by Lee Smolin
This doesn’t mean that there is some other fixed geometry that characterizes space—that space is like a sphere, or a saddle, instead of a plane. The point is that the geometry can be anything at all, because it evolves in time, responding to matter and force. Rather than a law stating what the geometry is, there is a law that governs how the geometry changes—just as Newton’s laws tell us not where objects are but how they move, by specifying what effects force has on their motion.
Before Einstein, geometry was thought to be part of the laws. Einstein revealed that the geometry of space is evolving in time, according to other, deeper laws.
It is important to absorb this point completely. The geometry of space is not part of the laws of nature. There is therefore nothing in those laws that specifies what the geometry of space is. Thus, before solving the equations of Einstein’s general theory of relativity, you don’t have any idea what the geometry of space is. You find out only after you solve the equations.
This means that the laws of nature have to be expressed in a form that does not assume that space has any fixed geometry. This is the core of Einstein’s lesson. We encapsulate it in a principle we described earlier, which is background independence. The principle states that the laws of nature can be specified completely without making any prior assumption about the geometry of space. In the old picture, in which geometry was fixed, it could be thought of as part of the background, the unchanging stage on which the pageant of nature unfolds. To say that the laws of physics are background-independent means that the geometry of space is not fixed but evolves. Space and time emerge from the laws rather than providing an arena in which things happen.
Another aspect of background independence is that there is no preferred time. General relativity describes the history of the world most fundamentally in terms of events and relationships between them. The principal relationships have to do with causality; one event may be in the chain of causes leading to another event. From this point of view, space is a secondary concept. The concept of space is in fact entirely dependent on the notion of time. Given a clock, we can think of all the events that are simultaneous with the clock striking noon. These make up space.
An important aspect of the general theory of relativity is that there is no preferred way to keep time. Any sort of clock will do, as long as it shows causes preceding effects. But because the definition of space depends on time, there are as many different definitions of space as there are of time. Just above, I spoke about the geometry of space evolving in time. That holds not for a single universal notion of time but for every possible notion of time. How all this works is part of the intricate beauty of Einstein’s general theory of relativity. For our purposes, it will be enough to remember that the equations of that theory tell us how the geometry of space evolves in time not just for one but for any possible definition of time.
Actually, background independence means even more than this. There are other aspects of nature that are fixed in the usual expressions of the laws of physics. But perhaps they shouldn’t be. For example, the fact that there are only three dimensions of space is part of the background. Might there be a deeper theory in which we don’t have to make any prior assumption about the number of spatial dimensions? In such a theory, the three dimensions might come out as the solution to some dynamical law. Perhaps, in such a theory, the number of spatial dimensions could even change in time. If we could invent such a theory, it might explain to us why our universe has three dimensions. This would constitute progress, for something that previously was simply assumed would finally be explained.
So the idea of background independence in its broadest terms is a piece of wisdom about how to do physics: Make better theories in which things that are now assumed are explained, by allowing such things to evolve subject to some new law. Einstein’s general theory of relativity did precisely that for the geometry of space.
The key question for a quantum theory of gravity is then the following: Can we extend to quantum theory the principle that space has no fixed geometry? That is, can we make quantum theory background-independent, at least with regard to the geometry of space? If we can do this, we will automatically merge gravity and quantum theory, because gravity is already understood to be an aspect of dynamical spacetime geometry.
There are then two approaches to merging gravity and quantum theory: those that achieve background independence and those that do not. The field of quantum gravity split along these lines all the way back in the 1930s, although most approaches studied today are background-independent. The one exception is the approach that most of today’s physicists study—string theory.
How it came to be that the highest achievement of the most famous scientist of the twentieth century has been virtually ignored by most of those clamoring to follow in his footsteps is one of the strangest stories in the history of science. But it is a story that must be told here, as it is central to the questions I raised in the Introduction. Indeed, you might wonder, given that Einstein’s general theory of relativity is so well accepted, why anyone would try to develop a new theory that did not take on board its central tenet. The answer is a story, and like many stories told in this book, it began with Einstein.
Already in 1916, Einstein realized that there were gravitational waves and that they carried energy. He noticed right away that consistency with atomic physics would require that the energy carried by gravitational waves be described in terms of quantum theory. In the very first paper ever written on gravitational waves, Einstein said that “it appears that the quantum theory would have to modify not only Maxwell’s theory of electrodynamics but also the new theory of gravitation.”1
Nevertheless, whereas Einstein was the first to state the problem of quantum gravity, his deepest insight has been ignored by most of those who have since worked on it. How could this be?
There is a reason, and it is that no one knew at the time how to go about directly applying the then developing quantum theory to general relativity. Instead, progress turned out to be possible by an indirect route. Those who wanted to apply quantum mechanics to general relativity faced two challenges. Besides background independence, they had to grapple with the fact that general relativity is a field theory. There are an infinite number of possibilities for the geometry of space and hence an infinite number of variables.
