by Lee Smolin
FIGURE 23
Quantum states of the geometry of space are expressed in loop quantum gravity in terms of loops. These states are exact solutions to the equations of quantum gravity, as long as there are no intersections or kinks in the loops.
It took us a few days of hard work to find still more solutions. We found that even if the loops intersected, we could still combine them to make solutions provided certain simple rules were obeyed. In fact, we could write down an infinite number of these states - all we had to do was draw loops and apply some simple rules whenever they intersected.
It took many years for us and others to work out the implications of what we had found in those few days. But even at the start we knew that we had in our grasp a quantum theory of gravity that could do what no theory before it had done - it gave us an exact description of the physics of the Planck scale in which space is constructed from nothing but the relationships among a set of discrete elementary objects. These objects were still Wilson’s and Polyakov’s loops, but they no longer lived on a lattice, or even in space. Instead, their interrelations defined space.
There was one step to go to complete the picture. We had to prove that our solutions really were independent of the background space. This required us to show that they solved an additional set of equations, known as diffeomorphism constraints, which expressed the independence of the theory from the background. These were supposed to be the easy equations of the theory. Paradoxically, the equations we had solved so easily, the so-called Wheeler-DeWitt equations, were supposed to be the hard ones. At first I was very optimistic, but it turned out to be impossible to invent quantum states that solved both sets of equations. It was easy to solve one or the other, but not both.
Back at Yale the next year, we spent many fruitless hours with Louis Crane trying to do this. We pretty much convinced ourselves it was impossible. This was very frustrating because it was easy to see what the result would be: if we could only get rid of the background, we would have a theory of nothing but loops and their topological relationships. It would not matter where in space the loops were, because the points in space would have no intrinsic meaning. What would matter would be how the loops intersected one another. It would also matter how they knotted and linked.
I realized this one day while I was sitting in my garden in Santa Barbara. Quantum gravity would be reduced to a theory of the intersecting, knotting and linking of loops. These would give us a description of quantum geometry on the Planck scale. From the work I had done with Paul and Ted, I also knew that the quantum versions of the Einstein equations we had invented could change the way the loops linked and knotted with one another. So the relationships among the loops could change dynamically. I had thought about intersecting loops, but I had never wondered about how loops could knot or link.
I went inside and called Louis Crane. I asked him whether mathematicians knew anything about how loops might knot and link. He said, yes, there is a whole field devoted to the subject, called knot theory. He reminded me that I had had dinner a few times in Chicago with one of its leading thinkers, Louis Kauffman. So the last step was to rid the theory of any dependence on where the loops were in space. This would reduce our theory to the study of knots, links and kinks, as James Hartle, one of the leading American relativists, teasingly began to call it shortly afterwards. But this was not so easy, and we were not able to take this step for over a year. We tried very hard, with Louis and others, but we could not do it.
The workshop at Santa Barbara had closed with a conference at which our new results were first presented. There I had met a young Italian scientist, Carlo Rovelli, who had just been awarded his Ph.D. We didn’t talk much, but shortly after he wrote to ask if he could come to visit us at Yale. He arrived that October, and took a room in Louis Crane’s apartment. The first day he was there I explained to him that there was nothing to do, because we were completely stuck. The work had looked promising, but Louis and I had found the last step to be impossible. I told Carlo he was welcome to stay, but perhaps given the sad state of the subject he might prefer to go back to Italy. There was an awkward moment. Then, looking for something to talk about, I asked him if he liked to sail. He replied that he was an avid sailor, so we abandoned science for the day and went straight to the harbour where the Yale sailing team kept its boats, and took out a sailing dinghy. We spent the rest of the afternoon talking about our girlfriends.
I didn’t see Carlo the next day. The day after that he appeared at the door to my office and said, ‘I’ve found the answer to all the problems.’ His idea was to make one more reformulation of the theory, so that the basic variables were nothing but the loops. The problem was that the theory up till then depended both on the loops and on the field flowing around the loop. Carlo saw that it was the dependence on the field that was making it impossible to proceed. He also saw how to get rid of it, by using an approach to quantum theory invented by his mentor, Chris Isham, at Imperial College. Carlo had found that applying it to the loops gave exactly what we needed. It took us no more than a day to sketch the whole picture. In the end we had a theory of the kind that Polyakov had spoken about as his great dream: a theory of pure loops which described an aspect of the real world in equations so simple they could be solved exactly. And when it was used to construct the quantum version of Einstein’s theory of gravity, the theory depended only on the relationships of the loops to one another - on how they knot, link and kink. Within days we had shown that one can construct an infinite number of solutions to all the equations of quantum gravity. For example, there is one solution for every possible way to tie a knot.
