Our Mathematical Universe

by Max Tegmark

  As we saw in the last chapter, the Copenhagen interpretation of quantum mechanics says that if your friend observes the object without telling you the outcome, then she’ll collapse the wavefunction so that the object is either here or there, and you simply don’t know which. In other words, the Copenhagen interpretation says that an observation somehow makes these off-diagonal numbers zero. I wondered if there might be some less mysterious physical process that did the same thing. If you have an isolated system that’s not interacting with anything else, then it’s easy to prove using the Schrödinger equation that those pesky numbers will never go away. But real systems are almost never isolated, and I asked myself what effect that would have. For example, as you read this sentence, air molecules and photons are constantly colliding with you. So if something is in two places at once, what happens to the two-by-two table of numbers describing it when something else bounces off it?

  This was one of those wonderful self-answering questions, and the rest was automatic. I simply considered the object and the colliding particle together as a single isolated system, and used the Schrödinger equation to calculate what would happen. A couple of hours later, I sat there with pages full of math symbols and gasped: those off-diagonal numbers changed to very close to zero, just as if the wavefunction had collapsed! It hadn’t really collapsed, of course, and those parallel universes were still alive and well, but here was a brand-new effect that looked like wavefunction collapse and smelled like wavefunction collapse, and just as a true collapse would have done, made it impossible to ever observe the object in two places at once. So the quantum weirdness doesn’t go away, it just gets censored!

  I concluded that quantum mechanics requires secrecy: an object can only be found in two places at once in a quantum superposition as long as its position is kept secret from the rest of the world. If the secret gets out, all quantum superposition effects become unobservable, and it’s for all practical purposes as if it’s either here or there and you simply don’t know which. If a lab technician measures the position and writes it down, the information is obviously out. But even if a single photon bounces off the object, the information about its whereabouts is out: it gets encoded in the subsequent position of the photon. As illustrated in Figure 8.5, a nanosecond later, the photon will be in two quite different places depending on the position of the object, so by measuring where the photon is, you’ll find out where the mirror is.

  Back in the beginning of the last chapter, I was wondering whether you needed a human observer to collapse the wavefunction, or if a robot would suffice. Now I was convinced that consciousness had nothing to do with it, since even a single particle could do the trick: a single photon bouncing off of an object had the same effect as if a person observed it. I realized that quantum observation isn’t about consciousness, but simply about the transfer of information. Finally I understood why we never see macroscopic objects in two places at once even if they’re in two places at once: it’s not because they’re big, but because they’re hard to isolate! A bowling ball outdoors typically gets struck by about 1020 photons and 1027 air molecules every second. It’s by definition impossible for me to see something without it getting struck by photons, since I can only see it when photons (light) bounce off it, so a bowling ball that’s in two places at once will have its quantum superposition ruined even before I have a chance to become consciously aware of it. In contrast, if you pump out as many air molecules as you can with a good vacuum pump, an electron can typically survive for about a second without colliding with anything, which is plenty enough time for it to demonstrate funky quantum-superposition behavior. For example, it takes only a quadrillionth as long (about 10−15 seconds) for an electron to orbit an atom, so there will be almost no effect on its ability to be on all sides of the atom at once.

  Moreover, if an air molecule bounces off of a bowling ball and encodes information about its position in its own position (as in Figure 8.5), this molecule will soon collide with many other molecules, which will get the information, too. It’s a lot like when Wikileaks posts classified information online: it gets copied, then the copies get copied, and soon the cat is so out of the bag that it’s in practice impossible to make the information secret again. And if you can’t make the information secret again, then the quantum superposition can’t be restored. Now I finally understood why Level III parallel universes stay parallel!

  I felt that I was on a roll that night. I also worked this stuff out in more quantitative detail. For example, most things can be not just in two places but in many, and I worked out this case, too, as illustrated in Figure 8.6. Basically, I discovered that a photon mostly destroys the quantum superposition, but lets a bit of it survive: a superposition only as wide as its wavelength. A photon of wavelength 0.0005 millimeter essentially acts like an observer who can only measure the position of something to an accuracy of 0.0005 millimeter. We saw in the last chapter that all particles act like waves and have a wavelength, and I showed that when any particle whatsoever bounces off something, quantum superpositions wider than the wavelength get destroyed.

  For years now, I’d known that I loved physics and wanted to dedicate my life to it. But I’d always wondered whether I had it in me to be able to contribute to it, as opposed to just learning about it and cheering it on from the sidelines of the field. As I finally drifted off to sleep that night, for the first time in my life, I thought: Yes I can! Might my discovery become known as the “Tegmark effect”? I knew that whatever happened, I’d never forget my excitement that evening. I felt so fortunate for all the opportunities I’d been given and for all the inspiring people who’d enabled me to join the great adventure of science. It seemed almost too good to be true. And it was.…

  Figure 8.5: If you take a flash photo in a dark room, the photons returning to your camera have encoded information about what’s in the room. The figure shows how even a single photon can “measure” things: after a photon has bounced off a mirror, it encodes information about the mirror’s position in its own position. If the mirror is at both A and B in a quantum superposition, then it doesn’t matter whether it’s a human or just a photon that finds out where it is: in either case, the quantum superposition is effectively destroyed.

