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How did space and time come about? How did they form the smooth four-dimensional emptiness that serves as a backdrop for our physical world? What do they look like at the very tiniest distances? Questions such as these lie at the outer boundary of modern science and are driving the search for a theory of quantum gravity—the long-sought unification of Einstein's general theory of relativity with quantum theory. Relativity theory describes how spacetime on large scales can take on countless different shapes, producing what we perceive as the force of gravity. In contrast, quantum theory describes the laws of physics at atomic and subatomic scales, ignoring gravitational effects altogether. A theory of quantum gravity aims to describe the nature of spacetime on the very smallest scales—the voids in between the smallest known elementary particles—by quantum laws and possibly explain it in terms of some fundamental constituents.

Superstring theory is often described as the leading candidate to fill this role, but it has not yet provided an answer to any of these pressing questions. Instead, following its own inner logic, it has uncovered ever more complex layers of new, exotic ingredients and relations among them, leading to a bewildering variety of possible outcomes.

Over the past few years our collaboration has developed a promising alternative to this much traveled superhighway of theoretical physics. It follows a recipe that is almost embarrassingly simple: take a few very basic ingredients, assemble them according to well-known quantum principles (nothing exotic), stir well, let settle—and you have created quantum spacetime. The process is straightforward enough to simulate on a laptop.

To put it differently, if we think of empty spacetime as some immaterial substance, consisting of a very large number of minute, structureless pieces, and if we then let these microscopic building blocks interact with one another according to simple rules dictated by gravity and quantum theory, they will spontaneously arrange themselves into a whole that in many ways looks like the observed universe. It is similar to the way that molecules assemble themselves into crystalline or amorphous solids.

Spacetime, then, might be more like a simple stir fry than an elaborate wedding cake. Moreover, unlike other approaches to quantum gravity our recipe is very robust. When we vary the details in our simulations, the result hardly changes. This robustness gives reason to believe we are on the right track. If the outcome were sensitive to where we put down each piece of this enormous ensemble, we could generate an enormous number of baroque shapes, each a priori equally likely to occur—so we would lose all explanatory power for why the universe turned out as it did.

Similar mechanisms of self-assembly and self-organization occur across physics, biology and other fields of science. A beautiful example is the behavior of large flocks of birds, such as European starlings. Individual birds interact only with a small number of nearby birds; no leader tells them what to do. Yet the flock still forms and moves as a whole. The flock possesses collective, or emergent, properties that are not obvious in each bird's behavior.

A Brief History of Quantum Gravity
Past attempts to explain the quantum structure of spacetime as a process of emergence had only limited success. They were rooted in Euclidean quantum gravity, a research program initiated at the end of the 1970s and popularized by physicist Stephen Hawking's best-selling book A Brief History of Time. It is based on a fundamental principle from quantum mechanics: superposition. Any object, whether a classical or quantum one, is in a certain state—characterizing its position and velocity, say. But whereas the state of a classical object can be described by a unique set of numbers, the state of a quantum object is far richer. It is the sum, or superposition, of all possible classical states.

For instance, a classical billiard ball moves along a single trajectory with a precise position and velocity at all times. That would not be a good description for how the much smaller electron moves. Its motion is described by quantum laws, which imply that it can exist simultaneously in a wide range of positions and velocities. When an electron travels from point A to point B in the absence of any external forces, it does not just take the straight line between A and B but all available routes simultaneously. This qualitative picture of all possible electron paths conspiring together translates into the precise mathematical prescription of a quantum superposition, formulated by Nobel laureate Richard Feynman, which is a weighted average of all these distinct possibilities.

With this prescription, one can compute the probability of finding the electron in any particular range of positions and velocities away from the straight path that we would expect if the electrons followed the laws of classical mechanics. What makes the particles' behavior distinctly quantum mechanical are the deviations from a single sharp trajectory, called quantum fluctuations. The smaller the size of a physical system one considers, the more important the quantum fluctuations become.

Euclidean quantum gravity applies the superposition principle to the entire universe. In this case, the superposition consists not of different particle paths but of different ways the entire universe could evolve in time—in particular, the various possible shapes of spacetime. To make the problem tractable, physicists typically consider only the general shape and size of spacetime, rather than every single one of its conceivable contortions [see "Quantum Cosmology and the Creation of the Universe," by Jonathan J. Halliwell; Scientific American, December 1991].

Euclidean quantum gravity took a big technical leap during the 1980s and 1990s with the development of powerful computer simulations. These models represent curved spacetime geometries using tiny building blocks, which, for convenience, are taken to be triangular. Triangle meshes can efficiently approximate curved surfaces, which is why they are frequently used in computer animations. For spacetime, the elementary building blocks are four-dimensional generalizations of triangles, called four-simplices. Just as gluing together triangles at their edges creates a two-dimensional curved surface, gluing four-simplices along their "faces" (which are actually three-dimensional tetrahedra) can produce a four-dimensional spacetime.

