Eleven Dimensions

IN MY OWN cosmological journey, it helped that my work was solely driven by my love of the subject I was studying. Somewhere along the way, I learned that such an approach, while not always easy or wise, is nevertheless privileged. Being a scientist for the love of science gave me a lifetime of childhood, the kind where you are not afflicted by assumptions or prejudice and where you have no favorite sides to pick, only an exciting new world awaiting your discovery.

When I was a teenager in Albania, there were two mandatory subjects for all university students, including those studying the sciences: first, the history of Marxism, and second, physical education (better known as PE). I abhorred them both, albeit for very different reasons.

I deserved to fail in both subjects. But if I had, I wouldn’t have received my degree in a subject I did care about: physics. Memorizing the names, dates, and places of Marxist history, especially given that much of what we were told were lies, was torture for me. I’ve never been good at memorizing. And I have a character flaw—I can happily work around the clock, with the kind of intense concentration like nothing else around me exists, on subjects that I like, but I am a terrible procrastinator on subjects that I don’t. So I left the Marxist-history exam preparation for the last day. When I finally forced myself to study, I used a trick I occasionally relied on: I stayed up studying all night before the test. I planned to be the first student to appear in front of the committee the next morning. I would take the exam while everything was still fresh in my mind, then go home, sleep, and forget everything I’d learned. My dad stayed up with me, quizzing me.

But it didn’t work. During my exam, as I stood before the committee, they told me they had a very easy question: In what year did Stalin die? I had absolutely no idea. Two of my classmates were waiting outside, and one climbed up onto the other one’s shoulders, pressed his face to the window, and held up five fingers, then three. But I learned this later; at the time, I was too embarrassed to look at them as the committee awaited my answer.

After a brief pause, I told the exam committee that I was sorry, but I couldn’t remember. Then I burst out laughing and ran out from the room while the committee members shouted after me, “Do you think this is funny? You failed. You don’t even know when the leader of the Soviet Union, Comrade Stalin, died?”

Outside, some of my professors had heard the noise and asked what happened. They entered, spoke to the committee, returned, and told me, “The committee has no doubt that of course you knew the answer, but exams can be hard psychologically. They recognized that you just had a temporary blank, and therefore they have decided to pass you.” I learned later that my professors had pressured and bargained with the history committee not to fail me, openly hinting that if any of the committee members’ relatives took math or physics classes, my professors would go easy on them.

Much the same thing happened with my PE class. I can still hear my PE teacher screaming at me when we were learning how to throw toy grenades and I threw mine right at my feet. “If this were a real grenade, you would have blown yourself and all of us to bits. We would all be killed, thanks to you!” Again, my wonderful math and physics professors had a little chat with the PE teacher, and I passed.

But those experiences made me inherently suspicious of the party line—any party line. It wasn’t just what we were forced to learn, it was also what was kept from us. The government felt threatened and could not allow people to peek beyond Albania’s borders into the wider world through books, television, and travel. Yet, via homemade antennas and other methods, some Albanians found a way to break those barriers.

Given those experiences, I found it hard to understand the argument that scientists could never explore the moment of creation or what came before it. I couldn’t accept this. So, as a young scientist not much older than Everett had been when he decided to buck convention, I chose to explore the confounding world of possibilities that were opening up before me.

By the second half of the twentieth century, the two great pillars of modern physics, quantum mechanics and general relativity, had been chiseled into their current forms. Their establishment, in turn, led to considerable advances in particle physics, which had been around for only several decades but which was poised to revolutionize the world of physics.

Particle physics studies the creation and interactions of subatomic species, known as elementary particles, in the universe. Nested in the theory of the universe is a theory of quantum forces and their particles that populate the universe: a standard model of particles. Particles are carriers of forces.* Throughout the 1970s and 1980s, particle physics seemed poised to unify the quantum forces into one theory. The spectacular strides in particle physics culminated in the unification of the three quantum forces into models of a grand unified theory. Not a small feat!

The unification of forces under one umbrella theory makes use of the fact that the strength of forces (determined by empirical constants, known as “coupling constants”) is not really constant; rather, they change (or, in physics terms, “run”) with energy. Their unification happens because at some energy scale—in this case, at only about ten thousand times less than the energy of the Big Bang—the forces become roughly of equal strength and thus are indistinguishable from one another.