  Figure 8.6: Your knowledge of the position of the fallen card can be described by a so-called density matrix, which can be represented as a bumpy surface as illustrated above. The height of the surface along the diagonal (dashed line) gives the probability that you’ll find the card in various places, whereas the height of the surface elsewhere, loosely speaking, specifies the amount of quantum weirdness, the extent to which the card is in more than one place at once. The left density matrix corresponds to the card being equally in both of the two places depicted underneath, in quantum superposition, as revealed by the two peaks labeled “Quantum interference.” After a photon bounces off of the card, decoherence eliminates these two peaks, giving the density matrix on the right, which corresponds to the card effectively being in only one of the two places, you simply not knowing which. The slight widths of these peaks correspond to some remaining quantum uncertainty around the face-up and face-down positions.

  Click here to see a larger image.

  Two weeks later, I’d expanded my calculations into a first draft of a paper which I called “Apparent Wave Function Collapse Caused by Scattering,” scattering being the technical term we use for the behavior of particles bouncing off stuff. This was the first time ever I was writing a paper for publication, and I felt like when I was a little kid on Christmas Eve. My left-handed handwriting had always been hideous (pretty much every school assignment would come back with comments such as “Work on neatness!”), and it was exciting to see my illegible scribbles transform into beautiful typeset equations. At the same time, it was funny how paranoid I was getting that someone had already discovered what I had and I’d somehow missed it. I figured that something this basic would have been mentioned in the textbooks and taught in my grad quantum
class if it were known, but nonetheless, I almost trembled each time I opened a suspicious reference during my literature search. So far, so good.…

  Anticipating my publishing debut, I even went ahead and changed my surname to something more unique, from my dad’s name Shapiro to my mom’s name Tegmark. I’d enjoyed having the name Shapiro back in Sweden, since it was so unusual: we used to be the only family in the whole country who had it. To my horror, I discovered that it was about as unique in international academia as Andersson had been back home. The last straw for me was when I did a database search for physics papers by “M. Shapiro” and got thousands of hits. There were even three M. Shapiros in my own Berkeley Physics Department, one of whom (Marjorie) taught me particle physics! In contrast, my mom and her relatives were the only Tegmarks on the planet, as far as I could tell. I was a bit concerned that my dad might misinterpret the name change as some sort of rejection of him, but when I asked him about this, he assured me he didn’t mind with a Shakespeare quote: “What’s in a name?”

  * * *

  1Density matrices are generalizations of wavefunctions. For every wavefunction, there’s a corresponding density matrix, and there’s a corresponding Schrödinger equation for density matrices. If you’re a mathematically inclined reader and think of the wavefunction ψ as a complex number ψi for each possible state i, then the corresponding density matrix is ρij = ψiψj*, where the star denotes complex conjugation. If you don’t know the wavefunction of an object, and know only the probability that it has certain particular wavefunctions, then you should use the density matrix that’s the weighted average of the density matrices corresponding to these wavefunctions.

  The Joys of Getting Scooped

  It wasn’t until a month later, after I’d returned from Christmas holidays in Sweden and was just about to submit the paper, that it all came crashing down. All that time. All that enthusiasm. All that fun. All that excitement. All that hope. And—boom!—it took just a few minutes for it all to go up in smoke. Who lit the match? Andy Elby, actually. By telling me what a Polish physicist named Wojciech Zurek had already done. Forget the Tegmark effect—it already had a name: decoherence. In fact, I soon learned that the German physicist Dieter Zeh had discovered the effect already back in 1970.

  At first I didn’t feel much, as usual when I get bad news. Then I joked about it with my friends Wayne, Justin and Ted. Then I went home, without realizing that I was really close to the edge, and got into a stupid argument with my girlfriend about something utterly trivial: she’d made just enough rice for herself and a girlfriend, handing me some frozen rice from the freezer instead. All of a sudden I felt so sad that I wanted to cry, but didn’t even manage to accomplish that.

  Gradually, I’ve come to totally change my feelings about getting scooped. First of all, the main reason I’m doing science is that I delight in discovering things, and it’s every bit as exciting to rediscover something as it is to be the first to discover it—because at the time of the discovery, you don’t know which is the case. Second, since I believe that there are other more advanced civilizations out there—in parallel universes if not in our own—everything we come up with here on our particular planet is a rediscovery, and that fact clearly doesn’t spoil the fun. Third, when you discover something for yourself, you probably understand it more deeply and you certainly appreciate it more. From studying history, I’ve also come to realize that a large fraction of all breakthroughs in science were repeatedly rediscovered—when the right questions are floating around and the tools to tackle them are available, many people will naturally find the same answers independently. From quantum class, I remember Eugene Commins’s deadpan quip: “It’s called the Klein-Gordon equation because it was discovered by Schrödinger.”