The tiny building blocks themselves have no direct physical meaning. If one could examine real spacetime with an ultrapowerful microscope, one would not see small triangles. They are merely approximations. The only physically relevant information comes from the collective behavior of the building blocks imagining that each one is shrunk down to zero size. In this limit, nothing depends on whether the blocks were triangular, cubic, pentagonal or any mixture thereof to start with.

The insensitivity to a variety of small-scale details also goes under the name of "universality." It is a well-known phenomenon in statistical mechanics, the study of molecular motion in gases and fluids; these substances behave much the same whatever their detailed composition is. Universality is associated with properties of systems of many interacting parts and shows up on a scale much larger than that of the individual constituents. The analogous statement for a flock of starlings is that the color, size, wingspan and age of individual birds are completely irrelevant in determining the flying behavior of the flock as a whole. Only a few microscopic details filter through to macroscopic scales.

Shriveling Up
With these computer simulations, quantum gravity theorists began to explore the effects of superposing spacetime shapes that classical relativity cannot handle—specifically, ones that are highly curved on very small distance scales. This so-called nonperturbative regime is precisely what physicists are most interested in but is largely inaccessible with the usual pen-and-paper calculations.

Unfortunately, these simulations revealed that Euclidean quantum gravity is clearly missing an important ingredient somewhere along the line. They found that nonperturbative superpositions of four-dimensional universes are inherently unstable. The quantum fluctuations of curvature on short scales, which characterize the different superposed universes contributing to the average, do not cancel one another out to produce a smooth, classical universe on large scales. Instead they typically reinforce one another to make the entire space crumple up into a tiny ball with an infinite number of dimensions. In such a space, arbitrary pairs of points are never more than a tiny distance apart, even if the space has an enormous volume. In some instances, space goes to the other extreme and becomes maximally thin and extended, like a chemical polymer with many branches. Neither of these possibilities remotely resembles our own universe.

Before we reexamine the assumptions that led physicists down this dead-end street, let us pause to consider an odd aspect of this result. The building blocks are four-dimensional, yet they collectively give rise to a space having an infinite number of dimensions (the crumpled universe) or two dimensions (the polymer universe). Once the genie is let out of the bottle by allowing large quantum fluctuations of empty space, even a very basic notion such as dimension becomes changeable. This outcome could not possibly have been anticipated from the classical theory of gravity, in which the number of dimensions is always taken as a given.

One implication may come as a bit of a disappointment to science-fiction aficionados. Science-fiction stories commonly make use of wormholes—thin handles attached to the universe that provide a shortcut between regions that would otherwise be far apart. What makes wormholes so exciting is their promise of time travel and faster-than-light transmission of signals. Although such phenomena have never been observed, physicists have speculated that wormholes might find a justification within the still unknown theory of quantum gravity. In view of the negative results from the computer simulations of Euclidean quantum gravity, the viability of wormholes now seems exceedingly unlikely. Wormholes come in such a huge variety that they tend to dominate the superposition and destabilize it, and so the quantum universe never gets to grow beyond a small but highly interconnected neighborhood.

What could the trouble be? In our search for loopholes and loose ends in the Euclidean approach, we finally hit on the crucial idea, the one ingredient absolutely necessary to make the stir fry come out right: the universe must encode what physicists call causality. Causality means that empty spacetime has a structure that allows us to distinguish unambiguously between cause and effect. It is an integral part of the classical theories of special and general relativity.

Euclidean quantum gravity does not build in a notion of causality. The term "Euclidean" indicates that space and time are treated equally. The universes that enter the Euclidean superposition have four spatial directions instead of the usual one of time and three of space. Because Euclidean universes have no distinct notion of time, they have no structure to put events into a specific order; people living in these universes would not have the words "cause" or "effect" in their vocabulary. Hawking and others taking this approach have said that "time is imaginary," in both a mathematical sense and a colloquial one. Their hope was that causality would emerge as a large-scale property from microscopic quantum fluctuations that individually carry no imprint of a causal structure. But the computer simulations dashed that hope.

Instead of disregarding causality when assembling individual universes and hoping for it to reappear through the collective wisdom of the superposition, we decided to incorporate the causal structure at a much earlier stage. The technical term for our method is causal dynamical triangulations. In it, we first assign each simplex an arrow of time pointing from the past to the future. Then we enforce causal gluing rules: two simplices must be glued together to keep their arrows pointing in the same direction. The simplices must share a notion of time, which unfolds steadily in the direction of these arrows and never stands still or runs backward. Space keeps its overall form as time advances; it cannot break up into disconnected pieces or create wormholes.

