by Lee Smolin
Indeed, over the last five years the climate of mutual ignorance and complacency that separated the string theorists from the loop quantum gravity people has begun to dissipate. The reason is that it has been becoming increasingly clear that each group has a problem it cannot solve. For string theory it is the problem of making the theory background independent and finding out what M theory really is. This is necessary both to unify the different string theories into a single theory and to make string theory truly a quantum theory of gravity. Loop quantum gravity is faced with the problem of how to show that a quantum spacetime described by an evolving spin network will grow into a large classical universe, which to a good approximation can be described in terms of ordinary geometry and Einstein’s theory of general relativity. This problem arose in 1995 when Thomas Thiemann, a young German physicist then working at Harvard, presented for the first time a complete formulation of loop quantum gravity which resolved all the problems then known to exist. Thiemann’s formulation built on all the previous work, to which he added some brilliant innovations of his own. The result was a complete theory which in principle should be able to answer any question. Furthermore, the theory could be derived directly from Einstein’s general theory of relativity by following a well defined and mathematically rigorous procedure.
As soon as we had the theory, we began calculating with it. The first thing to calculate was how a graviton might appear as a description of a small wave or disturbance passing through a spin network. Before this could be done, however, we had to solve a more basic problem, which was to understand how the geometry of space and time, which seems so smooth and regular on the scales we can see, emerges from the atomic description in terms of spin networks. Until this was done we would not be able to make sense of what a graviton is, as gravitons should be related to waves in classical spacetime.
This kind of problem, new to us, is very familiar to physicists who study materials. If I cup my hands together and dip them into a stream I can carry away only as much water as will fill the ‘cup’. But I can lift a block of ice just by holding it at its two sides. What is it about the different arrangements of the atoms in water and ice that accounts for the difference? Similarly, the spin networks that form the atomic structure of space can organize themselves in many different ways. Only a few of these ways will have a regular enough structure to reproduce the properties of space and time in our world.
What is remarkable - indeed, what is almost a miracle - is that the hardest problem faced by each group was precisely the key problem that the other had solved. Loop quantum gravity tells us how to make a background independent quantum theory of space and time. It offers a lot of scope to the M theorist looking for a way to make string theory background independent. On the other hand, if we believe that strings must emerge from the description of space and time provided by loop quantum gravity, we then have a lot of information about how to formulate the theory so that it does describe classical spacetime. The theory must be formulated in such a way that the gravitons appear not on their own, but as modes of excitations of extended objects that behave as strings.
It is then possible to entertain the following hypothesis: string theory and loop quantum gravity are each part of a single theory. This new theory will have the same relationship to the existing ones as Newtonian mechanics has to Galileo’s theory of falling bodies and Kepler’s theory of planetary orbits. Each is correct, in the sense that it describes to a good approximation what is happening in a certain limited domain. Each solves part of the problem. But each also has limits which prevent it from forming the basis for a complete theory of nature. I believe that this the most likely way in which the theory of quantum gravity will be completed, given the present evidence. In this penultimate chapter I shall describe some of this evidence, and the progress that has recently been made towards inventing a theory that unifies string theory and loop quantum gravity.
As a first step we can ask for a rough picture of how the two theories might fit together. As it happens, there is a very natural way in which strings and loops can emerge from the same theory. The key to this is a subtlety that I have so far only hinted at. Both loop quantum gravity and string theory describe physics on very small scales, roughly the Planck length. But the scale that sets the size of strings is not exactly equal to the Planck length. That scale is called the string length. The ratio of the Planck length to the string length is a number of great significance in string theory. It is a kind of charge, which tells us how strongly strings will interact with one another. When the string scale is much larger than the Planck length this charge is small and strings do not interact very much with one another.
We then can ask which scale is larger. There is evidence that, at least in our universe, the string scale is larger than the Planck scale. This is because their ratio determines the fundamental unit of electric charge, and that is itself a small number. We can then envisage scenarios in which loops are more fundamental. The strings will be descriptions of small waves or disturbances travelling through spin networks. Since the string scale is larger, we can explain the fact that string theory relies on a fixed background, as the needed background can be supplied by a network of loops. The fact that strings seem to experience the background as a continuous space is explained by them being unable to probe down to a distance where they can distinguish a smooth background from a network of loops (see Figure 38 on page 165).
One way to talk about this is that space may be ‘woven’ from a network of loops, as shown in Figure 38, just as a piece of cloth is woven from a network of threads. The analogy is fairly precise. The properties of the cloth are explicable in terms of the kind of weave, which is to say in terms of how the threads are knotted and linked with one another. Similarly, the geometry of the space we may weave from a large spin network is determined only by how the loops link and intersect one another.
