Our Mathematical Universe

by Max Tegmark

  In March 2008, Anthony told me about an idea (which I’ll explain in a moment) for a possible solution that his old Harvard professor David Layzer had suggested, and we spent two exciting hours in a Belmont café scribbling math symbols on the backs of napkins1—but in vain. We couldn’t manage to make the math work. But I couldn’t let go of the idea either. Two years later, I started obsessing about this again, and found a 1968 paper by the quantum-gravity theorist Jim Hartle that I felt contained another piece of the puzzle. But as I sat there in my Winchester apartment late in the evening of March 6, 2010, I just couldn’t get the pieces to fit together. Frustrated, I decided to take a thinking walk through town. To my amazement, after five minutes in the wintry air, it finally clicked! I suddenly realized how to solve both problems in one fell swoop: unify the two multiverse levels and understand the unequal probabilities. It kept me up until about three a.m., and consumed me all of the next day in that wonderful trancelike way that you have to experience to fully understand. I felt that it was one of my most exciting clicks since rediscovering decoherence nineteen years earlier, and couldn’t let go until I’d typed up a four-page skeleton paper for Anthony.

  Figure 8.11: How the Level I and Level III multiverses are unified. Each circle represents a planet where you’ve bet money that your quantum card will land face-up. Before the measurement, you’re in a neutral mood; afterward, you’re either happy because you won or sad because you lost. The card starts just ever so slightly tilted, so that you expect to win with probability 2/3. These planets are typically very far apart, say, a googolplex meters in various directions, but I’ve drawn them side by side in a straight line to illustrate the key points.

  Click here to see a larger image.

  Figure 8.11 illustrates the key idea. Suppose that you’re about to perform the Quantum Cards experiment with the card slightly tilted, so that you expect to see it fall face-up and win $100 with a probability of 2/3. In the old-school view (shown to the left in each rectangle in Figure 8.11), there’s one copy of you to start with and either one or two copies after the experiment, depending on whether the wavefunction collapses or not: if the Copenhagen interpretation is correct, there will be one definite outcome generated at random, while if Everett is correct, there will be two parallel universes, each containing a single copy of you: one where you’re happy about winning and one where you’re sad.

  Now let’s instead assume that the Level I multiverse from Chapter 6 exists, as modern cosmology suggests. This means that an infinite number of indistinguishable copies of you are about to perform the exact same experiment on other planets far, far away in space, illustrated by the strip of neutral faces in the figure. In my calculation, I applied the Schrödinger equation to the wavefunction describing the entire collection of particles making up all these copies of you and your experiment.

  What ends up happening? If the wavefunction collapses, then you get a single random outcome for the infinite space (the Level I multiverse), such that you’re happy on 2/3 of the planets and sad on 1/3 of them—that’s not surprising. If Everett is right in that there’s no collapse, then you end up with the entire infinite space in a quantum superposition of different states, each of which has you happy on some planets and sad on others. Now here’s the kicker: all of those states of space turn out to be indistinguishable from one another, with you being happy on exactly 2/3 of the infinitely many planets! Any finite sequence of planets with happy and sad outcomes in one of those states can be found somewhere else in space in each of the other states. You might think that there should also be states of space that are different, say, one where you’re happy on every single planet. However, using the Schrödinger equation and the mathematics of Hilbert space, I was able to prove that the wavefunction you actually get is equal to simply a superposition of infinitely many indistinguishable states. Anthony and I found this striking for several reasons.

  First of all, the great debate about whether the wavefunction collapses ends in a grand anticlimax: it simply doesn’t matter! Figure 8.11 illustrates that regardless of whether Everett is right or not, you’re happy on 2/3 of the planets. Indeed, both sides of the collapse debate emerge a bit bruised. The Copenhagen interpretation introduced this controversial collapse business to get rid of pesky parallel universes and obtain a unique outcome, yet you can see in the figure that this no longer helps: even with collapse, you still end up with parallel universes with both outcomes. The Everett interpretation had Level III (quantum) parallel universes as its hallmark, but you can see in the figure that you can safely ignore them, because they’re all indistinguishable. In this sense, the Level I and Level III multiverses are unified: as long as you have an infinite space with a Level I multiverse, you can ignore all its Level III parallel universes, since they’re in practice all just identical copies. Perhaps Level III can be unified with Level II as well, but we haven’t yet been able to prove that.

  Second, Figure 8.11 illustrates the origin of unequal probabilities by bringing Everett’s many worlds to our good old three-dimensional space: the different outcomes aren’t merely happening somewhere else in this hard-to-grasp mathematical Hilbert space, but also far away in our own space that we study with telescopes. Here, the key point is that after your card has fallen down but before you’ve opened your eyes and looked at it, you have no way of knowing which of your many copies you are, since they all feel subjectively indistinguishable up to this point. You therefore have to consider yourself to be a random member of this group of copies. Since you know that 2/3 of them will see the card face-up when they open their eyes, you’ll consider what you see as random, with a 2/3 probability of seeing face-up. This is analogous to the way that French noblemen originally introduced the notion of probability to optimize their gambling strategies: if, in a game, all you know is that you’ll be in one of many equally likely situations (corresponding to different ways that your cards could have been dealt, say), then you say that your probability of winning is simply the fraction of all the situations where you win.

