Three Roads to Quantum Gravity (26 page)

Two floors above them there would be a larger room where the ellipse theorists met. They would spend their time studying the orbits of planets, both in the real solar system and in imagined worlds of various dimensions. For them the key principle would be the great discovery by Kepler that planets move on elliptical orbits. They would be quite unconcerned with how bodies fall on Earth because they would share the view that only in the heavens could one see the true symmetries behind the world, uncontaminated by the complexities of the Earth, where so many bodies pushed on one another as they sought the centre. In any case they would be convinced that all motion, including that on Earth, must in the end reduce to complicated combinations of ellipses. They would assure sceptics that it was not yet time to study such problems, but when the time came they would have no problem explaining falling bodies in terms of the theory of ellipses.

Instead, they would focus their attention on the recent discovery of D-planets, which would have been found to follow parabolas rather than ellipses. So the definition of ellipse theory would be extended to include parabolas and other such curves such as hyperbolas. There would even be a conjecture that all the different orbits could be unified under one common theory, called C-theory. However, there was no agreed set of principles for C-theory, and most work on the subject required new mathematics that most physicists could not follow.

Meanwhile, another new form of mathematics was being invented by a brilliant mathematician and philosopher in Paris, René Descartes. He propounded a third theory, in which planetary orbits have to do with vortices.

It is true that while Galileo and Kepler did correspond, each seemed to show little interest in the key discoveries of the
other. They wrote to each other about the telescope and what it revealed, but Galileo seems never to have mentioned ellipses, and to have gone to his grave believing the planetary orbits were circles. Nor is there any evidence that Kepler ever thought about falling bodies or believed them to be relevant to explaining the motions of the planets. It took a young scientist of a later generation, Isaac Newton, born the year of Galileo’s death, to wonder whether the same force that made apples fall drew the Moon to the Earth and the planets to the Sun. So, while my story is fanciful, it really did happen that scientists with the stature of Galileo and Kepler each contributed an essential ingredient to a scientific revolution while remaining almost ignorant of and apparently uninterested in each other’s discoveries.

We can hope that it will take less time to bring the different pieces of the quantum theory of gravity together than it did for someone to see the relationship between the work of Kepler and Galileo. The simple reason is that there are many more scientists working now than there were then. Whereas Kepler and Galileo might each have complained, if asked, that they were too busy to look at what the other was doing, there are now plenty of people to share the work. However, there is now the problem of making sure that young people have the freedom to wander across boundaries established by their elders without fear of jeopardizing their careers. It would be naive to say this is not a significant issue. In many areas of science we are paying for the consequences of an academic system that rewards narrowness of focus over exploration of new areas. This underlines the fact that good science is, and will always be, as much a question of judgement and character as it is a question of cleverness.

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 how 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.