Three Roads to Quantum Gravity

by Lee Smolin

  The story of loop quantum gravity really begins in the 1950s with an idea that came from what would seem to be a totally different subject - the physics of superconductors. Physics is like this: the few really good ideas are passed around from field to field. The physics of materials such as metals and superconductors has been a very fertile source of ideas about how physical systems might behave. This is undoubtedly because in these fields there is a close interaction between theory and experiment which makes it possible to discover new ways for physical systems to organize themselves. Elementary particle physicists do not have access to such direct probes of the systems they model, so it has happened that on several occasions we have raided the physics of materials for new ideas.

  Superconductivity is a peculiar phase that certain metals can be put into in which their electrical resistance falls to zero. A metal can be turned into a superconductor by cooling it below what is called its critical temperature. This critical temperature is usually very low, just a few degrees above absolute zero. At this temperature the metal undergoes a change of phase something like freezing. Of course, it is already a solid, but something profound happens to its internal structure which liberates the electrons from its atoms, and the electrons can then travel through it with no resistance. Since the early 1990s there has been an intensive quest to find materials that are superconducting at room temperature. If such a material were to be found there would be profound economic implications, as it might greatly reduce the cost of supplying electricity. But the set of ideas I want to discuss go back to the 1950s, when people first understood how simple superconductors work. A seminal step was the invention of a theory by John Bardeen, Leon Cooper and John Schrieffer, known as the BCS theory of superconductivity. Their discovery was so important that it has influenced not only many later developments in the theory of materials, but also developments in elementary particle physics and quantum gravity.

  You may remember a simple experiment you did at school with a magnet, a piece of paper and some iron filings. The idea was to visualize the field of the magnet by spreading the filings on a piece of paper placed over the magnet. You would have seen a series of curved lines running from one pole of the magnet to the other (Figure 19). As your teacher may have told you, the apparent discreteness of the field lines is an illusion. In nature they are distributed continuously; they only appear to be a discrete set of lines because of the finite size of the iron filings. However, there is a situation in which the field lines really are discrete. If you pass a magnetic field through a superconductor, the magnetic field breaks up into discrete field lines, each of which carries a fundamental unit of magnetic flux (Figure 20). Experiments show that the amount of magnetic flux passing through a superconductor is always an integer multiple of this fundamental unit.

  FIGURE 19

  Field lines between two poles of an ordinary magnet, in air.

  FIGURE 20

  The magnetic field of a superconductor breaks up into discrete flux lines, each carrying a certain minimum amount of the field.

  This discreteness of the magnetic field lines in superconductors is a curious phenomenon. It is unlike the discreteness of the electric charge, or of matter, in that it has to do with a field that carries a force. Furthermore, it seems that we can turn it on and off, depending on the material the magnetic field is passing through.

  The electric field has field lines as well, although there is no equivalent of the iron filings experiment that allows us to see them. But in all circumstances we know about they are continuous: no material has been found which functions like an electric superconductor to break electric field lines into discrete units. But we can still imagine something like an electric superconductor, in which the field lines of the electric field would be quantized. This idea has been very successful in explaining a result from another seemingly unrelated subject: experiments show that protons and neutrons are each composed of three smaller entities called quarks.

  We have good evidence that there are quarks inside protons and neutrons, just as there are electrons, protons and neutrons inside the atom. There is one difference, however, which is that the quarks seem to be trapped inside the protons. No one has ever seen a quark moving freely, that was not trapped inside a proton, neutron or other particle. It is easy to free electrons from atoms - one needs only to supply a little energy, and the electrons jump out of the atom and move freely. But no one has found a way to free a quark from a proton or neutron. We say that the quarks are confined. What we then need to understand is whether there is a force that can act as the electric field does in holding electrons around the nucleus, but that does so in such a way that the quarks can never come out.

  From many different experiments we know that the force that holds the quarks together inside a proton is quite similar to the electric force. For one thing, we know that force is transmitted by a field that forms lines like electric and magnetic field lines. These lines connect charges which are carried by the quarks, just as electric field lines connect positive and negative electric charges. However, the force between quarks is rather more complicated than the electric force, for which there is only one kind of charge. Here there are three different varieties of charge, each of which can be positive or negative. These different charges are called colours, which is why the theory that describes them is called quantum chromodynamics, QCD for short. (This has nothing to do with ordinary colours, it is just a vivid terminology which reminds us there are three kinds of charge.) Imagine two quarks held together by some colour-electric field lines, as shown in Figure 21. Experiments show that when the two quarks are very close to each other they seem to move almost freely, as if the force between them is not very strong. But if an attempt is made to separate the two quarks, the force holding them together rises to a constant value, which does not fall off no matter how far apart they are pulled. This is very different from the electric force, which becomes weaker with increasing distance.