  Third, this allowed us to propose what we called the cosmological interpretation of quantum mechanics. Here we interpret the wavefunction for an object as describing not some funky imaginary ensemble of possibilities for what the object might be doing, but rather the actual spatial collection of identical copies of the object that exist in our infinite space. Moreover, quantum uncertainty that you experience simply reflects your inability to self-locate in the Level I multiverse, i.e., to know which of your infinitely many copies throughout space is the one having your subjective perceptions.

  In some fields, coauthors on a paper traditionally list their names in alphabetical order. In cosmology, however, we usually let the author ordering reflect who has contributed most to the paper. In most cases, it’s quite obvious who’s done most of the work, but this time it was unusually hard to say. By the time we were ready to submit this paper for publication, both Anthony and I had worked quite hard on it and made arguably equally important contributions. We had an amusing phone conversation about this where we both lauded the other one’s contributions while stubbornly holding back from offering each other to go first. I finally suggested a solution that we both liked: deciding the author order with a quantum random-number generator. In this particular universe, he’s the first author (http://arxiv.org/pdf/1008.1066.pdf), but if our paper is correct, then I’m first author not only in half of the Level III parallel universes where we used this procedure, but in half of the Level I parallel universes as well.

  In 2010, Alex Vilenkin invited me to give a talk about this paper at Tufts, and just as in the opening of Chapter 5, Alan Guth was in the audience. I kept getting flashbacks from thirteen years earlier of Alan’s head slumping toward his chest, and tried to mentally brace for the inevitable, since I couldn’t recall a single talk ever where he hadn’t fallen asleep. Then a miracle happened, which felt like the best endorsement that our paper could ever have received, and like the pinnacle of my scientific car
eer: Alan stayed awake through my entire talk!

  * * *

  1I find it odd that there’s so much talk about “back-of-the-envelope calculations” when, in my personal experience, most impromptu calculations are in fact done on napkins, despite their susceptibility to tearing and their generally inferior quality as a graphological medium.

  Shifting Views: Many Worlds or Many Words?

  So what should you make of all this quantum business? Should you believe in wavefunction collapse or in quantum parallel universes? Although quantum mechanics is arguably the most successful physical theory ever invented, the century-old debate about how it fits into a coherent picture of physical reality shows no sign of abating. A veritable zoo of interpretations of what’s going on has cropped up over the years, including the ensemble, Copenhagen, instrumental, hydrodynamic, consciousness, Bohm, quantum logic, Many-Worlds, stochastic mechanics, many-minds, consistent histories, objective collapse, transactional, modal, existential, relational, Montevideo and cosmological interpretations.1 Moreover, different proponents of a particular interpretation often disagree about its detailed definition. Indeed, there isn’t even consensus on which ones should be called interpretations.…

  You might figure that since the experts are still arguing about this about a century after quantum mechanics was discovered, with no consensus in sight, they’ll probably be arguing for another century as well. However, the whole context of the debate has changed in three important ways, involving theory, cosmology and technology, causing sociological changes that I find quite interesting.

  First of all, we’ve seen how the theoretical discoveries by Everett, Zeh and others have shown that even if you drop the controversial wavefunction-collapse postulate and keep only the simple bare-bones quantum mechanics where the Schrödinger equation always holds, then you’ll still subjectively feel like the wavefunction collapses when you make observations, obeying all the right probability rules, and you’ll remain blissfully unaware of any quantum parallel universes.

  Second, the cosmology discoveries that we covered in Chapters 5 and 6 have suggested that we’re stuck with parallel universes even if Everett is wrong. Moreover, we saw earlier how these Level I parallel universes elegantly merge with the quantum ones.

  Third, support for the idea that quantum gravity somehow collapses the wavefunction has itself collapsed, because of a string-theory breakthrough known as the AdS/CFT correspondence. The details of this acronym don’t matter for our discussion: the key point is that a mathematical transformation has been found showing that certain quantum-field theories with gravity can be reinterpreted as other quantum-field theories without gravity. Gravity clearly isn’t causing wavefunction collapse if the very presence of gravity is merely a matter of interpretation.

  Fourth, ever more accurate experiments have ruled out many attempts to explain away the quantum weirdness. For example, could the apparent quantum randomness be replaced by some kind of unknown quantity stored inside particles, so-called hidden variables? The Irish physicist John Bell showed that, in this case, quantities that could be measured in certain difficult experiments would inevitably disagree with the standard quantum predictions. After many years, technology finally improved to the point that these experiments could be done, and the hidden-variable explanation was ruled out.