  I’ve rediscovered many other things since, and what you usually find is that you’ve rediscovered all the basic stuff, and that you’ve also worked out some interesting details others hadn’t and vice versa, enabling you to still salvage a toned-down publication that acknowledges the prior work and adds something to it. This time it was almost spooky: I’d make a top-ten list of natural sources of decoherence, from obvious stuff such as air and sunlight to hard-to-shield things such as background radioactivity and neutrinos from the Sun—and then I found a beautiful paper by Zeh and his student Erich Joos from six years earlier with a virtually identical table. I still had enough new stuff in my paper (http://arxiv.org/pdf/gr-qc/9310032.pdf) to manage to get it published in a less prestigious journal, but if I’d hoped to start my publishing career with a great splash, this felt like more of a belly flop.

  In hindsight, the most hilarious scooping I’ve ever had wasn’t this first one, but in 1995, when I’d invented a technique for measuring the quantum state (wavefunction or density matrix) of a particle. I’ll never forget how my jaw dropped the night I was going to submit it and stood there like an idiot, staring at a published article in the empty library: these guys hadn’t just scooped me, but they’d made a really elaborate and pedagogical figure that was virtually identical to my plot, and they’d coined exactly the same obscure name as I had for the technique: phase-space tomography. All I could do was exclaim “HURF!”—a special word my brother Per and I have invented which really captured the moment.

  I eventually got to meet many of these intimidating anonymous competitors, and discovered that they were all really nice people. Zeh and Zurek both sent me encouraging emails about my work and invited me to visit them and give talks. In 2004, I visited Wojciech Zurek in Los Alamos and discovered one of the most amazing perks of being a scientist: you get invited to visit exotic places where you spend all your time talking with fascinating people—and you get to call it work! And they even pay for your trip! Wojciech Zurek had big burly hair and a wild impish glint in his eyes, revealing his taste for adventure in both research and recreation. He once persuaded me to climb beneath a rock overhang in the cordoned-off area next to Iceland’s mighty Gullfoss waterfall and go within a meter of the falling water—when the cascade suddenly shifted direction, I wondered how many parallel universes had just lost two decoherence theorists in one fell swoop. When I visited Dieter Zeh and his group in Heidelberg in 1996, I was struck by how few accolades he’d gotten for his hugely important discovery of decoherence. Indeed, his curmudgeonly colleagues in the Heidelberg Physics Department had largely dismissed his work as too philosophical, even though their department was located on “Philosopher Street.” His group meetings had been moved to a church building, and I was astonished to learn that the only funding that he’d been able to get to write the first-ever book on decoherence came from the German Lutheran Church.

  This really drove home to me that Hugh Everett was no exception: studying the foundations of physics isn’t a recipe for glamour and fame. It’s more like art: the best reason to do it is because you love it. Only a small minority of my physics colleagues choose to work on the really big questions, and when I meet them, I feel a real kinship. I imagine that a group of friends who’ve passed up on lucrative career options to become poets might feel a similar bond, knowing that they’re all in it not for the money but for the intellectual adventure.

  Whenever the person next to me on the plane asks me science questions, I’m reminded of the correct way to think about competition and getting scooped. There in the airplane seat, I’m the ambassador from Physics Land, taking great joy and pride in describing not what I’ve personally done, but what we physicists as a community have done. Sometimes I scoop them, more often they scoop me, but the key point is that together we can learn from each other, inspire each other, and accomplish more than any single person could in their wildest dreams. It’s a wonderful community, and I feel extremely fortunate to get to be part of it.

  Why Your Brain Isn’t a Quantum Computer

  “Sir Roger Penrose is incoherent, and Max Tegmark says he can prove it.” Whoa! I was reading the first line of a news article in the February 4, 2000, issue of the journal Science, and felt r
ather taken aback. I’d never called this famous mathematical physicist incoherent, but journalists tend to like both conflict and puns, and I’d written a paper (http://arxiv.org/abs/quant-ph/9907009) arguing that one of Penrose’s ideas was killed by decoherence.

  In recent years, there’s been a surge of interest in building so-called quantum computers, which would exploit the weirdness of quantum mechanics to solve certain problems faster. For example, if you bought this book online, your credit-card number was encrypted with a method based on the fact that multiplying two 300-digit prime numbers together is quick, but factoring the resulting 600-digit number (figuring out which two numbers it’s the product of) is hard, and would take longer than the age of our Universe on today’s best computers. If a large quantum computer can be built, then a hacker could use it to get the answer quite quickly and steal your money, using a quantum algorithm invented by my MIT colleague Peter Shor. As quantum-computing pioneer David Deutsch puts it, “Quantum computers share information with huge numbers of versions of themselves throughout the multiverse,” and can get answers faster here in our Universe by, in a sense, getting help from these other versions. A quantum computer could also simulate the behavior of atoms and molecules quite efficiently, replacing measurements in chemistry labs in the same way that simulations on traditional computers have replaced measurements in wind tunnels. Many modern computers calculate faster by using multiple processors in parallel. A quantum computer can be thought of as the ultimate parallel computer, using the Level III multiverse as a computational resource and in a certain limited sense running different parallel calculations in parallel universes.