Three Roads to Quantum Gravity Read online

  This is perhaps not so surprising - it seems not very different from the present state of cancer research or evolutionary theory. Because the problem is hard, it might be expected that, like climbers confronting a virgin peak, different people would attempt different approaches. Of course, some of these approaches will turn out to be total failures. But, at least in the case of quantum gravity, several approaches seem recently to have led to genuine discoveries about the nature of space and time.

  The most compelling developments, taking place as I write, have to do with bringing together the different lessons that have been learned by following the different approaches, so that they can be incorporated into a single theory - the quantum theory of gravity. Although we do not yet have this single theory in its final form, we do know a lot about it, and this is the basis of what I shall be describing in the chapters to come.

  I should warn the reader that I am by temperament a very optimistic person. My own view is that we are only a few years away from having the complete quantum theory of gravity, but I do have friends and colleagues who are more cautious. So I want to emphasize that what follows is a personal view, one that not every scientist or mathematician working on the problem of quantum gravity will endorse. I should also add that there are a few mysteries that have yet to be solved. The final stone that finishes the arch has yet to be found.

  Furthermore, I must emphasize that so far it has not been possible to test any of our new theories of quantum gravity experimentally. Until very recently it was even believed that the quantum theory of gravity could not be tested with existing technology, and that it would therefore be many years into the future before the theory could be confronted with data from experimental science. However, it now appears that this pessimism may have been short-sighted. Philosophers of science such as Paul Feyerabend have stressed that new theories often suggest new kinds of experiment which may be used to test them. This is very definitely happening in quantum gravity. Very recently, new experiments have been proposed which it appears will make it possible to test at least some of the theory’s predictions in the very near future. These new experiments will employ existing technology, but used in surprising ways, to study phenomena that would not have been thought, on the basis of the old theories, to have anything to do with quantum gravity. This is indeed a sign of real progress. However, we must never forget that until the experiments are performed it will always be possible that, as beautiful and compelling as the new theories may seem, they are simply wrong.

  During the past few years there has been a growing sense of excitement and confidence among many of the people working on quantum gravity. It is hard to avoid the feeling that we are indeed closing in on the beast. We may not have it in our net, but it feels as if we have it cornered and we have seen, with our flashlights, a few glimpses of it.

  Among the many different paths to quantum gravity, most recent traffic, and most progress, has been along three broad roads. Given that quantum gravity is supposed to arise from a unification of two theories - relativity and quantum theory - two of these paths are perhaps not unexpected. There is the route from quantum theory, in which most of the ideas and methods used were developed first in other parts of quantum theory. Then there is the road from relativity, along which one starts with the essential principles of Einstein’s theory of general relativity and seeks to modify them to include quantum phenomena. These two roads have each led to a well worked-out and partly successful theory of quantum gravity. The first road gave birth to string theory, while the second led to a seemingly different theory (although with a similar name) called loop quantum gravity.

  Both loop quantum gravity and string theory agree on some of the basics. They agree that there is a physical scale on which the nature of space and time is very different from that which we observe. This scale is extremely small, far out of the reach of experiments done with even the largest particle accelerators. It may in fact be very much smaller than we have so far probed. It is usually thought to be as much as 20 orders of magnitude (i.e. a factor of 1020) smaller than an atomic nucleus. However, we are not really sure at which point it is reached, and recently there have been some very imaginative suggestions that, if they bear fruit, will bring quantum gravity effects within the range of present-day experimental capabilities.

  The scale where quantum gravity is necessary to describe space and time is called the Planck scale. Both string theory and loop quantum gravity are theories about what space and time are like on this tiny scale. One of the stories I shall be telling is how the pictures that each theory gives us are converging. Not everyone yet agrees, but there is more and more evidence that these different approaches are different windows into the same very tiny world.

  Having said this, I should confess my own situation and bias. I was one of the first people to work on loop quantum gravity. The most exhilarating days of my life (apart from the purely personal) were those when, all of a sudden, after months of hard work, we suddenly understood one of our theory’s basic lessons. The friends I did that with are friends for life, and I feel equal affection and hope for the discoveries we made. But before then I worked on string theory and, for the past four years, most of my work has been in the very fertile domain that lies between the two theories. I believe that the essential results of both string theory and loop quantum gravity are true, and the picture of the world I shall be presenting here is one that comes from taking both seriously.

