Our Mathematical Universe

by Max Tegmark

  As you remember, we encountered parallel universes in Chapter 6 as well, but of a different kind. To avoid confusing ourselves with an overdose of parallel universes, let’s review the terminology we agreed on in Chapter 6. By our Universe, we mean the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang, with its classical observed properties (which galaxies are where, what the history books say, etc.). In Chapter 6, we called other such spherical regions far away in our large or infinite space Level I parallel universes or Level II parallel universes, depending on whether they had our effective laws of physics or not. Let’s call the quantum parallel universes that Everett discovered Level III parallel universes, and the collection of all of them the Level III multiverse. Where are these parallel universes? Whereas the Level I and Level II kinds are far away in our good old three-dimensional space, the Level III ones can be right here as far as these three dimensions are concerned, but separated from us in what mathematicians call Hilbert space, an abstract space with infinitely many dimensions where the wavefunction lives.3

  After being dismissed and almost completely ignored for a decade, Everett’s version of quantum mechanics first began to get popularized by the famous quantum-gravity theorist Bryce DeWitt, who called it the Many Worlds interpretation—a name that stuck. When I later met Bryce, he told me he’d at first complained to Hugh Everett, saying that he liked his math, but was really bothered by the gut feeling that he just didn’t feel like he was constantly splitting into parallel versions of himself. He told me that Everett had responded with a question: “Do you feel like you’re orbiting the Sun at thirty kilometers per second?” “Touché!” Bryce had exclaimed, and conceded defeat on the spot. Just as classical physics predicts both that we’re zooming around the Sun and that we won’t feel it, Everett showed that collapse-free quantum physics predicts both that we’re splitting and that we won’t feel it.

  Sometimes it’s hard to reconcile what I believe with what I feel. Fast-forward to May 1999, and I’m waiting for the stork to arrive with my first son. I feel anxious, and hope that the delivery will end well. But at the same time, my physics calculations have convinced me that it will both end well and end badly, in different parallel universes. And in that case, what do I mean by hoping? Perhaps I mean that I hope that I’ll end up in one of those parallel universes where things went well? No, that’s nonsense, since I’ll end up in all of these parallel universes, and am jubilant in some and devastated in others. Hmmm. Perhaps I mean that I hope that the delivery will go well in most of the parallel universes? No, that’s nonsense as well, since the percentage where things go well can in principle be calculated using the Schrödinger equation, and it’s illogical to have hopes about something that’s already predetermined. But apparently—and perhaps fortunately—my emotions aren’t completely logical.

  * * *

  1His thesis finally went online in 2008, and you can read it here: http://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf. The notion that at certain magic instances, reality undergoes some sort of metaphysical split into two branches that subsequently never interact isn’t only a misrepresentation of Everett’s thesis, but also inconsistent with Everett’s postulate that the wavefunction never collapses, since the subsequent developments could in principle make the branches interfere with each other. According to Everett, there is, was and always will be only one wavefunction, and only decoherence calculations (which I’ll explain later in this chapter), not postulates, can tell us when it’s a good approximation to treat two branches as non-interacting.

  2In practice, this unstable card will of course get toppled in no time by a tiny air current, so it would be better to take a normal card with a thick bottom edge and use a quantum device such as Schrödinger’s radioactive-atom trigger to nudge it one way or the other.

  3The wavefunction corresponds to a single point in this infinite-dimensional space, and the Schrödinger equation says that this point will orbit around the center of the space at a fixed distance.

  The Illusion of Randomness

  I had more questions. It was well known that if you repeated a quantum experiment many times, you’d typically get different results seemingly at random: for example, you can measure the spin direction of lots of identically prepared atoms in such a way that you’ll get a seemingly random sequence of results—say “clockwise,” “counterclockwise,” “clockwise,” “clockwise,” “counterclockwise,” etc. Quantum mechanics won’t predict the outcomes, merely the probability of different outcomes. But this probability business was all part of the collapse postulate from the Copenhagen interpretation, so after Everett dropped it, how could he get quantum mechanics to predict anything random? There’s nothing random at all about the Schrödinger equation: if you know the wavefunction of our Universe right now, it will in principle let you predict what the wavefunction will be at any time in the future.

  In the fall of 1991, I signed up for an unusual course on the interpretation of quantum mechanics that was taught by a fellow grad student, Andy Elby. His dorm room used to be next to my girlfriend’s, and his door would be adorned with helpful advice such as “How to procrastinate in 7 easy steps.” Like me, he was very interested in what quantum mechanics really meant, and as part of his course, he let me give two lectures about Everett’s work. This was an exciting rite of passage for me, since it was my first time ever giving a talk about physics, and I spent much of it on how Everett explained randomness. First of all, if you do the Quantum Cards experiment (Figure 8.1), both copies of you afterward (each effectively in a separate parallel universe) will see a definite outcome. Both will feel that this outcome was random in the sense that there was no way to predict it: for any predicted outcome, the opposite outcome also occurred in an equally real universe. Now what about probabilities—where do they come sailing in from? Well, if you repeat this experiment with four cards, there will be 24 = 16 outcomes (Figure 8.2), and in most of these cases, it will appear to you that queens occur randomly, with roughly 50% probability. Only in two of the sixteen cases will you get the same result all four times. As you repeat the experiment more and more times, things start getting interesting. According to a 1909 theorem by the French mathematician Émile Borel, you’ll observe queens 50% of the time in almost all cases (in all cases except for what mathematicians call a set of measure zero) if you repeat the card experiment infinitely many times. Almost all of the copies of you in the final superposition will therefore conclude that the laws of probability apply even though the underlying physics (the Schrödinger equation) isn’t random at all.

