by Lee Smolin
The loop approach to quantum gravity is now a thriving field of research. Many of the older ideas, such as supergravity and the study of quantum black holes, have been incorporated into it. Connections have been discovered to other approaches to quantum gravity, such as Alain Connes’s non-commutative approach to geometry, Roger Penrose’s twistor theory and string theory.
One lesson we have learned from this experience is the extent to which science progresses quickly when people with different backgrounds and educations join forces to push back the frontiers. The relationship between theoretical physicists and mathematical physicists is not always smooth. It is rather like the relationship between the scouts who first explore the frontier, and the farmers who come after them and fence the land and make it productive. The mathematical farmers need to tie everything down, and determine the exact boundaries of an idea or a result, while we physicist scouts like our ideas when they are still a bit wild and untamed. Each has a tendency to think that they did the essential part of the work. But something we and the string theorists have both learned is that in spite of their different ways of working and thinking, it is essential that mathematicians and physicists learn to communicate and work with one another. As happened with general relativity, quantum gravity requires new mathematics as much as it requires new concepts, ideas and ways of doing calculations. If we have made real progress it is because we have discovered that people can work together to invent something that no one person could have come up with alone.
In the end, what is most satisfying about the picture of space given by loop quantum gravity is that it is completely relational. The spin networks do not live in space; their structure generates space. And they are nothing but a structure of relations, governed by how the edges are tied together at the nodes. Also coded in are rules about how the edges may knot and link with one another. It is also very satisfying that there is a complete correspondence between the classical and quantum pictures of geometry. In classical geometry the volumes of regions and the areas of the surfaces depend on the values of gravitational fields. They are coded in certain complicated collections of mathematical functions, known collectively as the metric tensor. On the other hand, in the quantum picture the geometry is coded in the choice of a spin network. These spin networks correspond to the classical description in that, given any classical geometry, one can find a spin network which describes, to some level of approximation, the same geometry (Figure 27).
In classical general relativity the geometry of space evolves in time. For example, when a gravitational wave passes a surface, the area of that surface will oscillate in time. There is an equivalent quantum picture in which the structures of the spin networks may evolve in time in response to the passage of a gravitational wave. Figure 28 shows some of the simple steps by which a spin network evolves in time. If we let a spin network evolve, we get a discrete spacetime structure. The events of this discrete spacetime are the processes by which changes of the form shown in Figure 28 occur. We can draw pictures of evolving spin networks; they look like Figures 29 to 31. An evolving spin network is very like a spacetime, but it is discrete rather than continuous. We can say what the causal relations are among the events, so it has light cones. But it also has more, for we can draw slices through it that correspond to moments of time. As in relativity theory, there are many different ways of slicing an evolving spin network, so as to see it as a succession of states evolving in time. Thus, the picture of spacetime given by loop quantum gravity agrees with the fundamental principle that in the theory of relativity there are no things, only processes.
John Wheeler used to say that on the Planck scale spacetime would no longer be smooth, but would resemble a foam, which he called spacetime foam. In tribute to Wheeler, the mathematician John Baez has suggested that evolving spin networks be called spin foam. The study of spin foam has sprung up since the mid-1990s. There are several different versions presently under study, invented by Mike Reisenberger, by Louis Crane and John Barrett, and by Fotini Markopoulou-Kalamara. Carlo Rovelli, John Baez, Renate Loll and many of the other people who contributed to loop quantum gravity are now engaged in the study of spin foam. So this is presently a very lively area of research. Figure 32 shows a computer simulation of a world with one space and one time dimension, modelled upon ideas from spin foam theory. This is work of Jan Ambjørn, Kostas Anagnastopoulos and Renate Loll. These universes are very small, each edge corresponding to one Planck length. They do not always evolve smoothly; instead, from time to time the size of the universe jumps suddenly. These are the quantum fluctuations of the geometry. After many years, we have here a real quantum theory of the geometry of spacetime.
FIGURE 28
Simple stages by which a spin network can evolve in time. Each one is a quantum transition of the geometry of space. These are the quantum theoretic analogues of the Einstein equations. [From F. Markopoulou, ‘Dual formulation of spin network evolution’, gr-qc/9704013. All the papers referenced here as gr-qc/xxxx are available at xxx.lanl.gov.]
FIGURE 29
Two pictures of quantum spacetimes. Each event in the quantum spacetime is a simple change in the quantum geometry of space, corresponding to one of the moves shown in Figure 28. According to loop quantum gravity, this is what spacetime looks like if we examine it on a time scale of 10-43 of a second and a length scale of 10-33 of a centimetre. The upper picture shows a single elementary move. The lower one shows a combination of two elementary moves. [From C. Rovelli, ‘The projector on physical states in loop quantum gravity’, gr-qc/9806121.]
FIGURE 30
Another of the elementary moves for quantum transitions among spin networks, together with the spacetime picture that represents it. [From R. de Pietri, ‘Canonical loop quantum gravity and spin foam models’, gr-qc/ 9903076.]