The three unified forces, out of the four known to exist in our universe, are Maxwell’s electromagnetic force, which describes light and electromagnetic interactions; the weak force, which is responsible for particle and radioactive decay; and the strong nuclear force, which binds quarks together to make protons, neutrons, and atomic nuclei. The unification of the fourth known force, the gravitational force, with the other three was the final barrier left between science and the discovery of a mega-theory. How hard could that be?

It couldn’t be that hard for the brightest minds. Hawking, who, along with many others, spent a good part of his life working toward the theory of everything, had predicted that it would be discovered before the year 2000.

But yet again, nature refused to cooperate and reveal its theory of everything. Quantum theory and Einstein’s theory of relativity still could not be unified; they remained locked in the same battle of dueling answers for the workings of the universe that had raged since their respective discoveries.

So theoretical physicists did what they have done for years: they went back to the mental drawing board and began envisioning not a bigger world but an even smaller one. The result, called string theory, was developed by a number of prominent physicists over several decades in the second half of the twentieth century. In the simplest terms, string theory reduces the world to one-dimensional objects, replacing point-like particles with extended one-dimensional strings. There are two possible kinds of strings: open strings, where the two ends are free, and closed strings, where the two ends come togther in a loop. They are too small to be observed. But, string theorists argue, these strings are the essential building blocks of the universe. They are what ultimately made all the elementary particles in the universe; they are the threads from which the very fabric of space-time is woven.

String theory was intended to be the elusive theory of everything that connected the three quantum forces to the fourth, gravitational force. But in string theory, achieving the unification of forces in a mathematially consistent way is possible only if the world is enlarged by introducing additional spatial dimensions to the universe. String theorists postulated that, in much the same way as the vibrations made by an instrument’s strings give rise to musical notes, the vibrations of tiny string filaments gave rise to different particles; each elementary particle was determined by the subtle intonations of its underlying closed strings. Thus, every particle that had been discovered could be reconceptualized as a “note” in nature’s grand symphony.

We are used to thinking of all elementary particles, such as the ones that make up an atom (electrons, protons, neutrons), as point-like particles. But if we were to look at these objects under a microscope with sufficient magnification to observe individual atoms, instead of a particle, we would see a bundle of waves tightly packed together into a tiny wave packet (as you might expect, if you recall the concept of wave-particle duality from our tour of the quantum world).

But suppose we get our hands on an even more powerful microscope, one that can probe sizes much smaller than the size of an atom—one that allows us to zoom in at scales of 10^-33 centimeters, or a Planck length, the smallest scale at which our quantum and relativistic theories can still be trusted. At this level, instead of glimpsing point-like particles and bundles of waves, we should see loops of vibrating strings. Furthermore, each frequency of each string should correspond to a certain amount of energy. (Einstein’s famous equation E = mc^2 means that the Planck energy E = hv produced by the vibrations’ frequency is converted into the mass of the particle.) When vibrating at one frequency, a one-dimensional closed string would correspond to an electron, but vibrating at another (higher) frequency, that same one-dimensional string would produce a proton, and vibrating in yet another frequency, it would produce a graviton (a hypothetical particle that mediates the force of gravity). That is, the type of vibrations of a single string are what determines the mass of each type of elementary particle.


Figure 8. A point particle, in the left panel, is spread into a bundle of waves packed together, shown in the wave packet in the middle panel. But what appears as a point particle to us is, according to string theory, actually the vibration of a closed string, shown in the right panel. The frequency of vibration determines the mass of the particle.

In string theory, the layers of vibrations of the strings—like the notes played simultaneously by different instruments in an orchestra, to stay with our music analogy—combine in a variety of ways to fill every cell of the cosmos, all the way down to the smallest scales. But we can produce the music of the spheres only if we can make this framework harmoniously hang together mathematically. The consistency in that celestial melody comes at a heavy price. Because string theory needs to be reduced to our four-dimensional world.

