by Lee Smolin
The idea of duality is still a major driving force behind research in elementary particle physics and string theory. Duality is the very simple view that there are two ways of looking at the same thing - either in terms of strings or in terms of fields. But so far no one has been able to show that duality is applicable in ordinary QCD. It has been shown to be valid in very specialized theories which depend on very specific simplifying assumptions. Either the dimensionality of space is reduced from three to one, or a great deal of additional symmetry is added, which leads to a theory that can be understood much more easily. But even if it has not yet solved the problem that inspired its invention, duality has turned out to be a central concept in quantum gravity. How this happened is a very typical tale of how good scientific ideas can spread far from their point of origin, for I rather doubt that either Wilson or Polyakov originally considered how their idea might be applied to a quantum theory of gravity.
Like many good ideas, this one needed several goes to get it right. Inspired by what I had heard from Wilson and Polyakov, and further lessons on lattice theories I got from Gerard ’t Hooft, Michael Peskin and Stephen Shenker during my first year of graduate school, I set out to formulate quantum gravity in terms of Wilson’s lattice theory. Using some ideas borrowed from several people, I was able to concoct such a theory, which enabled me to spend a year or so learning the various techniques developed by Polyakov, Wilson and others by applying them to my version of quantum gravity. I wrote up and sent out a long paper about it and waited for a reaction. As was common in those days, the only response was a stack of postcards from far-away places requesting copies of the paper. There was of course the inevitable request from the U.S. Army research lab, which reminded us that someone somewhere was being paid to think about the possible military applications of whatever young graduate students were up to. It is strange to recall those days, not so long ago, when we typed our papers on IBM Selectric typewriters, got a professional in the basement to draw the illustrations, and then stuffed the copies individually into envelopes and mailed them out. These days we write our papers on laptops and upload them to electronic archives from where they are immediately available on the Internet. I doubt many of our current students have seen either an IBM Selectric typewriter or a postcard requesting a preprint. Many have never even gone to the library to read a paper printed in a journal.
A few months later I realized that the paper was basically wrong. It was a brave attempt, but fatally flawed. Still, it got me a few invitations to conferences. I don’t think Stephen Hawking was very happy when I used the occasion of his invitation to give a talk at a conference he organized to explain why making a lattice theory of gravity was not a very smart thing to do. Some people seemed to like the idea, but I did not see what else I could do - it was a bad idea, and I had the responsibility to explain why.
At another conference I left a copy of the paper in the mailbox of someone called Ashok Das, who had told me he was interested in doing something similar. Bryce DeWitt, who is justly thought of as the father of serious research in quantum gravity, looked for his mail in the same box and assumed that my paper was intended for him. I’m sure he saw all its shortcomings, but he was still kind enough to ask me to join him as a postdoc. I owe my career to Bryce’s mistake. At that time I was being told that I had committed professional suicide by working on quantum gravity and that I was unlikely to get any job at all.
What was wrong with my first paper was that Wilson’s lattice was an absolute, fixed structure and thus clashed with the relational nature of Einstein’s theory of gravity. So my theory did not contain gravity and had nothing at all to do with relativity. To fix this, the lattice itself would have to become a dynamical structure which could evolve in time. The key lesson I learned from this failed attempt was that one cannot fashion a successful quantum theory of gravity out of objects moving against a fixed background.
At about this time I met Julian Barbour, a physicist and philosopher who lives in a little village near Oxford. Julian had left the academic world after his Ph.D. in order to have the freedom to think deeply about the nature of space and time. He supported himself by translating Russian scientific journals into English and, away from the usual pressures of academic life, he used his considerable linguistic skills to read deeply into the history of our understanding of space and time. He had understood from his study the importance of the idea that space and time are relational, and he had then applied this wisdom to modern physics. He was I believe the first person to gain a deep understanding of the role this idea plays in the mathematical structure of Einstein’s theory of relativity. In a series of papers, first alone and then with an Italian friend, Bruno Bertotti, he showed how to formulate mathematically a theory in which space and time were nothing but aspects of relationships. Had Leibniz or anyone else done this before the twentieth century, it would have changed the course of science.