  Could it be that there’s a small correction to the Schrödinger equation that we haven’t discovered yet, but which causes quantum superpositions to break down for sufficiently large objects? Back when quantum mechanics was born, there were indeed many physicists who believed that quantum mechanics would prove to work only on the atomic scale. Well, no longer! The simple double-slit interference experiment (Figure 7.7), hailed by Feynman as the mother of all quantum effects, has been successfully repeated for objects larger than individual elementary particles: atoms, small molecules and even the soccer ball–shaped carbon-60 “Bucky Ball” molecule. Back in grad school, I asked my classmate Keith Schwab if he thought one could experimentally demonstrate that a macroscopic object was in two places at once. Amazingly, two decades later, he runs his own lab at Caltech working on doing exactly this, with a metal rod containing many billions of atoms. Indeed, his Santa Barbara colleague Andrew Cleland has already done this with a metal paddle large enough to see with the naked eye. Anton Zeilinger’s group in Vienna has even started discussing doing it with a virus. If we imagine, as a thought experiment, that this virus has some primitive kind of consciousness, then the Many-Worlds interpretation seems unavoidable: extrapolation to superpositions involving other sentient beings such as humans would then be merely a quantitative rather than a qualitative one. Zeilinger’s group has also demonstrated that counterintuitive quantum properties of photons persisted while they traveled 89 kilometers through space—hardly a microscopic distance. So I feel that the experimental verdict is in: the world is weird, and we just have to learn to live with it.

  Indeed, many people have warmed up to quantum weirdness, for reasons that aren’t philosophical but financial: this very weirdness may offer useful new technologies. According to a recent estimate, more than a quarter of the U.S. gross national product is now based on inventions made possible by quantum mechanics, from lasers to computer chips. Indeed, fledgling technologies such as quantum cryptography and quantum computing explicitly exploit the Level III multiverse and work only if the wavefunction doesn’t collapse.

  These breakthroughs in theory, cosmology and technology have caused a notable shift in views. When I give talks, I like to know what the people in my audience think. When I asked them which interpretation of quantum mechanics they identified most closely with, here’s what they said, first at a 1997 quantum-mechanics conference at UMBC in Maryland, then at a 2010 talk I gave at the Harvard Physics Department:

  Interpretation Maryland 1997 Harvard 2010

  Copenhagen 13 0

  Everett 8 16

  Bohm 4 0

  Consistent histories 4 2

  Modified dynamics 1 1

  None of the above/undecided 18 16

  Total votes 48 35

  Although these polls were highly informal and unscientific, and clearly don’t survey a representative sample of all physicists, they nonetheless indicate a rather striking shift in opinion: after reigning supreme for decades, the Copenhagen interpretation saw its approval rating drop below 30% in 1997 and to 0% (!) in 2010. In contrast, after being proposed in 1957 and going virtually unnoticed for about a decade, Everett’s Many-Worlds interpretation survived twenty-five years of fierce criticism and occasional ridicule to top the 2010 poll. It’s also worth noting that there’s a large fraction of undecided voters, suggesting that the quantum-mechanics debate is still in full swing.

  The Austrian animal behaviorist Konrad Lorenz mused that important scientific discoveries go through three phases: first they’re completely ignored, then they’re violently attacked, and finally they’re brushed aside as well known. The poll suggests that after spending the 1960s in phase 1, Everett’s parallel universes have now shifted to somewhere between phase 2 and phase 3.

  To me, this shift means that it’s time to update the quantum textbooks to mention decoherence (many still don’t) and to make clear that the Copenhagen interpretation is better thought of as the Copenhagen approximation: even though the wavefunction probably doesn’t collapse, it’s a very useful approximation to do the calculations as if it does collapse when you make an observation.

  All physics theories have two parts: mathematical equations and words that tell us what they mean. Although above I rattled off the names of over a dozen interpretations of quantum mechanics, many of them differ only in the “words” part. To me, the most interesting question is what the math part is, and specifically whether the simplest math of all (just the Schrödinger equation with no exceptions) is enough. So far, there isn’t a shred of experimental evidence to the contrary, yet many of the quantum interpretations add a lengthy “words” part to talk away the parallel universes.
So when you pick your own favorite interpretation, it really comes down to what bothers you most: a profusion of worlds or a profusion of words. When the time came to write a paper for the proceedings of that 1997 Maryland conference, I called it “The Interpretation of Quantum Mechanics: Many Worlds or Many Words?” in an attempt to tease some of my colleagues. I was expecting to get flamed with plenty of hate mail from them as a result, but I have to hand it to them: even though I think they’re wrong about quantum mechanics, they do have a good sense of humor.…

  In Chapter 7, we talked about how everything is made of particles, and how particles are in a sense purely mathematical objects. In this chapter, we’ve seen that in quantum mechanics, there’s something that is arguably even more fundamental: the wavefunction and the infinite-dimensional place called Hilbert space where it lives. The particles can be created and destroyed, and can be in several places at once. In contrast, there is, was and always will be only one wavefunction, and it’s the object that moves through Hilbert space as dictated by Schrödinger’s equation. But if the ultimate physical reality corresponds to the wavefunction, then what sort of beast is a wavefunction? What’s it made of? What’s Hilbert space made of? As far as we know, nothing: they seem to be purely mathematical objects! So once again, as we attempt to dig deeper in search of the underlying physical reality, we’ve found a hint that the bedrock itself is purely mathematical. We’ll take this idea much further in Chapter 10.