We humans experience the volume of space in three dimensions: height, width, and length. With the addition of time—which Einstein placed on an equal footing with the other dimensions in his general relativity theory—we have a total of four dimensions (space-time).

Human perception can accommodate a four-dimensional reality. But string theory requires us to assume a world composed of eleven dimensions. This mind-boggling idea originated with Edward Witten, a pioneer of string theory and a renowned physicist and mathematician at the Institute for Advanced Study at Princeton, who in 1995 realized that previous versions of string theory could be unified under an umbrella theory he named M-theory. M-theory tells us that besides the familiar four dimensions of time, width, height, and length, there are an additional seven dimensions hidden in the cosmos.

Of course, the circuitry of our brains is not wired to comprehend seven extra dimensions. So we have to reframe the concept in our minds. As the great surrealist painter René Magritte, perhaps best known for his self-portrait in a bowler hat with a green apple representing his face, once said, “Everything we see hides another thing.” With this statement as our guide, we can start to wrap our human minds around the eleven dimensions that string theory requires.

In an effort to make eleven-dimensional space-time understandable, let’s try our own thought experiment using an analogy with paintings and artists. Artists are able to construct a three-dimensional representation of the world on a two-dimensional plane, the canvas, through the clever use of focal points and perspective. Borrowing from their methods, we can visualize M-theory’s additional spatial dimensions.

Pick your favorite representational painting that captures three dimensions and look closely at the distance between two objects that appear to be situated at different locations. If you were to shine a light on each object, the light rays are the perspective, and the point where these light rays converge is the focal or vanishing point. Note how the vanishing points of the two objects in the painting are misaligned to be out of focus with each other. This subtle misalignment in the painting is what renders the third dimension possible, because our perception interprets it as depth. While it is impossible for depth to be included on a two-dimensional canvas, the artist has nevertheless managed to draw it by visually tricking our brains.

Exactly the same trick can be used to transcend the three-dimensional canvas of our minds to envision additional dimensions. Although we know that technically a macroscopic fourth spatial dimension does not exist in our universe, we can imagine one.

To grasp how this might work, hold a sheet of paper in a vertical position and at a right angle to your body, as depicted in figure 9. All you should be able to see is a line, not the plane of paper, because from this angle, the length of the paper is at 90 degrees and cannot be seen, and the width of that piece of paper is too small to observe from far away, so therefore it is also hidden from view. All we can see is the height of the piece of paper. For the sake of argument, let’s assume that there are more dimensions in addition to the known three dimensions of our universe hiding behind the width and the length of this piece of paper (which, although they are hidden from view, we know exist, because we know we live in a universe with three spatial dimensions). Imagine a fourth and—why not?—a fifth spatial dimension are nested within the width or length of the paper and hidden from view, as in figure 9. These additional dimensions are so microscopically small and curled up that we cannot see them with the naked eye or with our current microscopes, and yet we suspect mathematically that they are there.

To imagine a sixth dimension, suppose (hypothetically) that what is hidden behind the fourth dimension of figure 9 is an object with three-dimensional volume instead of the two-dimensional piece of paper in our illustration. In this case, we would have two more dimensions hiding behind the fourth one, which brings us to a total of six dimensions.

We can repeat this mental exercise to imagine still more hidden dimensions until we eventually get to the eleven dimensions of the M-theory space-time. Thus, much like uncovering tinier and tinier versions of nesting Russian matryoshka dolls, we can extrapolate the other seven extra dimensions of M-theory curled up and hidden from view inside the cosmos by continuing to delve deeper into the layers of space-time.

In sum, the basic concepts of string theory are that point particles hide a vibrating, closed string at Planck length, and space-time hides an additional seven dimensions—but only, crucially, at the kind of subatomic scales that escape testability.

String theory is an interesting exercise—but it is a mathematical exercise. How can we possibly know if it physically exists? String theory can achieve this goal only if it ultimately reproduces and explains the universe in which we exist.