As I described in chapter 4, as soon as quantum mechanics was completely formulated, physicists began to apply it to field theories, such as the electromagnetic field. These are formulated in a fixed-spacetime background, so the issue of background independence does not arise. But they gave physicists experience with handling the problem of an infinite number of variables.
The first big success of quantum field theory was QED, the unification of Maxwell’s theory of electromagnetism with quantum theory. It is remarkable that in their first paper on QED, in 1929, Werner Heisenberg and Wolfgang Pauli, two of the founders of quantum mechanics, were already contemplating extending their work to quantum gravity. They apparently felt it would not be too hard, because they write that the “quantization of the gravitational field, which appears to be necessary for physical reasons, may be carried out without any new difficulties by means of a formalism fully analogous to that applied here.”2
More than seventy-five years later, we can only marvel at the extent to which two such brilliant people underestimated the difficulty of the problem. What could they have been thinking? Well, I know, because many people have since had the same thought, and the dead end it leads to has been thoroughly explored.
What Heisenberg and Pauli were thinking was that when gravitational waves are very weak, they can be seen as tiny ripples disturbing a fixed geometry. If you drop a stone into a pond on a still morning, it causes tiny ripples that barely disturb the flat surface of the water, so it is easy to think that the ripples move on a fixed background given by that surface. But when water waves are strong and turbulent, as near a beach on a stormy day, it makes no sense to see th
em as disturbances of something fixed.
General relativity predicts that there are regions of the universe where the geometry of spacetime evolves turbulently, like waves crashing on a beach. But Heisenberg and Pauli thought it would be simpler to first study cases in which the gravitational waves are extremely weak and can be seen as tiny ripples on a fixed background. This allowed them to apply the same methods they had developed to study quantum electromagnetic fields moving on a fixed background of spacetime. And in fact it was not difficult to apply quantum mechanics to very weak gravitational waves moving freely. The result was that each gravitational wave could be seen quantum mechanically, as a particle called the graviton—analogous to the photon, which is the quantum of the electromagnetic field. But at the next step, they faced a big problem, because gravitational waves interact with each other. They interact with anything that has energy, and they themselves carry energy. This problem does not occur with electromagnetic waves, because though photons interact with electric and magnetic charges, they are not themselves charged, so they go right through one another. This important difference between the two kinds of waves is what Heisenberg and Pauli missed.
Describing the self-interaction of gravitons consistently turned out to be a tough nut to crack. We now understand that the failure to solve this problem is a consequence of not taking Einstein’s principle of background independence seriously. Once the gravitational waves interact with one another, they can no longer be seen as moving on a fixed background. They change the background as they travel.
A few people already understood this in the 1930s. Probably the first PhD thesis ever written on the problem of quantum gravity was the 1935 dissertation of the Russian physicist Matvei Petrovich Bronstein. Those who recall him think of him as one of the two most brilliant Soviet physicists of his generation. He wrote in a 1936 paper that “the elimination of the logical inconsistencies [requires] rejection of our ordinary concepts of space and time, modifying them by some much deeper and nonevident concepts.” Then he quoted a German proverb, “Let him who doubt it pay a Thaler.”3 Bronstein’s view was championed by a brilliant young French physicist, Jacques Solomon.
By now almost everyone who thinks seriously about quantum gravity agrees with Bronstein, but it has taken seventy years. One reason is that even such brilliant minds as Bronstein and Solomon could not escape the insanity of their time. A year after Bronstein wrote the paper I just quoted, he was arrested by the NKVD, and he was executed by a firing squad on February 18, 1938. Solomon became a member of the French Resistance and was killed by the Germans on May 23, 1942. Their ideas were lost to history. I have worked on the problem of quantum gravity all my life and I learned of them only while finishing this book.
The work of Bronstein was forgotten, and most physicists returned to the study of quantum field theory. As I described in chapter 4, it took until the late 1940s for QED to be developed. This success then inspired a few people to take up again the challenge of unifying gravity with quantum theory. Right away, two opposing camps sprang up. One of them followed Bronstein in taking the background independence of general relativity seriously. The other ignored background independence and followed Heisenberg and Pauli’s route in their efforts to apply quantum theory to gravitational waves seen as moving on a fixed background.
Since background independence is one of the principles of general relativity, it would seem sensible to incorporate it into attempts to unify that theory with quantum theory. But as it turned out, things were not so simple. A few people—like the British physicist P.A.M. Dirac, and Peter Bergmann, a German who had begun his career as an assistant to Einstein in Princeton—did attempt to construct a background-independent theory of quantum gravity. They found it an arduous task. Such attempts did not bear fruit until the mid 1980s, but since then there has been a lot of progress in understanding quantum gravity from a background-independent point of view. Most quantum-gravity theorists now work on one of several background-independent approaches. We’ll return to these later in the book, for they constitute the most important alternatives to string theory.