  Figure 8.2: The origin of quantum probabilities. According to quantum physics, a card perfectly balanced on its edge will by symmetry fall down in both directions at once, in what’s known as a superposition. If you’ve bet money on the queen landing face-up, the state of the world will become a superposition of two outcomes: your smiling with the queen face-up and your frowning with the queen face-down. In each case, you’re unaware of the other outcome and feel as if the card fell randomly. If you repeat this experiment with four cards, there will be 2 × 2 × 2 × 2 = 16 outcomes (see figure). In most of these cases, it will appear to you that queens occur randomly, with about 50% probability. Only in two of the sixteen cases will you get the same result all four times. If you repeat 400 times, most of the 2400 outcomes have about 50% queens (top right). According to a famous theorem, you’ll observe queens 50% of the time in almost all cases in the limit where you repeat the card experiment infinitely many times. Almost all of the copies of you in the final superposition will therefore conclude that the laws of probability apply even though the underlying physics isn’t random and, as Einstein put it, “God doesn’t play dice.”

  Click here to see a larger image.

  In other words, the subjective perception of a copy of you in a typical parallel universe is a seemingly random sequence of wins and losses, behaving as if generated through a random process wi
th probabilities of 50% for each outcome. This experiment can be made more rigorous if you take notes on a piece of paper, writing “1” every time you win and “0” every time you lose, and place a decimal point in front of it all. For example, if you lose, lose, win, lose, win, win, win, lose, lose and win, you’d write “.0010111001.” But this is just what real numbers between zero and one look like when written out in binary, the way computers usually write them on the hard drive! If you imagine repeating the Quantum Cards experiment infinitely many times, your piece of paper would have infinitely many digits written on it, so you can match each parallel universe with a number between zero and one. Now what Borel’s theorem proves is that almost all of these numbers have 50% of their decimals equal to 0 and 50% equal to 1, so this means that almost all of the parallel universes have you winning 50% of the time and losing 50% of the time.1 It’s not just that the percentages come out right. The number “.010101010101 …” has 50% of its digits equal to 0 but clearly isn’t random, since it has a simple pattern. Borel’s theorem can be generalized to show that almost all numbers have random-looking digits with no patterns whatsoever. This means that in almost all Level III parallel universes, your sequence of wins and losses will also be totally random, without any pattern, so that all that can be predicted is that you’ll win 50% of the time.

  It gradually hit me that this illusion of randomness business really wasn’t specific to quantum mechanics at all. Suppose that some future technology allows you to be cloned while you’re sleeping, and that your two copies are placed in rooms numbered 0 and 1 (Figure 8.3). When they wake up, they’ll both feel that the room number they read is completely unpredictable and random. If in the future, it becomes possible for you to upload your mind to a computer, then what I’m saying here will feel totally obvious and intuitive to you, since cloning yourself will be as easy as making a copy of your software. If you repeated the cloning experiment from Figure 8.3 many times and wrote down your room number each time, you’d in almost all cases find that the sequence of zeros and ones you’d written looked random, with zeros occurring about 50% of the time.

  Figure 8.3: The illusion of randomness occurs whenever you clone yourself, so there’s really nothing specifically quantum-mechanical about it. If some future technology allows my son Philip to be cloned while he’s asleep, and his two copies are placed in rooms numbered 0 and 1, then they’ll both feel that the room number they read on awakening is completely unpredictable and random.

  In other words, causal physics will produce the illusion of randomness from your subjective viewpoint in any circumstance where you’re being cloned. The fundamental reason that quantum mechanics appears random even though the wavefunction evolves deterministically is that the Schrödinger equation can evolve a wavefunction with a single you into one with clones of you in parallel universes.

  So how does it feel when you get cloned? It feels random! And every time something fundamentally random appears to happen to you, which couldn’t have been predicted even in principle, it’s a sign that you’ve been cloned.

  Hugh Everett’s work is still controversial, but I think that he was right and that the wavefunction never collapses. I also think that he’ll one day be recognized as a genius on par with Newton and Einstein—at least in most parallel universes. Unfortunately, in this particular universe, his work was almost completely dismissed and ignored for over a decade. He didn’t get a job in physics, became rather bitter and withdrawn, smoked and drank too much, and died of an early heart attack in 1982. I’ve learned more about him recently because I got to meet his son, Mark, at the shooting of a TV documentary called Parallel Worlds, Parallel Lives. The producer wanted me to explain his dad’s work to him, and I felt lucky and honored: back when I stood there in that radical Berkeley bookstore, I couldn’t in my wildest dreams have imagined that I’d one day get this personal connection to one of my physics superheroes. Mark is a rock star, and if you’ve seen Shrek, you’ve heard him sing. His dad’s fate has really tormented his family, and you can hear it in many of his songs. He and his sister had almost no contact with their dad even though they lived together. His sister committed suicide, leaving a note saying that she was going to visit her dad in a parallel universe.

