Our Mathematical Universe

by Max Tegmark

  Before building such a machine, major engineering hurdles need to be overcome, such as isolating the quantum information well enough that decoherence doesn’t ruin quantum superpositions. There’s still a long way to go: whereas the computer in your cell phone probably stores billions of bits of information (zeros and ones), the state-of-the-art quantum computers in labs around the world can store only a handful. However, Penrose and others made a shocking suggestion: perhaps you already have a quantum computer—in your head! They suggested that our brains (or at least parts of them) are quantum computers, and that this is a key to understanding consciousness.

  Since decoherence spoils quantum effects, I decided to use the decoherence formulas that I’d been scooped on to check whether Penrose’s idea really worked. I first did the math for neurons (Figure 8.7), the hundred billion or so nerve cells that, like wires, transmit electrical signals in your brain. Neurons are thin and long: if you laid yours out in a row, they’d wrap around Earth about four times. They transmit electrical signals by transporting sodium and potassium atoms which each have an electron missing (and therefore have a positive electric charge). If you connect a voltmeter to a resting neuron, you’ll measure 0.07 volts between the inside and outside of the cell. If one end of the neuron gets triggered to lower this voltage, then voltage-sensitive gates in the cell wall open up, charged sodium atoms start gushing through, this voltage drops further and more atoms gush in. This chain reaction, called firing, propagates down the length of the neuron at a speed of up to 200 miles per hour, while about a million sodium atoms enter the cell. The axon quickly recovers, and fast neurons can repeat this firing process over a thousand times per second.

  Figure 8.7: Schematic illustration of a neuron (left), a section of its long wire-shaped part called the axon (center), and a piece of its axon membrane (right). The axon is typically insulated with an insulation material called myelin that has small bare patches every half millimeter or so where voltage-sensitive sodium and potassium gates are concentrated. If the neuron is in a superposition of firing and not firing, then roughly a million sodium atoms (with chemical symbol Na) are in a superposition of being inside and outside the cell (right).

  Now suppose that your brain really is a quantum computer, and that neuron firing is in some way involved in the computation. Then an individual neuron must be able to be in a superposition of firing and not firing, which means that about a million sodium atoms are in two places at once, both inside and outside the neuron. As we discussed above, a quantum computer only works as long as its state is kept secret from the outside world, so how long could a neuron keep secret whether it was firing or not? When I plugged in the numbers, the answer I got was “not very long at all,” or to be more specific, about 10−20 (ten billionths of a trillionth) of a second. That’s how long it would typically take before a random water molecule bumped into one of the million sodium atoms and discovered where it was, thereby destroying the quantum superposition. I also did the math for another model that Penrose had proposed, where the quantum computation was done not by neurons but by microtubules, parts of the scaffolding in cells, and found that they suffered decoherence after about 10−13 (100 quadrillionths) of a second. For my thoughts to correspond to a quantum computation, they’d need to finish before decoherence kicked in, so I’d need to be able to think fast enough to have 10,000,000,000,000 thoughts each second. Perhaps Roger Penrose can think that fast, but I sure can’t.…

  It’s really not that surprising that your brain doesn’t work as a quantum computer: my colleagues who are trying to build quantum computers go to great lengths to fight decoherence, typically isolating their devices in a cold, dark vacuum to keep their states secret from the rest of the world, while your brain is a warm and wet place whose parts aren’t isolated. However, some people complained about my paper, and I got to experience my first scientific controversy. In particular, Stuart Hameroff, one of the quantum-consciousness pioneers, said he felt I’d “laid a stink bomb in the field” and caused problems for quantum-consciousness researchers. “Are you a hit man for scientific orthodoxy?” he asked me.

  I found this rather ironic, since I’m normally on the opposite side from scientific orthodoxy, and tend to instinctively side with the underdog who pursues contrarian ideas. Also, I hadn’t made these calculations hoping for a particular result, but simply to find out what the answer was. In fact, I’d probably have been happier with the opposite conclusion, since it would have felt really cool to have my own quantum computer. With two coauthors, Hameroff went on to publish a critique of my paper which I felt was flawed,1 and I couldn’t help feel that sometimes scientists get attached to an idea with an almost religious fervor, so that no facts can dissuade them. I wondered if the impressive-sounding technical terminology was really just an attempt to rationalize this argument: “Consciousness is a mystery and quantum mechanics is a mystery, so they must be related.”

  I finally met Stuart Hameroff in 2009 and found him to be quite a jovial and friendly fellow. We had lunch together in New York and, interestingly enough, weren’t able to pinpoint a single calculation or measurement that we disagreed on, so we just politely agreed to disagree on what this all meant for consciousness.