Is the theory right? We do not know yet. In the end, it will be decided by experiments designed to test the predictions the theory makes about the discreteness of area and volume and other measures of spacetime geometry. I do want to emphasize that although it follows directly from the combination of the principles of general relativity and quantum theory, loop quantum gravity does not need to be the complete story to be true. In particular, the main predictions of the theory, such as the quantization of area and volume, do not depend on many details being correct, only on the most general assumptions drawn from quantum theory and relativity. The predictions do not constrain what else there can be in the world, how many dimensions there are or what the fundamental symmetries are. In particular, they are completely consistent with the basic features of string theory, including the existence of extra dimensions and supersymmetry. I know of no reason to doubt their truth.
Of course, in the end experiment must decide. But can we really hope for experimental confirmation of the structure of space on the Planck scale, 20 orders of magnitude smaller than the proton? Until very recently most of us were sceptical about whether we might see such tests in our lifetime. But now we know we were being too pessimistic. A very imaginative young Italian physicist, Giovanni Amelino-Camelia, has pointed out that there is a way to test the predictions that the geometry of space is discrete on the Planck scale. His method uses the whole universe as an instrument.
FIGURE 31
Another picture of a quantum spacetime, showing the causal futures of the events where the spin networks change. These are drawn as light cones, as in Chapter 4. [From F. Markopoulou and L. Smolin, ‘The causal evolution of spin networks’, gr-qc/9702025.]
When a photon travels through a discrete geometry it will suffer small deviations from the path that classical physics predicts for it. These deviations are caused by the interference effects that arise when the photon’s associated wave is scattered by the discrete nodes of the quantum geometry. For photons that we can detect, these effects are very, very small. What no one before Amelino-Camelia had thought of, though, is that the effects accumulate when a photon travels very long distances. And we can detect photo
ns that have travelled across large fractions of the observable universe. He proposes that by carefully studying images taken by satellites of very violent events such as those that create X-ray and gamma-ray bursts, it may be possible to discover experimentally the discrete structure of space.
FIGURE 32
Computer model of a quantum spacetime, showing a universe with one space and one time dimension. The structures shown exist on scales of 10-33 of a centimetre and 10-43 of a second. We see that the quantum geometry fluctuates very strongly because of the uncertainty principle. As with the position of an electron in an atom, for such small universes the quantum fluctuations in the size of the universe are very important as they are as large as the universe itself. [These simulations are the work of Jan Ambøjrn, Kostas Anagnastopoulos and Renate Loll. They can be seen at their Web page, http://www.nbi.dk/~konstant/homepage/lqg2/.]
If these experiments do show that space has an atomic structure on the Planck scale, it will surely be one of the most exciting discoveries of early twenty-first century science. By developing these new methods we may be able to look at pictures of the discrete structure of space, just as we are now able to study pictures of arrays of atoms. And if the work I have described in the last two chapters is not completely irrelevant, what we shall see are Wilson’s and Polyakov’s loops, organized into Penrose’s spin networks.
CHAPTER 11
THE SOUND OF SPACE IS A STRING
I am convinced that the hardest thing about doing science is not that it sometimes demands a certain level of skill and intelligence. Skills can be learned, and as for intelligence, none of us is really smart enough to get anywhere on our own. All of us, even the most independent, manage to carry our work through to completion because we are part of a community of committed and honest people. When we are stuck, most of us look for a way out in the work of others. When we are lost, most of us look to see what others are doing. Even then, we often get lost. Sometimes even whole groups of friends and colleagues get lost together. Consequently, the hardest thing about science is what it demands of us in terms of our ability to make the right choice in the face of incomplete information. This requires characteristics not easily measured by tests, such as intuition and a person’s faith in themself. Einstein knew this, which is why he told John Wheeler, in a remark that Wheeler has often repeated, how much he admired Newton’s courage and judgement in sticking with the idea of absolute space and time even though all his colleagues told him it was absurd. The idea is absurd, as Einstein knew better than anyone. But absolute space and time was what was required to make progress at the time, and to see this was perhaps Newton’s greatest achievement.
Einstein himself is often presented as the prime example of someone who did great things alone, without the need for a community. This myth was fostered, perhaps even deliberately, by those who have conspired to shape our memory of him. Many of us were told a story of a man who invented general relativity out of his own head, as an act of pure individual creation, serene in his contemplation of the absolute as the First World War raged around him.
It is a wonderful story, and it has inspired generations of us to wander with unkempt hair and no socks around shrines like Princeton and Cambridge, imagining that if we focus our thoughts on the right question we could be next great scientific icon. But this is far from what happened. Recently my partner and I were lucky enough to be shown pages from the actual notebook in which Einstein invented general relativity, while it was being prepared for publication by a group of historians working in Berlin. As working physicists it was clear to us right away what was happening: the man was confused and lost - very lost. But he was also a very good physicist (though not, of course, in the sense of the mythical saint who could perceive truth directly). In that notebook we could see a very good physicist exercising the same skills and strategies, the mastery of which made Richard Feynman such a great physicist. Einstein knew what to do when he was lost: open his notebook and attempt some calculation that might shed some light on the problem.