  Apart from string theory and loop quantum gravity, there has always been a third road. This has been taken by people who discarded both relativity and quantum theory as being too flawed and incomplete to be proper starting points. Instead, these people wrestle with the fundamental principles and attempt to fashion the new theory directly from them. While they make reference to the older theories, these people are not afraid to invent whole new conceptual worlds and mathematical formalisms. Thus, unlike the other two paths, which are trodden by communities of people each large enough to exhibit the full spectrum of human group behaviour, this third path is followed by just a few individuals, each pursuing his or her own vision, each either a prophet or a fool, who prefers that essential uncertainty to the comfort of travelling with a crowd of like-minded seekers.

  The journey along the third path is driven by deep, philosophical questions such as, ‘What is time?’ or, ‘How do we describe a universe in which we are participants?’ These are not easy questions, but some of the greatest minds of our time have chosen to attack them head-on, and I believe that there has been great progress along this path too. New and, in some cases, quite surprising ideas have been discovered, which I believe are up to the task of answering these questions. I believe that they provide the conceptual framework that is allowing us to take the next step - to proceed to a quantum theory of gravity.

  It has also happened that someone on this third road discovered a mathematical structure which at first seemed unconnected to anything else. Such results are often dismissed by the more conservative members of the field as having no possible connection to reality, but these critics have sometimes had to eat their words when the same structure surprisingly turns up on one of the first two roads as the answer to what seemed an otherwise intractable problem. This of course only proves that fundamental questions are hardly ever solved by accident. The people who discovered these structures are among the true heroes of this story. They include Alain Connes, David Finkelstein, Christopher Isham, Roger Penrose and Raphael Sorkin.

  In this book we shall walk down all three roads. We shall discover that they are closer than they seem - linked by paths, little used and perhaps a bit overgrown, but passable nevertheless. I shall argue that, if we put together the key ideas and discoveries from all the roads, a definite picture emerges of what the world is like on the Planck scale. My intention here is to display this picture and, by doing so, to show how close we are to the solution of the problem of quantum gravity.

  I have tried to aim this book at the intellig
ent layperson, interested in knowing what is going on at the frontiers of physics. I have not assumed any previous knowledge of relativity or quantum theory. I believe that the reader who has not read anything previously on these subjects will be able to follow this book. At the same time I have introduced ideas from relativity and quantum theory only when they are needed to explain something. I could have said much more about most of the subjects I mention, even at an introductory level. But to have included a complete introduction to these subjects would have resulted in a very long book, and this would have defeated my main goal. Fortunately, there are many good introductions to these subjects for the layperson. At the end of this book there are some suggestions for further reading for those who want to know more.

  I must also emphasize that in most cases I have not given proper credit to the inventors of the ideas and discoveries I present. The knowledge we have about quantum gravity has not come out of the head of two or three neo-Einsteins. Rather, it is the result of several decades of intense effort by a large and growing community of scientists. In most cases to name only a few people would be a disservice to both the community of scientists and to the reader, as it would reinforce the myth that science is done by a few great individuals in isolation. To come anywhere near the truth, even about a small field like quantum gravity, one has to describe the contributions of scores of people. There are many more people to name than could be kept track of by the reader encountering these ideas for the first time.

  For a few episodes with which I was involved enough to be confident of knowing what happened, I have told the stories of how the discoveries were made. Because people are most interesting when one tells the truth about them, in these cases I am happy to introduce some very human stories to illustrate how science actually gets done. Otherwise I have stayed away from telling the stories of who did what, for I would inevitably have got some of it wrong, in spite of having been a close observer of the subject for the last two decades.

  In taking the liberty of telling a few stories I also take a risk, which is that the reader will get the impression that I believe my own work to be more important than the work of other people in the field. This is not true. Of course, I do believe in the approach I pursue in my own research, otherwise I would not have a point of view worth forming a book around. But I believe that I am also in a position to make a fair appraisal of the strengths and weaknesses of all the different approaches, not only those to which I’ve contributed. Above all else, I feel very privileged to be part of the community of people working on quantum gravity. If I were a real writer, skilled in the art of conveying character, I would like nothing better than to describe some of the people in this world I most admire, from whom I continue to learn, every chance I get. But given my limited skills I shall stick to a few stories about people and incidents I know very well.

  When our task is done, someone will write a good history of the search for quantum gravity. Whether this will be in a few years, as I believe, or in many decades, as some of my more pessimistic colleagues expect, it will be a story in which the best human virtues, of courage, wisdom and vision, are mixed with the most ordinary sort of primate behaviour, expressed through the rituals of academic politics. I hope that story will be written in a style that celebrates both sides of our very human occupation.