To reproduce a four-dimensional universe out of an eleven-dimensional world, string theorists concentrated their efforts on minimizing or simplifying the seven extra dimensions to return M-theory to a four-dimensional universe. They did so by conceptualizing the added seven dimensions as infinitesimally tiny while keeping the three dimensions of our known world large. This made the extra seven dimensions, though still part of the fundamental building blocks of nature, completely invisible to “normal” creatures familiar with a classical world only, one containing a large three-dimensional volume of space and a fourth dimension of time. Put differently, after having struggled to visualize an additional seven dimensions of space, you are about to scrap them to recover your real-life, four-dimensional world.


Figure 9. Cartoon illustration of the perspective explained in the adjacent text. The person standing in front the paper can see its length and height but not its depth, and thus he is not aware of the existence of this third dimension. This person would miss out on the existence of additional dimensions and structures if more of these were nested and hidden from view within the depth of the paper.

In mathematical terminology, this curling up of the extra dimensions is known as the compactification of space. But implementing compactification is challenging. If, at the fundamental level, nature has ten spatial dimensions and one time dimension, then what happens to all the other “stuff,” such as particles, quantum fields, currents, fluxes, and forces that must fill the extra space in those dimensions, just as particles, quantum fields, currents, fluxes, and forces fill the volume of our four-dimensional world?

Compactifying the extra dimensions is akin to taking a cube full of stuff and trying to squash it down all the way until it completely flattens to its base. Now imagine that cube is filled with quantum stuff, like the fields and the particles and the fluxes and the currents. Every time it is squashed down, the quantum stuff inside is excited, and the interior energy of the cube is sedimented at its base.

These variations on the energy contents inside the cube in our example, the fluctuations from compactification, are nothing more than the quantum fluctuations we came across before. They are best pictured as invisible springs inside the cube, very much like a spring mattress. Suppose a large person volunteers to lie on top of the mattress and compress it until it flattens. Then, as the quantum contents of the cube—the springs in our metaphor—are squashed down, they get further engaged or “excited” and strained.

To add to the difficulty, imagine the height of the cube is seven-dimensional and the base of the cube is three-dimensional. As we squash the cube, all the quantum material inside its volume is compressed and excited along the seven-dimensional “height.” This heap of energy has to be dumped somewhere when the cube is flattened, and that somewhere can only be on the remaining three-dimensional surface of the base.

At this stage, if we could zoom in using our super-microscope again, we would expect a glimpse of the hidden world like the one in figure 10. Pictorially, that is how, after the massive mathematical exercise of compactification, string theorists believe they have created a four-dimensional world (as in our universe) that contains matter and energy.

The hope was that once this exercise was successfully completed, string theorists would have derived a unified theory of everything for our origin and would thus be able to write the final chapter in physics.

But this is not what happened. What actually occurred around 2004 was a lot better—or a lot worse, depending on whom you ask.


Figure 10. Shown in the picture at the bottom are bundles of strings inside the four-dimensional “compressed spring box,” which is what is left after the additional seven dimensions are compactified. According to string theory, pictorially, this is what the space-time of our universe would look like if we could probe scales of, say, 10^(-30) cm. Instead of seeing point particles or empty space-time, we would observe bundles of strings. The volume of the seven additional dimensions of the box at the top is compressed (represented by the arrow) and hidden from view.

* * *

At first, the outcome of compactification struck the scientific community as a disaster.

Compactifying, a process to get rid of the extra seven dimensions and reduce the volume of space-time from eleven dimensions to four, indeed produces a universe like ours. But it also gave physicists far more than they had wished for.

It turns out there are a great many ways of curling up the extra dimensions and even more ways of combining layers of fluctuations of the quantum stuff in that extra volume. And for each possible option, another potential energy well,* known as landscape vacua, that can potentially ignite a Big Bang is produced. At present, using the mathematical process of compactification, string theorists have found approximately 10^600 (10 with 600 zeros behind it) possibilities. This vast collection of potential Big Bang energies, the collection of about 10^600 vacua obtained through the process I have just described, is known as the landscape of string theory.

The discovery of the landscape of string theory shook the world of theoretical physics to its core. The decades of efforts by string theorists to mathematically reduce the eleven-dimensional world of string theory to obtain solutions that describe a single four-dimensional universe had inadvertently unleashed the scenario of a virtual universe-making factory that could act as an incubator for many potential Big Bang energies from which billions of baby universes could possibly spring into existence.

