quantum field theory and fixed-background general relativity. With fixed background I mean that the curvature of spacetime does not change.

Detecting particles where there were none… where does the energy come from?

From the force which keeps you accelerated! That’s true: whoever is pushing you would feel a certain drag, because some of the energy is being wasted in a creation of particles.

It's hot when I accelerate!! Ayayay!!!

It’s hot when I accelerate!! Ayayay!!!

I see \hbar appeared in the formula for the Unruh temperature. Is it a purely quantum phenomenon?

Yes, although there is a wave-like explanation to (most of) it. Whenever you move with respect to a wave source with constant speed, you will see its frequency Doppler-shifted. If you move with acceleration, the frequency will change in time. This change of frequency in time causes makes you lose track of phase, and really observe a mixture of frequencies. If you multiply frequencies by hbar, you get energies, and the result is just a thermal (Bose-Einstein) distribution!

But, really… is it quantum or not?

Yes. What is a particle? What is a vacuum? A vacuum is just the quantum state for matter which has the minimum energy, the ground state. Particles are excitations above it. All observers are equipped with a Hamiltonian, which is just a certain “way to measure energies”. Special relativity implies that all inertial observers must see the same vacuum. If the quantum state has minimal energy for an observer at rest, it will have minimal energy for all of them. But, what happens to non-inertial observers? They are equipped with a Hamiltonian, a way to measure energies, which is full of weird inertial forces and garbage. It’s no big wonder that, when they measure the energy of the vacuum, they find it’s not minimal. And, whenever it’s not minimal, it means that it’s full of particles. Yet… why a thermal distribution?

Is all this related to quantum information?

Short story: yes. As we explained in the previous post, an accelerated observer will always see an horizon appear behind him. Everything behind the horizon is lost to him, can not affect him, he can not affect it. There is a net loss of information about the system. This loss can be described as randomness, which can be read as thermal.

Long story. In quantum mechanics we distinguish two types of quantum states: pure and mixed. A pure quantum state is maximally determined, the uncertainty in its measurements is completely unavoidable. Now imagine a machine that can generate quantum systems at two possible pure states A and B, choosing which one to generate by tossing a coin which is hidden to you. The quantum system is now said to be in a mixed state: it can be in any two pure states, with certain probabilities. The system is correlated with the coin: if you could observe the coin, you would reduce your uncertainty about the quantum state.

The true vacuum, as measured by inertial observers, is a pure state. Although it is devoid of particles, it can not be said to be simple in any sense. Instead, it contains lots of correlations between different points of space. Those correlations, being purely quantum, are called entanglement. But, besides that, they are quite similar to the correlations between the quantum state and the coin.

When the horizon appears to the accelerated observer, some of those correlations are lost forever. Simply, because some points are gone forever. Your vacuum, therefore, will be in a mixed state as long as you do not have access to those points, i.e.: while the acceleration continues.

Where do we physicists use to find mixed states? In systems at a finite temperature. Each possible pure state gets a probability which depends on the quotient between its energy and the temperature. The thermal bath plays the role of a hidden coin. So, after all, it was not so strange that the vacuum, as measured by the accelerated observer, is seen as a thermal state.

Thermal dependence with position

As we explained in the previous post, the acceleration of different points in the reference frame of the (accelerated) observer are different. They increase as you approach the horizon, and become infinite there. That means that it will be hotter near the horizon, infinitely hotter, in fact.

After our explanation regarding the loss of correlations with points behind the horizon, it is not hard to understand why the Unruh effect is stronger near it. Those are the points which are more strongly correlated with the lost points.

But from a thermodynamic point of view, it is very strange to think that different points of space have different temperatures. Shouldn’t they tend to equilibrate?

No. In general relativity, in curved spacetime we learn that a system can be perfectly at thermal equilibrium with different local temperatures. Consider the space surrounding a heavy planet. Let us say that particles near the surface at at a given temperature. Some of them will escape to the outer regions, but they will lose energy in order to do so, so they will reach colder. Thus, in equilibrium systems, the temperature is proportional to the strength of gravity… again, acceleration. Everything seems to come together nicely.

And Hawking radiation?

Hawking predicted that, if you stand at rest near a black hole, you will detect a thermal bath of particles, and it will get hotter as you approach the event horizon. Is that weird or not? To us, not any more. Because in order to remain at rest near a black hole, you need a strong supporting force behind your feet. You feel a strong acceleration, which is… your weight. The way to feel no acceleration is just to fall freely. And, in that case, you would detect no Hawking radiation at all. So, Hawking radiation is just a particular case of Unruh effect.

There is the feeling in the theoretical physics community that the Unruh effect is, somehow, more fundamental than it seems. This relation between thermal effects and acceleration sounds so strange, yet everything falls into its place so easily, from so many different points of view. It’s the basis of the so-called black hole information paradox, which we will discuss some other day. There have been several attempts to take Unruh quite seriously and determine a new physical theory, typically a quantum gravity theory, out of it. The most famous may be the case of Verlinde’s entropic gravity. But that’s enough for today, isn’t it?

For references, see: Crispino et al., “The Unruh effect and its applications”.

I’ll deliver a talk about our proposal for a quantum simulator of the Unruh effect in Madrid, CSIC, C/ Serrano 123, on Monday 14th, at 12:20. You are all very welcome to come and discuss!

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_lift

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

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.

rindler2

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.

rindler3

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.

rindler4

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…