We may then imagine a string as a large loop which makes a kind of embroidery of the weave. From a microscopic point of view, the string can be described by how it knots the loops in the weave. But on a larger scale we would see only the loop making up the string. If we cannot see the fine weave that makes up space, the string will appear against a background of some apparently smooth space. This is how the picture of strings against a background space emerges from loop quantum gravity.
If this is right, then string theory will turn out to be an approximation to a more fundamental theory described in terms of spin networks. Of course, just because we can argue for a picture like this does not mean that it can be made to work in detail. In particular, it may not work for any version of loop quantum gravity. To make the large loops behave as strings we may have to choose the details of the loop theory carefully. This is good, not bad, for it tells us how information about the world already revealed by string theory may be coded in such a way that it becomes part of the fundamental theory that describes the atomic structure of space and time. At present, a programme of research is under way to unify string theory and loop quantum gravity using essentially this idea. Very recently this has led to the discovery of a new theory that appears to contain within it both string theory and a form of loop quantum gravity. It looks promising to some of us but, as it is work in progress, I can say no more about it here.
However, if this programme does work it will exactly realize the idea of duality I discussed in Chapter 9. It will also realize the aims of Amitaba Sen, for the whole loop approach arose out of his efforts to understand how to quantize supergravity, which is now understood to be closely related to string theory.
While my hypothesis is certainly not proven, evidence has been accumulating that string theory and loop quantum gravity may describe the same world. One piece of evidence, discussed in the last chapter, is that both theories point to some version of the holographic principle. Another is that the same mathematical ideas structures keep appearing on both sides. One example of this is a structure called non-commutative geometry. This is an idea about ho
w to unify quantum theory with relativity that was invented by the French mathematician Alain Connes. The basic idea is very simple: in quantum physics we cannot measure the position and velocity of a particle at the same time. But if we want to we can at least determine the position precisely. However, notice that a determination of the position of a particle actually involves three different measurements, for we must measure where the particle is relative to a set of three axes (these measurements yield the three components of the position vector). So we may consider an extension of the uncertainty principle in which one can measure only one of these components precisely at any one time. When it is impossible to measure two quantities simultaneously, they are said not to commute, and this idea leads to a new kind of geometry which is labelled non-commutative. In such a world one cannot even define the idea of a point where something may be exactly located.
Alain Connes’s non-commutative geometry thus gives us another way to describe a world in which the usual notion of space has broken down. There are no points, so it does not even make sense to ask if there are an infinite number of points in a given region. What is really wonderful, though, is that Connes has found that large pieces of relativity theory, quantum theory and particle physics can be carried over into such a world. The result is a very elegant structure that seems also to penetrate to several of the deepest problems in mathematics.
At first, Connes’s ideas were developed independently of the other approaches. But in the last few years people have been surprised to discover that both loop quantum gravity and string theory describe worlds in which the geometry is non-commutative. This gives us a new language in which to compare the two theories.
One way to test the hypothesis that strings and loops are different ways of describing the same physics is to attack a single problem with both methods. There is an obvious target: the problem of giving a description of a quantum black hole. From the discussion in Chapters 5 to 8, we know that the main objective is to explain in terms of some fundamental theory where the entropy and temperature of a black hole come from, and why the entropy is proportional to the area of the black hole’s horizon. Both string theory and loop quantum gravity have been used to study quantum black holes, with spectacular results coming on each side in the last few years.
The main idea on each side is the same. Einstein’s theory of general relativity is to be thought of as a macroscopic description, obtained by averaging over the atomic structure of spacetime, in exactly the same way that thermodynamics is obtained by applying statistics to the motion of atoms. Just as a gas is described roughly in terms of continuous quantities such as density and temperature, with no mention of atoms, in Einstein’s theory space and time are described as continuous, and no mention is made of the discrete, atomic structure that may exist on the Planck scale.
Given this general picture, it is natural to ask whether the black hole’s entropy is a measure of the missing information that could be obtained from an exact quantum description of the geometry of space and time around a black hole. The fact that the entropy of a black hole is proportional to the area of its horizon should be a huge clue to its meaning. String theory and loop quantum gravity have each found a way to use this clue to construct a description of a quantum black hole.