A few weeks later we went to Syracuse University, which by then was the centre of work stemming from Ashtekar’s and Sen’s discoveries, and Carlo gave the first seminar on the new theory of quantum gravity. On the way to the airport we were rear-ended by a guy in a very flashy car. No one was hurt, and the rear bumper of my old Dodge Dart was barely scratched, but his Maserati was wrecked. Still, we made it. The next day Carlo had a high fever, but he got through the seminar, and at the end there was a long, appreciative silence. Abhay Ashtekar said it was the first time he had seen something that might be the quantum theory of gravity. A few weeks after that I gave the second seminar on the new theory, in London, in front of Chris Isham, on my way to a conference in India.
In India, two ancient cultures met when I introduced the conference organizer to Carlo, who had decided impulsively to jump on a plane and come, though he had no invitation. The distinguished gentleman looked at his long hair and the sandals and clothes he had picked up while wandering alone for two days through the back streets of Bombay, and sputtered, ‘Mr Rovelli, but didn’t you get my letter saying the meeting is closed?’ Carlo smiled and replied, ‘No, but didn’t you get mine?’ He was given the best room in the hotel, and Air India put him in first class on his flight home to Rome.
Thus was born what is now called loop quantum gravity. It took several years of work, first with Carlo and then as part of a growing community of friends and colleagues, to unravel the meaning of the solutions to the quantum gravity equations we had found. One straightforward consequence is that quantum geometry is indeed discrete. Everything we had done had been based on the idea of a discrete line of force, as in a magnetic field in a superconductor. Translated into the loop picture of the gravitational field, this turns out to imply that the area of any surface comes in discrete multiples of simple units. The smallest of these units is about the Planck area, which is the square of the Planck length. This means that all surfaces are discrete, made of parts each of which carries a finite amount of area. The same is true of volume.
To arrive at these results we had to find a way to eliminate the infinities that plague all expressions in quantum theories of fields. I had an intuition, stemming from my past conversations with Julian Barbour, and the work I had done with Louis Crane, that the theory should have no infinities. Many physicists have speculated that the infinities come
from some mistaken assumption about the structure of space and time on the Planck scale. From the older work it was clear to me that the wrong assumption was the idea that the geometry of spacetime was fixed and non-dynamical. When calculating the measures of geometry, such as area and volume, one had to do it in just the right way to eliminate any possible contamination from non-dynamical, fixed structures. Exactly how to do this was a technical exercise that cannot be explained here. But in the end it did turn out that as long as one asks a physically meaningful question, there will be no infinities.
In my experience it really is true that as a scientist one has only a few good ideas. They are few and far between, and come only after many years of preparation. What is worse, having had a good idea one is condemned to years of hard work developing it. The idea that area and volume would be discrete had come to me in a flash as I was trying to calculate the volume of some quantum geometry, while I was sitting for an hour in a noisy room in a garage waiting for my car to be fixed. The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units. These units correspond to the smallest areas and volumes that can exist. I showed these calculations to Carlo, and a few months later, during a period we spent working together at the University of Trento, in the mountains of north-east Italy, he invented an argument that showed that the basic unit of area could not be taken to zero. This meant there was no way to avoid the conclusion that if our theory were true, space had an ‘atomic’ structure.
I well remember our work in Trento for another reason. In the previous year one of our students, Bernd Bruegmann, had come to my office with a very disturbed look on his face. His thesis problem was to apply the new methods from loop quantum gravity to QCD on a lattice, and see whether the properties of protons and neutrons would emerge. While doing so he did what good scientists should do, but which we had not, which was to check the literature thoroughly. He had found a paper in which methods very similar to ours had already been applied to QCD by two people we had never heard of, Rodolfo Gambini and Anthony Trias, who were working in Montevideo and Barcelona.
Scientists are human, and we all suffer from the need to feel that what we do is important. Pretty much the worst thing that can happen to a scientist is to find that someone has made the same discovery before you. The only thing worse is when someone publishes the same discovery you made, after you’ve published it yourself, and does not give you adequate credit. It was true that we had discovered the method of working with loops in the realm of quantum gravity rather than in QCD, but there was no avoiding the fact that the method we had developed was quite close to the one that Gambini and Trias had already been using for several years in their work on QCD. Even though they had been publishing in the Physical Review, which is a major journal, we had somehow missed seeing their work.
With a heavy heart we did the only thing we could, which was to sit down and write them a very apologetic letter. We heard nothing from them until one afternoon in Trento, when Carlo got a phone call from Barcelona. Our letter had finally reached them. They had tracked us down to Trento, and asked if we would still be there tomorrow. The next morning they arrived, having driven most of the night across France and northern Italy. We spent a wonderful day showing each other our work, which was thankfully complementary. They had applied the method to QCD, while we had applied it to quantum gravity. Anthony Trias did most of the talking, while Rodolfo Gambini sat at the back of the room and at first hardly said anything. But we soon found that Rodolfo was a creative scientist of the first order. Just how creative we learned over the next few months, as he quickly invented a new approach to doing calculations in loop quantum gravity.
