[tt] Using Causality to Solve the Puzzle of Quantum Spacetime: Scientific American

[Brian Atkins]

http://www.sciam.com/article.cfm?id=the-self-organizing-quantum-universe&print=true

Using Causality to Solve the Puzzle of Quantum Spacetime
A new approach to the decades-old problem of quantum gravity goes back to basics 
and shows how the building blocks of space and time pull themselves together

By Jerzy Jurkiewicz, Renate Loll and Jan Ambjorn

Editor's Note: Click here for the web animations mentioned in the article

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".

-- 
Brian Atkins
Singularity Institute for Artificial Intelligence
http://www.singinst.org/