  Figure 8.4: Hugh Everett’s rock-star son, Mark, pondering his dad’s theory with me in 2007.

  Since I believe that Hugh Everett’s parallel universes are real, I can’t help thinking about what they’re like. In our Universe, he was rejected from the Princeton Physics Department for grad school, went to the Math Department, and transferred to Physics a year later. Because of his limited time, his quantum work was his only work. In many other universes, I think he was admitted to the Princeton Physics Department from the start and had time to make his mark with more mainstream research first, making his subsequent quantum ideas harder to ignore. This launched him on a career similar to that of Einstein, whose special theory of relativity was also met with initial suspicion (especially coming from a guy working outside academia as a patent clerk), but couldn’t be ignored because Einstein had already made a name for himself with previous discoveries. Just as Einstein stayed in academia and went on to discover general relativity, Everett, too, got the stability of a professorship and made further breakthroughs as remarkable as his first—ah, how I wonder what he discovered.…

  One event that I think Everett would have enjoyed took place in late August 2001, at Martin Rees’s house in Cambridge, where he’d gathered many of the world’s leading physicists for an informal meeting about parallel universes and related topics. To me, this was the first time when parallel universes started feeling scientifically respectable (albeit still controversial). I think many participants stopped feeling guilty and embarrassed about harboring such interests once they saw who else was there, and jokingly said things like, “Uh … what are you doing at a suspect meeting like this?…” During a long and intense group discussion about parallel universes, I suddenly realized that part of the discord was caused by mere misunderstandings rooted in crude language usage: different people were using the term parallel universe to refer to several quite different ideas! Wait, I thought, there are two—no three—different kinds! No—four! After thinking it through, I raised my hand and proposed the Level IV multiverse classification scheme that I’m using in this book.

  As brilliant as it was, Everett’s thesis left one important question unanswered: if a large object can really be in two places at once, why don’t we ever observe that? Sure, if you measure its position, the two copies of you in the two resulting parallel universes will each find it in a definite place. But that answer turns out not to be good enough, because careful experiments show that large objects never act like they’re in two places at once, even if you don’t look at them. In particular, they never display wavelike properties that make so-called quantum interference patterns. It wasn’t just Everett’s thesis that lacked an answer to this puzzle—there were no answers in my textbooks either.

  * * *

  1It’s interesting to note that Borel’s theorem made a strong impression on many mathematicians of the time, some of whom felt that the whole concept of probability was too philosophical to qualify as rigorous mathematics. Suddenly Borel confronted them with a theorem at the heart of classical mathematics that could be reinterpreted in terms of probabilities even though the theorem itself never mentioned probabilities at all. Borel would undoubtedly have been interested to know that his work showed the emergence of probabilities “out of the blue” not only in mathematics, but in physics as well.

  Quantum Censorship

  Holy guacamole! It works!!! It’s late November 1991 in Berkeley, it’s dark outside, and I’m at home at my desk frantically scribbling math symbols on a piece of paper. I felt a surge of excitement of a kind I’d never experienced before. Wow. Can it really be that I—little inconsequential me—have just discovered something really important? I just have to find out.

  I think that often, in science, th
e hardest part isn’t finding the right answer, but finding the right question. If you hit on a really interesting and well-formulated physics question, then it can take on a life of its own, automatically telling you what calculation you need to do to answer it, and the rest is almost automatic: even if the math takes hours or days, it feels a lot like mechanically pulling in a fishing line to see what you’ve caught. I’d just stumbled upon one of those lucky questions.

  I’d learned that the business about the wavefunction collapsing could be elegantly summarized mathematically in terms of a table of numbers called a density matrix in quantum-physics jargon, which encodes not only the state of something (its wavefunction), but also my perhaps incomplete knowledge of what the wavefunction is.1 For example, if something can be in only two different places, my knowledge about it can be described by a two-by-two table of numbers, as in these two examples:

  In both cases, the probability that I’ll find it in either place is 0.5, and that’s encoded by the two numbers on the diagonal of each matrix (the 0.5 in the upper left corner and the 0.5 in the lower right corner). The other two numbers in each table, “the off-diagonal elements of the density matrix” as we call them in geek-speak, encode the difference between quantum and classical uncertainty: when they, too, equal 0.5, we have a quantum superposition on our hands (Schrödinger’s cat is dead and alive, say), but when they equal zero, we’re effectively dealing with good old classical uncertainty, such as when I can’t remember where my keys are. So if you manage to replace these off-diagonal numbers by zeros, then you’ve turned and into or and collapsed the wavefunction!