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  1They claimed that the microtubule model I’d tested wasn’t from Roger Penrose’s book, but in 2006, Stuart graciously acknowledged that it was. They also argued that my calculation must be flawed because the decoherence time scale that I derived decreased as you lower the temperature of the brain, whereas you might intuitively expect the opposite. The point they overlooked is that as soon as you drop the absolute temperature by about 10%, below 0 degrees Celsius, your brain freezes and the decoherence time grows dramatically. The slight decrease in decoherence time for tiny temperature reductions reflects the well-known fact that things are more likely to bump into each other as you lower the temperature, just as slow neutrons are more likely than fast ones to strike targets in a nuclear reactor. They also argued that the brain might perform quantum computations using other mechanisms, but without specifying such mechanisms with enough detail that I could test them, and that there might be other quantum effects in the brain that weren’t computations, which I’d never disagreed with in the first place.

  Subject, Object and Environment

  I have a confession to make: my brain-decoherence calculation was just an excuse. It wasn’t the real reason I wrote that paper. I had an idea that I was really excited about and really wanted to publish, but figured that it would be viewed as too philosophical to get accepted for publication. So I came up with what I called my Trojan Horse strategy: hiding the philosophical part that I wanted to sneak past the referees behind pages and pages of respectable-looking equations. Amusingly, this strategy worked in the sense that the paper got accepted, but failed in the sense that people paid attention only to the masking material: the business about the brain not being a quantum computer.

  Figure 8.8: It’s convenient to decompose the world into three parts: the part corresponding to your subjective perceptions (the subject), the part being studied (the object), and everything else (the environment). As indicated, the interactions between these three parts cause qualitatively very different effects, providing a unified picture including both decoherence and apparent wavefunction collapse.

  So what was my hidden message? It was a unified way of thinking about quantum reality, as illustrated in Figure 8.8. Feynman had emphasized that quantum mechanics splits our Universe into two parts: the object under consideration and everything else (referred to as the environment). However, I felt that an important piece of the quantum puzzle was missing here: your mind. As Everett’s work had shown, understanding the process of observation requires us to include a third part of our Universe as well: your mental state as an observer, labeled subject in Figure 8.8.1

  If you’re not a physicist, it might seem funny that people still talk so little about the mind in the physics community, given all the fuss
about observations in quantum mechanics. After all, talking about observations without mentioning the mind feels a bit like discussing nearsightedness without mentioning the eye. I think the explanation is that, since we don’t understand how consciousness works, most physicists feel uncomfortable even talking about it, fearing that they’ll get regarded as too philosophical. Personally, I feel that just because we don’t understand something doesn’t mean that we can ignore it and still expect to get correct answers.

  I’ll talk a lot more about our mind in the next chapter. However, to understand Figure 8.8, the details of how your mind works don’t matter at all: the only assumption I’m making here is that your subjective consciousness results in some way from the remarkably complicated motions of the particles that make up your brain, and that these particles obey the Schrödinger equation just as all other particles do.

  In my Trojan Horse paper, I split the Schrödinger equation into pieces: three governing the three parts of our Universe (subject, object and environment), and additional pieces governing the interactions between these parts. I then analyzed the effects of these different parts of the equation, and showed that one part gave the stuff my textbooks covered, one part gave Everett’s many worlds, one part gave Zeh’s decoherence, and one part gave something new. Standard textbooks have focused only on the part of the Schrödinger equation that governs the object (an atom, say), in the reductionist spirit that things should be analyzable by themselves without worrying about the greater whole that they’re part of. The interaction between the subject and the object gives Everett’s parallel universes (this page), spreading quantum superpositions from the object to you, the subject. The interaction between the environment and the object gives the decoherence (this page), explaining why large objects such as a queen of hearts never show signs of strange quantum behavior such as being in two places at once. It’s normally hopeless to eliminate this decoherence in practice, but even in a thought experiment where you could (say, by repeating the Quantum Cards experiment in a dark, cold room with no air, with only a single photon striking the card then being seen by your eye), it wouldn’t make any difference: since the card is in two places at once, so is the photon, so at least one neuron in your optical nerves would enter a superposition of firing and not firing while you looked at the card, and as we saw earlier, this superposition would decohere in about 10−20 seconds.

  That decoherence still doesn’t fully explain why you never perceive quantum weirdness, though, since your thought processes (the internal dynamics of the subject) could create weird superpositions of familiar mental states. Fortunately, here the third interaction in Figure 8.8 comes to the rescue: the interaction between the subject and the environment. The fact that neurons decohere much faster than they can process information means that if the complex neuron-firing patterns in your brain have anything to do with consciousness, then decoherence in the brain will prevent you from experiencing weird superpositions.

