Three Roads to Quantum Gravity (17 page)

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.

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 continuous, 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.

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 single 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.