  Each of the following chapters is devoted to one step in our search for the quantum theory of gravity. We begin with four basic principles that determine how we approach our enquiry into the nature of space, time and the cosmos. These make up the first part, called ‘Points of departure’. With this preparation we turn to the second part, ‘What we have learned’, in which I shall describe the main conclusions that have so far been arrived at on the three roads to quantum gravity. These combine to give us a picture of what the world is like on the smallest possible scales of space and time. From there we turn to the last part, a tour of ‘The present frontiers’ of the subject. We shall introduce a new principle, called the holographic principle, which may very well be the fundamental principle of quantum gravity. The next chapter is a discussion of how the different approaches to quantum gravity may be coming together into one theory which seems to have the possibility of answering, at least for the foreseeable future, our questions about the nature of space and time. I end with a reflection on the question of how the universe chose the laws of nature.

  We begin at the beginning, with the first principle.

  I

  POINTS OF DEPARTURE

  CHAPTER 1

  THERE IS NOTHING OUTSIDE THE UNIVERSE

  We humans are the species that makes things. So when we find something that appears to be beautifully and intricately structured, our almost instinctive response is to ask, ‘Who made that?’ The most important lesson to be learned if we are to prepare ourselves to approach the universe scientifically is that this is not the right question to ask. It is true that the universe is as beautiful as it is intricately structured. But it cannot have been made by anything that exists outside it, for by definition the universe is all there is, and there can be nothing outside it. And, by definition, neither can there have been anything before the universe that caused it, for if anything existed it must have been part of the universe. So the first principle of cosmology must be ‘There is nothing outside the universe’.

  This is not to exclude religion or mysticism, for there is always room for those sources of inspiration for those who seek them. But if it is knowledge we desire, if we wish to understand what the universe is and how it came to be that way, we need to seek answers to questions about the things we see when we look around us. And the answers can involve only things that exist in the universe.

  This first principle means that we take the universe to be, by definition, a closed system. It means that the explanation for anything in the universe can involve only other things that also exist in the universe. This has very important consequences, each of which will be reflected many times in the pages that follow. One of the most important is that the definition or description of any entity inside the universe can refer only to other things in the universe. If something has a position, that position can be defined only with respect to the other things in the universe. If it has a motion, that motion can be discerned only by looking for changes in its position with respect to other things in the universe.

  So, there is no meaning to space that is independent of the relationships among real things in the world. Space is not a stage, which might be either empty or full, onto which things come and go. Space is nothing apart from the things that exist; it is only an aspect of the relationships that hold between things. Space, then, is something like a sentence. It is absurd to talk of a sentence with no words in it. Each sentence has a grammatical structure that is defined by relationships that hold between the words in it, relationships like subject-object or adjective-noun. If we take out all the words we are not left with an empty sentence, we are left with nothing. Moreover, there are many different grammatical structures, catering for different arrangements of words and the various relationships between them. There is no such thing as an absolute sentence structure that holds for all sentences independent of their particular words and meanings.

  The geometry of a universe is very like the grammatical structure of a sentence. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships that hold between the things in the universe. If you change a sentence by taking some words out, or changing their order, its grammatical structure changes. Similarly, the geometry of space changes when the things in the universe change their relationships to one another.

  As we understand it now, it is simply absurd to speak of a universe with nothing in it. That is as absurd as a sentence with no words. It is even absurd to speak of a space with only one thing in it, for then there would be no relationships to define where that one thing is. (Here the analogy breaks down because there do exist sentenc
es of one word only. However, they usually get their meaning from their relationships with adjacent sentences.)

  The view of space as something that exists independent of any relationships is called the absolute view. It was Newton’s view, but it has been definitively repudiated by the experiments that have verified Einstein’s theory of general relativity. This has radical implications, which take a lot of thinking to get used to. There are unfortunately not a few good professional physicists who still think about the world as if space and time had an absolute meaning.

  Of course, it does seem as though the geometry of space is not affected by things moving around. When I walk from one side of a room to the other, the geometry of the room does not seem to change. After I have crossed the room, the space within it still seems to satisfy the rules of Euclidean geometry that we learned in school, as it did before I started to move. Were Euclidean geometry not a good approximation to what we see around us, Newton would not have had a chance. But the apparent Euclidean geometry of space turns out to be as much an illusion as the apparent flatness of the Earth. The Earth seems flat only when we can’t see the horizon. Whenever we can see far enough, from an aircraft or when we gaze out to sea, we can easily see that this is mistaken. Similarly, the geometry of the room you are in seems to satisfy the rules of Euclidean geometry only because the departures from those rules are very small. But if you could make very precise measurements you would find that the angles of triangles in your room do not sum to exactly 180 degrees. Moreover, the sum actually depends on the relation of the triangle to the stuff in the room. If you could measure precisely enough you would see that the geometries of all the triangles in the room do change when you move from one side of it to the other.