  There is a simple way to picture what is happening. Imagine that the two quarks are connected by a length of string. This string has the peculiar property that it can be stretched however far we want. But to separate the quarks we must stretch the string, and this requires energy. No matter how long the string already is, we are going to have to put more energy in to stretch it more. To put energy into the string we must pull on it, which means that there is a force between the quarks. No matter how far apart the quarks are, to pull them farther apart you must stretch the string more, which means that there is always a force between them. As shown in Figure 21, no matter how far apart they may be they are still connected to each other by the string. This stringy picture of the force that holds quarks together is very successful, and explains the results of many experiments. But it brings with it a question: what is the string made of? Is it itself a fundamental entity, or is it composed of anything simpler? This is a question that generations of elementary particle physicists have worked to answer.

  The one big clue we have is that the string stretched between two quarks behaves just like a line of magnetic flux in a superconductor. This suggests a simple hypothesis:

  FIGURE 21

  Quarks are held together by strings made of quantized flux lines of a field, called the QCD field, which are analogous to the quantized magnetic flux lines in a superconductor (Figure 20). As the quarks are pulled farther apart, the flux lines are stretched, and the force between the quarks is the same no matter how far apart they are. The result is that the quarks cannot be pulled apart.

  perhaps empty space is very like a superconductor, except that what ends up discrete is the lines of force holding the colour charges of quarks together rather than the lines of magnetic flux. In this picture the lines of force between the coloured charges on the quarks are analogous to the electric rather than the magnetic field. So this hypothesis can be put very succinctly as follows: empty space is a colour-electric superconductor. This has been one of the most seminal ideas in elementary particle physic
s over the last few decades. It explains why quarks are confined in protons and neutrons, as well as many other facts about elementary particles. But what is really interesting is that the idea, clear as it is, contains a puzzle, for it can be looked it in two quite different ways.

  One can take the colour-electric field as the fundamental entity, and then try to understand the picture of a string stretched between the quarks as a consequence of space having properties that make it something like an electric version of a superconductor. This is the route taken by those physicists who work on QCD. For them, the key problem is to understand why empty space has properties that make it behave in certain circumstances like a superconductor. This is not as crazy as it sounds. We understand that in quantum theory space must be seen to be full of oscillating random fields, as discussed in Chapter 6. So we may imagine that these vacuum fluctuations sometimes behave like the atoms in a metal in a way that leads to large-scale effects like superconductivity.

  But there is another way to understand the picture of quarks held together by stretched strings. This is to see the strings themselves as fundamental entities, rather than as made up of the force lines of some field. This picture led to the original string theory. According to the first string theorists, the string is fundamental and the field is only an approximate picture of how the strings behave in some circumstances.

  We thus have two pictures. In one, the strings are fundamental and the field lines are an approximate picture. In the other, the field lines are fundamental and the strings are the derived entities. Both have been studied, and both have had some success in explaining the results of experiments. But surely only one can be right? During the 1960s there was only one picture - the string picture. During this period were planted the seeds that would lead, two decades later, to the invention of string theory as a possible quantum theory of gravity. QCD was invented in the 1970s and quickly superseded the string picture as it seemed more successful as a fundamental theory. But string theory was revived in the mid- 1980s, and now, as we enter the twenty-first century, both theories are thriving. It may still be that one is actually more fundamental than the other, but we have not yet been able to decide which.

  There is a third possibility, which is that both the string picture and the field picture are just different ways of looking at the same thing. They would then be equally fundamental, and no experiment could decide between them. This possibility excites many theorists, as it challenges some of our deepest instincts about how to think about physics. It is called the hypothesis of duality.

  I should emphasize that this hypothesis of duality is not the same as the wave-particle duality of quantum theory. But it is as important as that principle or the principle of relativity. Like the principles of relativity and quantum theory, the hypothesis of duality tells us that two seemingly different phenomena are just two ways of describing the same thing. If true, it has profound implications for our understanding of physics.

  The hypothesis of duality also addresses an issue that has plagued physics since the middle of the nineteenth century, that there seem to be two kinds of things in the world: particles and fields. This dualistic description seems necessary because, as we have known since the nineteenth century, charged particles do not interact directly with one another. Instead, they interact via the electric and magnetic fields. This is behind many observed phenomena, including the fact that it takes a finite speed for information to travel between particles. The reason is that the information travels via waves in the field.

  Many people have been troubled by the need to postulate two very different kinds of entities to explain the world. In the nineteenth century people tried to explain fields in terms of matter. This was behind the famous aether theory, which Einstein so effectively quashed. Modern physicists try instead to explain particles in terms of fields. But this does not eliminate all the problems. Some of the most serious of these problems have to do with the fact that the theory of fields is full of infinite quantities. They arises because the strength of the electric field around a charged particle increases as one gets closer to the particle. But a particle has no size, so one can get as close as one likes to it. The result is that the field approaches infinity as one approaches the particle. This is responsible for many of the infinite expressions that arise in the equations of modern physics.

