Three Roads to Quantum Gravity (18 page)

For me, as for many people working in this area, the turning point came with the revival of string theory as a possible quantum theory of gravity. I shall come to string theory in the next chapter. For now I shall say only that, having experienced the invention and failure of a whole series of wrong approaches to quantum gravity, I, along with many other physicists, was quite optimistic about what string theory could do for us. At the same time, I was also completely convinced that no theory could succeed if it was based on things moving in a fixed background spacetime. And however successful string theory was at solving certain problems, it was still very much a theory of this kind. It differed from a conventional theory only inasmuch as the objects moving in the background were strings rather than particles or fields. So it was clear right away, to me and to a few others, that while
string theory might be an important step towards a quantum theory of gravity, it could not be the complete theory. But nevertheless, as it did for many other physicists, string theory changed the direction of my research. I began to look for a way to make a background independent theory which would reduce to string theory as an approximation useful in situations in which spacetime could be regarded as a fixed background.

To get inspiration for this project I recalled the seminar given by Polyakov that had so excited me as a beginning graduate student. I wondered whether I could use the method he had used, which was to try to express QCD in terms in which the fundamental objects were loops of colour-electric flux. I needed a theory in which there was no lattice to get in the way, and he had worked without a lattice. I worked for about a year on this idea, with Louis Crane. I was then a postdoc at the University of Chicago, and Louis Crane was a graduate student. He was older than me, but he had actually been a child prodigy, perhaps the last of a distinguished series of scientists and scholars that the University of Chicago admitted to college in their early teens. He had suffered the misfortune of being expelled from graduate school for leading a strike against the invasion of Cambodia, and it had taken him ten years to find his way back to graduate school. Louis has since become one of a handful of mathematicians who has made significant creative contributions to the development of our ideas about quantum gravity. Some of his contributions have been absolutely seminal for developments in the field. I was very fortunate to become his friend at that time, and count myself lucky to be his friend still.

Louis and I worked on two projects. In the first we tried to formulate a gravitational theory based on the dynamics of interacting loops of quantized electric flux. We failed to formulate a string theory, and as a result we published none of this work, but it was to have very important consequences. In the second project we showed that a theory in which spacetime was discrete on small scales could solve many of the problems of quantum gravity. We did this by studying the implications of the hypothesis that the structure of spacetime
was like a fractal at Planck scales. This overcame many of the difficulties of quantum gravity, by eliminating the infinities and making the theory finite. We realized during that work that one way of making such a fractal spacetime is to build it up from a network of interacting loops. Both collaborations with Louis Crane persuaded me that we should try to construct a theory of spacetime based on relationships among an evolving network of loops. The problem was, how should we go about this?

This was how things stood when a discovery was made that completely changed how we understand Einstein’s theory of general relativity.

D
uring the year I was working with Louis, a young postdoc named Amitaba Sen published two papers which excited and mystified many people. We read them with a great deal of interest, for what Sen was doing was attempting to make a quantum theory from supergravity. Embedded in the papers were several remarkable formulas in which Einstein’s theory of gravity was expressed in a much simpler and more beautiful set of equations than Einstein had used. Several of us spent many hours discussing what would happen if we could somehow find a way to base quantum gravity on this much simpler formulation. But none of us did anything at the time.

The one person who took Sen’s equations seriously was Abhay Ashtekar. Abhay had been trained as a classical relativist, and early in his career had done important work in that area, but more recently he had set his sights on a quantum theory of gravity. Being mathematically inclined, Ashtekar saw that Sen’s equations contained the core of a complete reformulation of Einstein’s general theory of relativity, and by a year later he had done just that: fashioned a new formulation of general relativity. This did two things: it vastly simplified the mathematics of the theory, and expressed it in a mathematical language which was very close to that used in QCD. This was exactly what was needed to transform quantum gravity into a real subject, one in which it would in time become possible to do calculations that
yielded definite predictions about the structure of space and time on the Planck scale.

I invited Abhay to give a talk about this at Yale, where I had just become an assistant professor. At the talk was a graduate student named Paul Renteln, from Harvard, who had also been studying Sen’s papers. It was clear to us that Ashtekar’s formulation would be the key to further progress. Afterwards, I drove Abhay to the airport in Hartford. On the hour’s drive between New Haven and Hartford my car had not one but two flat tyres - and Abhay still just caught his flight. He had to hitch a ride for the last few miles, while I waited on the side of the road for help.

When I finally got home I sat down immediately and began to apply to the new formalism of Sen and Ashtekar the methods Louis Crane and I had developed during our unsuccessful attempts to re-invent string theory. A few weeks later there began a semester-long workshop in quantum gravity at the Institute for Theoretical Physics in Santa Barbara. By another piece of luck, I had convinced the authorities at Yale to let me spend a semester there, just after they had hired me. As soon as I got there I recruited two friends, Ted Jacobson and Paul Renteln. We found right away that a very simple picture of the quantum structure of space emerged if we used something very like the electric superconductor picture for the flux lines of the gravitational field. At first I worked with Paul. Fearful of the infinities that come with continuous space, we used a lattice, much like Wilson’s lattice. We found that the new form of the Einstein equations implied very simple rules for how the loops interact on the lattice. But we ran into the same problem as I had ten years before: how to get rid of the background imposed by the use of a fixed lattice.

Ted Jacobson suggested that we try to follow Polyakov and work without a lattice. I have already described the result, in Chapter 3. The next day we were standing in front of a blackboard, staring at something which it had never occurred to us, nor anyone else, to even look for. These were exact solutions to the full equations of the quantum theory of gravity.

What we had done was to apply the usual methods for constructing a quantum theory to the simple form of the equations for general relativity that Sen and Ashtekar had discovered. These led to the equations for the quantum theory of gravity. These equations had first been written down in the 1960s, by Bryce DeWitt and John Wheeler, but we found new forms for them which were dramatically simpler. We had to plug into these equations formulas that described possible quantum states of the geometry of space and time. On an impulse I tried something that Louis Crane and I had played with, which was to build these states directly from the expressions Polyakov used to describe the quantized loops of electric fields. What we found was that, as long as the loops did not intersect, they satisfied the equations. They looked like the loops in
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.