  This subject-environment interaction also helps tie up another loose end. Wojciech Zurek had continued his decoherence research beyond what I’d rediscovered, and shown that decoherence does one more important thing for us: not only does it explain why large objects never seem to be in two places at once, but it also explains why conventional states (such as being in only one place) are so special: out of all the states that quantum mechanics allows for large objects, these conventional states are the ones that are most robust to decoherence, and therefore the ones that survive. It’s a bit like why deserts tend to have cacti rather than roses: they’re the most robust to the environment. In fact, a paper on this very topic, which I’d written together with my dad, was the reason Wojciech invited me to give that talk in Los Alamos.

  Now, some decoherence can be reduced using clever lab equipment such as vacuum pumps and extreme refrigerators, but we can never turn off the decoherence of our neurons. We don’t know how our minds work, but we do know for sure that all information that ever reaches our mind from the outside world must first pass through neurons from our sensory organs, for example the optical nerves from our eyes and the cochlear nerves from our ears, which all decohere ridiculously fast. So by the time we become subjectively aware of any observation about the outside world, things have already decohered, guaranteeing that we’ll never perceive any quantum weirdness and explaining why we only perceive robust conventional states.

  Among all controversies in physics, a few are so grand that they tower over the rest and last for generations. The great controversy about how to interpret quantum mechanics is clearly one of them. Another involves the second law of thermodynamics. It states that the entropy of an isolated system never decreases, where entropy is a quantitative measure of our lack of information about a system—it’s essentially how many bits of information we’d need to specify its quantum state. On one hand, some scientists have elevated it to almost sacred status, and the great astrophysicist Sir Arthur Eddington had this so say: “The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations—then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” On the other hand, serious objections to the second law were made by physics titans such as Maxwell, Gibbs, Loschmidt and Poincaré, and there is still no consensus on whether they’ve all been satisfactorily resolved.

  The way I see it, these two great controversies of quantum mechanics and thermodynamics are linked, in the sense that they can both be resolved in one fell swoop if we use the standard quantum-mechanics definition of entropy (given by John von Neumann), reject wavefunction collapse, and consider all parts of reality: subject, object and environment.

  As Figure 8.8 summarizes, measurement and decoherence correspond to the object interacting with the subject and the environment, respectively. Although the processes of measurement and decoherence may appear different, entropy brings out an interesting parallelism between them involving the lack of information that we have about the object, which is a very important quantity that we call entropy in physics. If the object isn’t interacting with anything, its entropy stays constant: you know just as much about its state a second later, since you can calculate this state from the initial state using the Schrödinger equation. If the object interacts with you, then you typically get more information about it, and its entropy decreases—after opening your eyes in Figure 8.1, there are two copies of you, each seeing a different outcome, but both of whom know how the card fell in their parallel universe and have therefore acquired one additional bit of information about the card. If the object interacts with the environment, however, you typically lose information about it, so its entropy increases: if Philip knows where his Pokémon cards are, he’ll have less information about their whereabouts after Alexander messes with them. Similarly, if you know that a card is in the quantum state corresponding to being in two places at once, and then a person or a photon finds out where it is without informing you, then you’ve lost one bit of information about it: first you knew the quantum state, but now it’s effectively in one of two quantum states and you don’t know which. In summary, here’s how I informally think about this: the entropy of an object decreases while you look at it and increases while you don’t. Decoherence is simply a measurement that you don’t know the outcome of. More rigorously, we can reformulate the second law of thermodynamics in a more nuanced way:

  1. The object’s entropy can’t decrease unless it interacts with the subject.

  2. The object’s entropy can’t increase unless it interacts with the environment.

  The traditional formulation of the law simply corresponds to ignoring the subject. When I published a technical article about this
(http://arxiv.org/pdf/1108.3080.pdf),2 I included a mathematical proof of the second part (how decoherence increases entropy), but a rigorous proof of the first part (that on average, observation always reduces entropy) eluded me, even though my computer simulations strongly suggested that it was true. Then something wonderful happened, which reminded me of why I’m so fortunate to get to work at MIT: an enthusiastic twenty-year-old Armenian undergraduate student, Hrant Gharibyan, asked if I had any interesting problems he could work on. We teamed up, and he attacked my problem with great fervor, devouring math books like popcorn and mastering mathematical tools such as Schur products and spectral majorization, which aren’t known to most physicists, and which I’d only learned of from my dad, who’s a mathematician. And then one day when I saw Hrant, I knew from his triumphant smile that he’d solved the problem! We’re hoping to publish his proof as soon as I’m done with this book.

  * * *

  1Here I’m not referring to your entire brain, just to those aspects of it that correspond to your subjective conscious perceptions.

  2If you don’t mind the math, the article also explains how this result combined with inflation can explain how the entropy was so low in our early Universe, which in turn explains the so-called arrow of time (beautifully explained in the books by Sean Carroll and Dieter Zeh in the “Suggestions for Further Reading” section). It also provides a quantum-mechanical generalization of the standard procedure for updating our knowledge with new information, known as Bayes’s theorem.