If that seems like a lot to absorb, it is—even to seasoned physicists. However, there is a way we can envision a string-theory landscape as somewhat like the physical landscapes we are familiar with. It can have peaks and valleys, like a mountain range. But unlike the regular landscape on Earth, which exists in real space and time, the string-theory landscape exists in a space of energy—energies that represent the range of choices or possibilities that exist in this string world when trying to produce a four-dimensional universe. Just as on the familiar physical landscape a handful of marbles might roll down from a mountaintop and settle in a valley, in the string-theory landscape, a whole infant universe could settle in a place, a vacuum site. But the string-theory landscape contains a vast number of choices of vacuum energy sites from which our universe could have started. The new mystery in the context of the landscape discovery is which vacuum energy did our universe pick, and why did it pick that one?

The discovery of a string-theory landscape meant that, instead of one initial energy for one Big Bang type of inflation that resulted in one singular universe, there were an abundance of energies—many potential Big Bangs to start multiple, four-dimensional universes like ours. Instead of once and for all explaining our origin, this abundance of potential energies, trillions and trillions of possible starting energies so far (as illustrated earlier in this chapter), offered a mystifying multitude of possible origins. Simply put, a string-theory landscape provides a vast collection of initial energies—of potential Big Bang energies—capable of jump-starting multiple universes.

Unexpectedly, at the beginning of the twenty-first century, string theory had dealt a major blow to the simple vision of a single universe wrapped up in a theory of everything. By predicting a landscape of many possible worlds like our universe, string theory unintentionally traded the theory of everything for a theory of the multiverse, precipitating a major crisis in physics. For researchers around the world, this discovery seriously threatened the possibility that string theory would become the long-sought theory of everything.

Only a few years previously, in his book A Brief History of Time, Stephen Hawking had summarized the desire and the millennial efforts for the dream of a single universe described by the theory of everything and declared his advocacy for the theory of everything by paraphrasing Saint Augustine: “When asked: ‘What did God do before he created the universe?’ Augustine didn’t reply: ‘He was preparing Hell for people who asked such questions.’ Instead, he said that time was a property of the universe that God created, and that time did not exist before the beginning of the universe.” Still, while scientists were searching for a theory of everything and a singular universe, the landscape discovery implied that the world of physics was about to stumble away from its goal and into a terrifying inferno.

Physics was preparing for a paradigm shift, where every dream of a single universe wrapped in a theory of everything was about to be shattered into smithereens. It was far worse than the previous clashes among the quantum theorists, such as the fierce arguments between Einstein and Bohr over a deterministic classical universe versus an indeterminate quantum one; once, when a frustrated Einstein quipped that “God did not play dice when making the universe,” Bohr retorted, “Einstein, stop telling God what to do.”

Backing down was not an option unless the whole endeavor of string theory was to be discarded. Again and again in our theories of nature, each time we asked: Where do we come from? The answer was: A multiverse. Understandably, to most people working in physics, the premise of a string-theory landscape was nothing less than a full-blown crisis.

Indeed, the string-theory landscape did not come with a manual that explained why some of these possible universes might be better or fitter than others. It appeared that each one of the potential four-dimensional worlds out of the pool of 10^600 was an equally likely candidate to start our universe. It was like Schrödinger’s cat and the Hugh Everett paradox all over again. Moreover, since, bound by the speed-of-light limit, we could not observe beyond our universe’s own horizon, testing for and proving this new multiverse possibility seemed hopeless.

There was something else that made the multiverse scary for scientists: Hugh Everett’s fate. His example provided a poignant reminder of the risk of siding with a multiverse scenario of the cosmos.

To the majority of scientists in the theoretical physics community, the string-theory landscape seemed to be the worst of all worlds.

But not to me.

For me, the string-theory landscape arrived at exactly the moment when all my thought experiments about different starting points for different universes had been completely undone by the rules of entropy and the second law of thermodynamics. The discovery of the string-theory landscape was the third time that a physics breakthrough produced the same answer to the question of where we came from.

To my mind, the possibility of a multiverse could no longer be ignored.

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