  There are two ways to resolve this problem, and we shall see that both play a role in quantum gravity. One way is to deny that space is continuous, which then makes it impossible to get arbitrarily close to a particle. The other way is through the hypothesis of duality. What one can do is replace the particles by strings. This may work because from a distance one cannot really tell if something is a point or a little loop. But if the hypothesis of duality is true, then the strings and the fields may be different ways of looking at the same thing. In this way, by embracing the hypothesis of duality, several of the problems that have clouded our understanding of physics for almost two centuries may be resolved.

  I personally believe in this hypothesis. To explain why, I can tell the story of two seminars I attended just before and just after I started graduate school in 1976. I happened to have my interview at Harvard on the day that Kenneth Wilson was giving a talk about QCD. Wilson is one of the most influential theoretical physicists, responsible for several innovations, including the subject of that seminar. He had come up with a remarkable way to understand the electric-superconductor picture of empty space which has since been a major influence on the life’s work of many physicists, including myself.

  Wilson asked us to imagine that space is not continuous, but is instead represented by a kind of graph, with points connected in a regular arrangement by lines, as shown in Figure 22. We call such a regular graph a lattice. He suggested to us that the distance between the points of his lattice was very small, much smaller than the diameter of a proton. So it would be hard to tell from experiment that the lattice was there at all. But conceptually it made a huge difference to think of space as a discrete lattice rather than a continuum. Wilson showed us that there was a very simple way to describe the colour-electric field of QCD by drawing field lines on his lattice. Rather than trying to show that empty space was like a superconductor, he simply assumed that the field lines were discrete entities which could move around his lattice. He wrote down simple rules to describe how they moved and interacted with one another.

  Ken Wilson then argued in completely the opposite direction to everyone who had previously thought about these questions. He showed us that if there was one kind of electric charge, as in ordinary electricity, the field lines would have the tendency to group collectively in such a way that when they got very long they would lose the property of discreteness and behave like ordinary electric field lines. So he derived the ordinary experience of the world from his theory, rather than the reverse. But when there were three kinds of charge, as there were with quarks, then no matter how big they got, they would always stay discrete. And there would be a constant force between the quarks. The rules governing Wilson’s theory were very simple - so simple, in fact, that one could explain them to a child.

  FIGURE 22

  Quarks and strings as conceived of by Kenneth Wilson. Space is imagined as a lattice made of nodes connected by edges. The quarks can live only on the nodes of the lattice. The strings, or quantized tubes of flux of the field, connect the quarks but can exist only on the edges of the lattice. The distance between the nodes is assumed to be finite, but much smaller than the size of the proton. For simplicity, the lattice shown here is drawn in two dimensions only.

  Wilson’s loops, as everyone has called them since, later became a major theme of my life as a theoretical physicist. I don’t actually recall reflecting on the seminar afterwards, but I do recall its presentation very vividly. Nor did I then, so far as I remember, formulate the simple argument that came to me many years later: if physics is much simpler to describe under the assumption that space is discrete, rather than continuou
s, is not this fact itself a strong argument for space being discrete? If so, then might space look, on some very small scale, something like Wilson’s lattice?

  Next autumn I started graduate school, and later that year I came in one day to find a great buzz of excitement amongst the theorists. The Russian theorist Alexander Polyakov was visiting and was to give a talk that afternoon. In those days there were great schools of theoretical physics in the Soviet Union, but their members were seldom allowed to travel to the West. Polyakov was the most creative and most charismatic of them, and we all went along to his seminar. I recall someone with disarming warmth and informality, under which there was hidden (but not too well) someone with unlimited confidence.

  He began by telling us that he had dedicated his life to pursuing a foolish and quixotic vision, which was to re-express QCD in a form in which the theory could be solved exactly. His idea for doing this was to recast QCD completely as a theory of the dynamics of lines and loops of colour-electric flux. These were the same as Wilson’s loops, and indeed Polyakov had independently invented the picture of QCD on a discrete lattice. But in this seminar at least he worked without the lattice, to try to pull out from the theory a description in which the quantized loops of electric flux would be the fundamental entities. A physicist working without a lattice is something like a trapeze artist working without a net. There is an ever present danger that a false move will lead to a fatal result. In physics the fatalities arise from confrontations with infinite and absurd mathematical expressions. As we mentioned earlier, such expressions arise in all quantum theories based on continuous space and time. In his seminar Polyakov showed that despite these infinities, one could give physical meaning to loops of electric flux. If he did not succeed completely in solving the resulting equations, his seminar was altogether an assertion of faith in the hypothesis of duality - that the strings are as fundamental as the electric field lines.