As it happened, general relativity already existed, but - and this is a strange thing to say - it was widely misunderstood, even by many of the physicists who specialized in its study. Unfortunately, general relativity was commonly regarded as a machine that produces spacetime geometries, which are then to be treated as Newton treated his absolute space and time: as fixed and absolute entities within which things move. The question then to be answered was which of these absolute spacetimes describes the universe. The only difference between this and Newton’s absolute space and time is that there is no choice in Newton’s theory, while general relativity offers a selection of possible spacetimes. This is how the theory is presented in some textbooks, and there are even some philosophers, who should know better, who seem to interpret it this way. Julian Barbour’s important contribution was to show convincingly that this was not at all the right way to understand the theory. Instead, the theory has to be understood as describing a dynamically evolving network of relationships.
Julian was of course not the only person to learn to see general relativity in this way. John Stachel also came to this understanding, at least partly through his work as the first head of the project to prepare Einstein’s collected papers for publication. But Julian came to the study of general relativity equipped with a tool that no one else had - the general mathematical formulation of a theory in which space and time are nothing but dynamically evolving relationships. Julian was then able to show how Einstein’s theory of general relativity could be understood as an example of just such a theory. This demonstration laid bare the relational nature of the description of space and time in general relativity.
Since then, Julian Barbour has become known to most people working in relativity, and recently he has become even more widely known and appreciated as a result of the publication of his radical theories on the nature of time. But in the early 1980s few people knew of his work, and I was very fortunate to meet him shortly after I had realized that my lattice gravity theory was in trouble. During this meeting he explained to me the meaning of space and time in general relativity, and the role of the relational concept in it. This gave me the conceptual language to understand why my calculations were showing that gravity was nowhere to be found in the theory I had constructed. What I needed to do was invent something like Wilson’s lattice theory, but in which there was no fixed lattice, so that all the structures were dynamical and relational. A set of points connected by edges - in other words a graph - is a good example of a system defined by relationships. But what I had done wrong was to base the theory on a fixed graph. Instead, the theory should produce the graph, and it should not mirror any pre-existing geometry or structure. It should rather evolve according to rules as simple as those that Wilson had given for the motion of loops on his lattice. It was to be ten years before a way appeared which made this possible.
During those ten years I spent my time on a variety of unsuccessful attempts to apply techniques from particle physics to the problem. These techniques were all background dependent, in that they assumed that you could fix a singl
e classical spacetime geometry and study how quantized gravitational waves, called gravitons, move and interact on the background. We tried lots of different approaches, but they all failed. Besides this I wrote a few papers on supergravity, the new theory of gravity which had been invented by one of my advisors, Stanley Deser, and others. Those attempts also came to nothing. Then I wrote a few papers about the implications of the entropy of black holes, making various speculations about their connection with problems in the foundations of quantum mechanics. Looking at them now, it seems to me that these papers were the only interesting things I did during those years, but I have no evidence that very many people ever read them. Certainly there was no interest in, and no market for, young people applying ideas from quantum black holes to fundamental issues in quantum theory.
Looking back, I am quite puzzled about why I continued to have a career. One sure reason was because at that time very few people worked on quantum gravity, so there was little competition. I was not actually getting anywhere, but people seemed to be interested in at least that part of my work in which I tried to apply techniques from particle physics to quantum gravity, even if what I had to report was that these were not very smart things to do. No one else was getting very far either, so there was room for the kind of people who prefer trying new things to following the research programs of older people, or who thrive on stealing ideas from one field and applying them in another. I doubt very much that I would have a career in the present-day environment, which is much more competitive, and in which the jobs are controlled by older people who feel confident that they are working on the right approach to quantum gravity. This allows them - but I should really say ‘us’, for I am now one of the older people who hires postdocs - to feel justified in using the enthusiasm a young researcher shows for our own research program as a measure of that researcher’s promise.
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.
CHAPTER 10
KNOTS, LINKS AND KINKS
During 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 gravitatio
nal 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.