In string theory, good progress has been made by conjecturing that the missing information measured by the black hole’s entropy is a description of how the black hole was formed. A black hole is a very simple object. Once formed, it is featureless. From the outside one can measure only a few of its properties: its mass, electric charge and angular momentum. This means that a particular black hole might have been formed in many different ways: for example, from a collapsing star, or - in theory at least - by compressing, say, a pile of science-fiction magazines to an enormous density. Once the black hole has formed there is no way to look inside and see how it was formed. It emits radiation, but that radiation is completely random, and offers no clue to the black hole’s origin. The information about how the black hole formed is trapped inside it. So one may hypothesize that it is exactly this missing information that is measured by the black hole’s entropy.
Over the last few years string theorists have discovered that string theory is not just a theory of strings. They have found that the quantum gravity world must be full of new kinds of object that are like higher-dimensional versions of strings in that they extend in several dimensions. Whatever their dimension, these objects are called branes. This is shortened from ‘membranes’, the term used for objects with two spatial dimensions. The branes emerged when new ways to test the consistency of string theory were discovered, and it was found that the theory can be made mathematically consistent only by including a whole set of new objects of different dimensions.
String theorists have found that in certain very special cases black holes could be made by bringing together a collection of these branes. To do this they make use of a feature of string theory, which is that the gravitational force is adjustable. It is given by the value of a certain physical field. When this field is increased or decreased, the gravitational force becomes stronger or weaker. By adjusting the value of the field it is possible to turn the gravitational force on and off. To make a black hole they begin with the gravitational field turned off. Then they imagine assembling a set of branes which have the mass and charge of the black hole they want to make. The object is not yet a black hole, but they can turn it into one by turning up the strength of the gravitational force. When they do so a black hole must form.
String theorists have not yet been able to model in detail the process of the formation of the black hole. Nor can they study the quantum geometry of the resulting black hole. But they can do something very cute, which is to count the number of different ways that a black hole could be formed in this way. They then assume that the entropy of the resulting black hole is a measure of this number. When they do the counting, they get, right on the nose, the right answer for the entropy of the black hole.
So far only very special black holes can be studied by this method. These are black holes whose electric charges are equal to their mass. This is to say that the electrical repulsions of two of these black holes are exactly balanced by their gravitational attractions. As a result, one can put two of them next to each other and they will not move, for there is no net force between them. These black holes are very special because their properties are strongly constrained by the condition that their charge balances their mass. This makes it possible to get precise results, and, when this is possible the results are very impressive. On the other hand, it is not known how to extend the method to all black holes. Actually, string theorists can do a bit better than this, for the methods can be used to study black holes whose charges are close to their masses. These calculations also give very impressive results: in particular, they reproduce every last factor of 2 and π in the formula for the radiation emitted by these black holes.
A second idea about a black hole’s entropy is that it is a count, not of the ways to make a black hole, but of the information present in an exact description of the horizon itself. This is suggested by the fact that the entropy is proportional to the area of the horizon. So the horizon is something like a memory chip, with one bit of information coded in every little pixel, each pixel taking up a region 2 Planck lengths on a side. This picture turns out to be confirmed by calculations in loop quantum gravity.
A detailed picture of the horizon of a black hole has been developed using the methods of loop quantum gravity. This work started in 1995 when, inspired by the ideas of Crane, ’t Hooft, and Susskind, I decided to try to test the holographic principle in loop quantum gravity. I developed a method for studying the quantum geometry of a boundary or a screen. As I mentioned earlier, the result was that the Bekenstein bound was always satisfied, so that the information coded into the geometry on the boundary was always less than a certain number times its total area.
Meanwhile, Carlo Rovelli was developing a ro
ugh picture of the geometry of a black hole horizon. A graduate student of ours, Kirill Krasnov, showed me how the method I had discovered could be used to make Carlo’s ideas more precise. I was quite surprised because I had thought that this would be impossible. I worried that the uncertainty principle would make it impossible to locate the horizon exactly in a quantum theory. Kirill ignored my worries and developed a beautiful description of the horizon of a black hole which explained both its entropy and its temperature. (Only much later did Jerzy Lewandowksi, a Polish physicist who has added much to our understanding of loop quantum gravity, work out how the uncertainty principle is circumvented in this case.)
Kirill’s work was brilliant, but a bit rough. He was subsequently joined by Abhay Ashtekar, John Baez, Alejandro Corichi and other more mathematically minded people who developed his insights into a very beautiful and powerful description of the quantum geometry of horizons. The results can be applied very widely, and give a general and completely detailed description of what a horizon would look like were it to be probed on the Planck scale.