Since then Gambini has been one of the leaders in the field of quantum gravity, often working in collaboration with Jorge Pullin at Penn State University and a very good group of young people he trained in Montevideo. They have discovered many more solutions to the equations of quantum gravity, and resolved several important problems that came up along the way.
It also must be said that, despite his quiet nature, Rodolfo Gambini has been more or less single-handedly responsible for reviving physics in both Venezuela and Uruguay after its total destruction by the military dictatorships. Just what this meant was brought home to me the first time I visited Montevideo. It was the middle of winter, and we did physics with Rodolfo and his group in a run-down old convent, without heat or computers, fighting off the cold by drinking a continuous supply of matte (a kind of tea) that was kept hot over a Bunsen burner. Now the science departments at the University of Uruguay are housed in modern buildings and facilities, built with funds that Rodolfo raised in his spare time, while keeping up a continuous flow of new ideas and calculations.
One of the most beautiful results to have come from loop quantum gravity was the discovery that the loop states could be arranged in very beautiful pictures, which are called spin networks. These had actually been invented thirty years earlier by Roger Penrose. Penrose had also been inspired by the idea that space must be purely relational. Going directly to the heart of the matter, as is his nature, he had skipped the step of trying to derive a picture of relational space from some existing theory, as we had. Instead, having more courage, he had sought the simplest possible relational structure that might be the basis of a quantum theory of geometry. Spin networks were what he came up with. A spin network is simply a graph, such as those shown in Figures 24 to 27, whose edges are labelled by integers. These integers come from the values that the angular momentum of a particle are allowed to have in quantum theory, which are equal to an integer times half of Planck’s constant.
FIGURE 24
A spin network, as invented by Roger Penrose, also represents a quantum state of the geometry of space. It consists of a graph, together with integers on the edges. Only a few of the numbers are shown here.
FIGURE 25
A spin network can be made by combining loops.
I had known for a long time that Penrose’s spin networks should come into loop quantum gravity, but I had been afraid of working with them. When Penrose described them in his talks they always seemed so intricate that only he would be able to work with them without making mistakes. To do a calculation Penrose’s way, one has to add up long series of numbers which are each either +1, 0 or -1. If you miss one sign, you’re dead. Still, during a visit to Cambridge in 1994 I met Roger and asked him to tell me how to calculate with his spin networks. We did one calculation together, and I thought I had the hang of it. That was enough to convince me that spin networks would make it possible to calculate aspects of quantum geometry such as the smallest possible volume. I then showed what I had learned to Carlo, and we spent the rest of that summer translating our theory into the language of Penrose’s spin networks.
When we did this we found that each spin network gives a possible quantum state for the geometry of space. The integers on each edge of a network correspond to units of area carried by that edge. Rather than carrying a certain amount of electric or magnetic flux, the lines of a spin network carry units of area. The nodes of the spin networks also have a simple meaning: they correspond to quantized units of volume. The volume contained in a simple spin network, when measured in Planck units, is basically equal to the number of nodes of the network. It took much work and heartache to clarify this picture. Penrose’s method was invaluable but, as I had expected, it was not easy to work with. Along the way we learned the truth of something I once heard Richard Feynman say, which is that a good scientist is someone who works hard enough to make every possible mistake before coming to the right answer.
FIGURE 26
The quantization of space as predicted by loop quantum gravity. The edges of spin networks carry d
iscrete units of area. The area of a surface comes from the intersection of one edge of a spin network with it. The smallest possible area comes from one intersection, and is about 10-66 of a square centimetre. The nodes of the spin networks carry discrete units of volume. The smallest possible volume comes from one node, and is about 10-99 of a cubic centimetre.
FIGURE 27
A very large spin network can represent a quantum geometry that looks smooth and continuous when viewed on a scale much larger than the Planck length. We say that the classical geometry of space is woven by making it out of a very large and complex spin network. In the spin network picture, space only seems continuous - it is actually made up of building blocks which are the nodes and edges of the spin network.
Probably my worst moment in science came at a conference in Warsaw when a young physicist named Renate Loll, who had also been a student of Chris Isham in London, announced at the end of her talk that our calculation for the smallest possible volume was wrong. After a lot of argument it turned out that she was right, and we traced our error back to a single sign mistake. But remarkably, our basic pictures and results held up. They have since been confirmed by mathematical physicists who showed that the results we found are underpinned by rigorous mathematical theorems. Their work tells us that the spin network picture of quantum geometry is not just a product of someone’s imagination - rather, it follows directly from combining the basic principles of quantum theory with those of relativity.