But none of these promising signs were apparent when people started along the quantum-gravity road in the 1950s. The limited progress made with background-independent methods looked puny, compared with the great strides that were being made in QED. So until the late 1980s, most people took the other route, which was to attempt to apply the methods of QED to general relativity. This was perhaps understandable. After formulating QED, people knew a lot about background-dependent quantum theories, but no one knew anything about what a background-independent quantum theory might look like, if it existed at all.
Since this was the route that led to string theory, it is worth retracing. Because the work from the 1930s had been forgotten, it had to be rediscovered. The theory of gravitons was worked out again in a PhD thesis by Bryce DeWitt, who was a student of Julian Schwinger’s at Harvard in the late 1940s. For this and his many discoveries that followed, we regard DeWitt as one of the founders of the theory of quantum gravity.
But, as noted, a graviton theory was not enough. The graviton theory was fine as long as the gravitons just moved through space, but if that’s all they did, there was no gravity, and certainly no dynamical or curved geometry. So this was not a unification of general relativity or gravity with quantum theory, it was just a unification of weak gravitational waves with quantum theory. The problems with the theory of gravitons reemerged in the early 1950s, as soon as people began again to study how they might interact with one another. From then until the early 1980s, a lot of work was expended on this self-interaction problem to keep it from contradicting the principles of quantum theory. None of this work succeeded.
It might be useful to stop and think about what this means in human terms. We are talking about thirty years of continual hard work, involving many complicated calculations. Imagine doing your income tax every day, all day, for a week, and still not getting the calculations to add up consistently. You have an error somewhere, but you can’t find it. Now imagine a month spent like that. Can you stretch it to a year? Now imagine twenty years. Now imagine that there are a couple of dozen people around the world spending their time like this. Some are friends, some rivals. They all have their own schemes of how to make it work. Each scheme has so far failed, but if you were to try a slightly different approach, or combine two approaches, perhaps you might succeed. Once or twice a year, you go to an international conference, where you can present your new scheme to the other fanatics. This was the field of quantum gravity before 1984.
Richard Feynman was one of the first to attack this graviton problem. And why not? He had done such good work on QED, why shouldn’t he apply the same methods to quantum gravity? So in the early 1960s he took a few months off from particle physics to see if he could quantize gravity. To give you a sense of what a backwater quantum gravity was back then, here is a letter Feynman wrote to his wife in 1962 about a meeting in Warsaw where he was presenting his work:
I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments, this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here . . . and it is not good for my blood pressure. Remind me not to come to any more gravity conferences!4
Nevertheless, he made good progress and greatly clarified a technical issue having to do with probabilities, which are numbers between 0 and 1. Anything that is certain to happen is said to have probability 1, so the probability that anything at all happens is 1. Before Feynman did his work, no one could make the probabilities for various things to happen in quantum gravity add up to 1. Actually, Feynman made the probabilities add up only in the first level of approximation; a few years later, Bryce DeWitt figured out how to make it work at all levels. A year or so later, the same thing was figured out by two Russians, Ludwig Dmitrievich Faddeev and Victor Nicolaevich Popov. They couldn’t have known of DeWitt’s work, becaus
e the journal had sent his paper to an expert to review and the reviewer had taken more than a year to go over it. So, bit by bit, people solved some problems—but even if the probabilities could be made to add up to 1, the graviton theory as a whole never worked.
There were some side benefits of this work. The same method could be applied to the Yang-Mills theories that the standard model came to be based on. So by the time Steven Weinberg and Abdus Salam used those theories to unify the weak and electromagnetic interactions, the technology was in place to do real calculations. The results turned out better than in quantum gravity. As the Dutch theorist Gerard ’t Hooft finally proved in 1971, the Yang-Mills theories were completely sensible as quantum theories. Indeed, like others before him, ’t Hooft was studying Yang-Mills theory partly as a warm-up to an attack on the problem of quantum gravity. So the thirty years of work on quantum gravity was not a completely wasted effort; at least it enabled us to do particle physics sensibly.
But there was no saving quantum gravity. People tried all sorts of approximation methods. Since the standard model of particle physics made sense, many methods were developed to probe different features of it. One by one, each of these was tried on the problem of quantum gravity. Each failed. No matter how you organized the quantum theory of gravitational waves, as soon as you put in the fact that they interacted with one another, infinite quantities raised their head. No matter how you turned the problem around, the infinities could not be tamed. More years of work, more papers, more PhD theses, more presentations at conferences. Same situation. The bottom line is that by 1974 it was clear that a background-dependent approach to combining general relativity with quantum theory did not make sense.
There was, however, one thing that could be done with background-dependent methods. Rather than trying to quantize gravity and so understand the effect that quantum theory has on gravitational waves, we could turn the problem around and ask what effects gravity might have on quantum phenomena. To do that, we could study quantum particles moving in spacetimes where gravity is important, such as black holes or an expanding universe. Beginning in the 1960s, a lot of progress was made in this direction. It is an important direction, because some of the discoveries led to puzzles that later approaches, such as string theory, aimed to solve.