So we turned the pages with anticipation. But still he gets nowhere. What does a good physicist do then? He talks with his friends. All of a sudden a name is scrawled on the page: ‘Grossmann!!!’ It seems that his friend has told Einstein about something called the curvature tensor. This is the mathematical structure that Einstein had been seeking, and is now understood to be the key to relativity theory.
Actually I was rather pleased to see that Einstein had not been able to invent the curvature tensor on his own. Some of the books from which I had learned relativity had seemed to imply that any competent student should be able to derive the curvature tensor given the principles Einstein was working with. At the time I had had my doubts, and it was reassuring to see that the only person who had ever actually faced the problem without being able to look up the answer had not been able to solve it. Einstein had to ask a friend who knew the right mathematics.
The textbooks go on to say that once one understands the curvature tensor, one is very close to Einstein’s theory of gravity. The questions Einstein is asking should lead him to invent the theory in half a page. There are only two steps to take, and one can see from this notebook that Einstein has all the ingredients. But could he do it? Apparently not. He starts out promisingly, then he makes a mistake. To explain why his mistake is not a mistake he invents a very clever argument. With falling hearts, we, reading his notebook, recognize his argument as one that was held up to us as an example of how not to think about the problem. As good students of the subject we know that the argument being used by Einstein is not only wrong but absurd, but no one told us it was Einstein himself who invented it. By the end of the notebook he has convinced himself of the truth of a theory that we, with more experience of this kind of stuff than he or anyone could have had at the time, can see is not even mathematically consistent. Still, he convinced himself and several others of its promise, and for the next two years they pursued this wrong theory. Actually the right equation was written down, almost accidentally, on one page of the notebook we looked at it. But Einstein failed to recognize it for what it was, and only after following a false trail for two years did he find his way back to it. When he did, it was questions his good friends asked him that finally made him see where he had gone wrong.
Nothing in this notebook leads us to doubt Einstein’s greatness - quite the contrary, for in this notebook we can see the trail followed by a great human being whose courage and judgement are strong enough to pull him through a thicket of confusion from which few others could have emerged. Rather, the lesson is that trying to invent new laws of physics is hard. Really hard. No one knew better than Einstein that it requires not only intelligence and hard work, but equal helpings of insight, stubbornness, patience and character. This is why all scientists work in communities. And that makes the history of science a human story. There can be no triumph without an equal amount of foolishness. When the problem is as hard as the invention of quantum gravity, we must respect the efforts of others even when we disagree with them. Whether we travel in small groups of friends or in large convoys of hundreds of experts, we are all equally prone to error.
Another moral has to do with why Einstein made so many mistakes in his struggle to invent general relativity. The lesson he had such trouble learning was that space and time have no absolute meaning and are nothing but systems of relations. How Einstein himself learned this lesson, and by doing so invented a theory which more than any other realizes the idea that space and time are relational, is a beautiful story. But it is not my place here to tell it - that must be left to historians who will tell it right.
The subject of this chapter is string theory, and I begin it with these reflections for two reasons. First, because the main thing that is wrong with string theory, as presently formulated, is that it does not respect the fundamental lesson of general relativity that spacetime is nothing but an evolving system of relationships. Using the terminology I introduced in earlier chapt
ers, string theory is background dependent, while general relativity is background independent. At the same time, string theory is unlikely to be in its final form. Even if, as is quite possible, string theory is ultimately reformulated in a background independent form, history may record that Einstein’s view of Newton applies also to the string theorists: when it was necessary to ignore fundamental principle in order to make progress, they had the courage and the judgement to do so.
The story of string theory is not easy to tell, because even now we do not really know what string theory is. We know a great deal about it, enough to know that it is something really marvellous. We know much about how to carry out certain kinds of calculations in string theory. Those calculations suggest that, at the very least, string theory may be part of the ultimate quantum theory of gravity. But we do not have a good definition of it, nor do we know what its fundamental principles are. (It used to be said that string theory was part of twenty-first-century mathematics that had fallen by luck into our hands in the twentieth century. This does not sound quite as good now as it used to.) The problem is that we do not yet have string theory expressed in any form that could be that of a fundamental theory. What we have on paper cannot be considered to be the theory itself. What we have is no more than a long list of examples of solutions to the theory; what we do not yet have is the theory they are solutions of. It is as if we had a long list of solutions to the Einstein equations, without knowing the basic principles of general relativity or having any way to write down the actual equation that defines the theory.
Or, to take a simpler example, string theory in its present form most likely has the same relationship to its ultimate form as Kepler’s astronomy had to Newton’s physics. Johannes Kepler discovered that the planets travel along elliptical orbits, and he was able to use this principle together with two other rules he discovered to write down an infinite number of possible orbits. But it took Newton to discover the reason why the planetary orbits are ellipses. This allowed him to unify the explanation of the motions of the planets with many other observed motions, such as the parabolic trajectories that Galileo had discovered are followed by projectiles on the Earth. Many more examples of solutions to string theory have recently been discovered, and the virtuosity required to construct these solutions in the absence of a fundamental principle is truly humbling. This has made it possible to learn a lot about the theory, but so far, at least, it does not suffice to tell us what the theory is. No one has yet had that vital insight that will make it possible to jump from the list of solutions to the principles of the theory.