After we formulated this strategy in 1998, we demonstrated in highly simplified models that causal gluing rules lead to a large-scale shape different from that of Euclidean quantum gravity. That was encouraging but not yet the same as showing that these rules are enough to stabilize a full four-dimensional universe. Thus, we held our breath in 2004 when our computer was about to give us the first calculations of a large causal superposition of four-simplices. Did this spacetime really behave on large distances like a four-dimensional, extended object and not like a crumpled ball or polymer?

Imagine our elation when the number of dimensions came out as four (more precisely, as 4.02 ± 0.1). It was the first time anyone had ever derived the observed number of dimensions from first principles. To this day, putting causality back into quantum-gravitational models is the only known cure for the instabilities of superposed spacetime geometries.

Spacetime at Large
This simulation was the first in an ongoing series of computational experiments whereby we have attempted to extract the physical and geometric properties of quantum spacetime from the computer simulations. Our next step was to study the shape of spacetime over large distances and to verify that it agrees with reality—that is, with the predictions of general relativity. This test is very challenging in nonperturbative models of quantum gravity, which do not presume a particular default shape for spacetime. In fact, it is so difficult that most approaches to quantum gravity—including string theory, except for special cases—are not sufficiently advanced to accomplish it.

It turned out that for our model to work we needed to include from the outset a so-called cosmological constant, an invisible and immaterial substance that space contains even in the complete absence of other forms of matter and energy. This requirement is good news, because cosmologists have found observational evidence for such energy. What is more, the emergent spacetime has what physicists call a de Sitter geometry, which is exactly the solution to Einstein's equations for a universe that contains nothing but the cosmological constant. It is truly remarkable that by assembling microscopic building blocks in an essentially random manner—without regard to any symmetry or preferred geometric structure—we end up with a spacetime that on large scales has the highly symmetric shape of the de Sitter universe.

This dynamical emergence of a four-dimensional universe of essentially the correct physical shape from first principles is the central achievement of our approach. Whether this remarkable outcome can be understood in terms of the interactions of some yet to be identified fundamental "atoms" of spacetime is the subject of ongoing research.

Having convinced ourselves that our quantum-gravity model passed a number of classical tests, it was time to turn to another kind of experiment, one that probes the distinctively quantum structure of spacetime that Einstein's classical theory fails to capture. One of the simulations we have performed is a diffusion process—that is, we let a suitable analogue of an ink drop fall into the superposition of universes and watch how it spreads and is tossed around by the quantum fluctuations. Measuring the size of the ink cloud after a certain time allows us to determine the number of dimensions in space.

The outcome is pretty mind-boggling: the number of dimensions depends on the scale. In other words, if we let the diffusion go on for just a short while, spacetime appears to have a different number of dimensions than when we let it run for a long time. Even those of us who specialize in quantum gravity can scarcely imagine how spacetime could smoothly change its dimension depending on the resolution of one's microscope. Evidently, a small object experiences spacetime in a profoundly different way than a large object does. To that object, the universe has something akin to a fractal structure. A fractal is a bizarre kind of space where the concept of size simply does not exist. It is self-similar, which means that it looks the same on all scales. This implies there are no rulers and no other objects of a characteristic size that can serve as a yardstick.

How small is "small"? Down to a size of about 10 meter, the quantum universe at large is well described by the classical, four-dimensional de Sitter geometry, although quantum fluctuations become increasingly significant. That one can trust the classical approximation to such short distances is rather astonishing. It has important implications for the universe both very early in its history and very far into its future. At both these extremes the universe is effectively empty. Early on, gravitational quantum fluctuations may have been so enormous that matter barely registered; it was a tiny raft tossed on a roiling ocean. Billions of years from now, because of the universe's rapid expansion, matter will be so diluted that it likewise will play little or no role. Our technique may explain the shape of space in both cases.

On still shorter scales, quantum fluctuations of spacetime become so strong that classical, intuitive notions of geometry break down altogether. The number of dimensions drops from the classical four to a value of about two. Nevertheless, as far as we can tell, spacetime is still continuous and does not have any wormholes. It is not as wild as a burbling spacetime foam, as the late physicist John Wheeler and many others imagined. The geometry of spacetime obeys nonstandard and nonclassical rules, but the concept of distance still applies. We are now in the process of probing even finer scales. One possibility is that the universe becomes self-similar and looks the same on all scales below a certain threshold. If so, spacetime does not consist of strings or atoms of spacetime, but a region of infinite boredom: the structure found just below the threshold will simply repeat itself on every smaller scale, ad infinitum.

It is difficult to imagine how physicists could get away with fewer ingredients and technical tools than we have used to create a quantum universe with realistic properties. We still need to perform many tests and experiments—for example, to understand how matter behaves in the universe and how matter in turn influences the universe's overall shape. The holy grail, as with any candidate theory for quantum gravity, is the prediction of observable consequences derived from the microscopic quantum structure. That will be the ultimate criterion for deciding whether our model really is the correct theory of quantum gravity.

Note: This story was originally published with the title, "The Self-Organizing Quantum Universe".