by Lee Smolin
One of the rules that Loll and Ambjørn impose is that each quantum spacetime has to be seen as a sequence of possible spaces that succeed one another, like the ticks of a universal clock. The time coordinate, it is argued, is arbitrary, as in general relativity, but the fact that the history of the world can be seen as a succession of geometries that succeed one another in time is not.
Fig. 14. A model quantum universe according to the causal dynamical triangulation program. The figure depicts the history of a model quantum universe with three dimensions of space, one of which runs horizontally, and one of time, which runs vertically. Courtesy of Renate Loll
Given that restriction, plus a few simple rules, they have significant evidence that classical spacetime, with its three dimensions of space and one of time, emerges from a simple game of block piling. This is the best evidence yet in a background-independent quantum theory of gravity that a classical spacetime with three dimensions of space can emerge from a purely quantum world based only on discreteness and causality. In particular, it was shown by Ambjørn and others that if no restriction respecting causality is put in, no classical spacetime geometry emerges.
One consequence of these results is that some of the most widely believed ideas about quantum gravity are in fact wrong. For example, Stephen Hawking and others used to argue that causal structure was inessential, and that calculations could be done in quantum gravity by ignoring the differences between time and space—differences that exist even in relativity theory—and treating time as if it were just another dimension of space. This was what Hawking meant by those mysterious references, in his Brief History of Time, to time being “imaginary.” Ambjørn and Loll’s results show that this idea is wrong.
Before their work, other people had investigated the idea that the fundamental building blocks of spacetime involved causality, but no one had hit upon a theory from which classical spacetime could be shown to emerge. One such formulation, called causal set theory, took the fundamental units of spacetime to be naked events, whose only attributes were lists of other events that could have caused them and that they could have caused. These ideas are even simpler than Loll and Ambjørn’s models, because there is no requirement for a global succession in time. It has so far not been possible to show the emergence of classical spacetime from this theory.
There has been one major triumph of causal set theory, however, which is that it seems to solve the cosmological constant problem. By simply assuming that a classical world emerges from causal set theory, the Syracuse University physicist Rafael D. Sorkin and collaborators predicted that the cosmological constant would be about as small as has since been observed. As far as I’m aware, this is so far the only clean solution to the cosmological constant problem. This solution alone, plus the attraction of a theory based on such simple assumptions, makes this a research program that deserves continued support.
The English mathematical physicist Roger Penrose has also proposed an approach to quantum spacetime based on the principle that what is really fundamental is relations of causality. His approach is called twistor theory. He and a few adherents have been working on this since the 1960s. It is based on a reversal of the usual way of seeing events in spacetime. Traditionally, one sees what happens as primary and the relationships between what happens as secondary. Thus the events are real and the causal relations between the events are simply properties of the events. Penrose found that this way of looking at things can be reversed. You can take the elementary causal processes as fundamental and then define events in terms of coincidences between causal processes. More specifically, you can make a new space, consisting of all the light rays in space-time. You can then translate all of physics into this space of light rays. The result is an incredibly beautiful construction, which Penrose calls twistor space.
For the first twenty years after Penrose proposed it, twistor theory developed rapidly. In surprising and beautiful ways, many of the basic equations of physics could be rewritten in terms of twistor space. It really did seem as if you could see the light rays as the most fundamental thing, with space and time just an aspect of relations among them. There was also progress in unification, because equations describing the various kinds of particles take on the same simple form when written in terms of twistor space. Twistor theory partly realizes the idea that spacetime may emerge from another structure. The events of our spacetime turn out to be certain surfaces suspended in the twistor space. The geometry of our spacetime also emerges from structures in twistor space.
But there are problems with this picture. The main one is that twistor space is understood only in the absence of quantum theory. And while twistor space is very different from spacetime, it is a smooth geometrical structure. No one yet knows what a quantum twistor space looks like. Whether quantum twistor theory will make sense, and whether spacetime will emerge from it, has yet to be shown.
The center of twistor theory in the 1970s was Oxford, and I was one of many who were drawn to spend time there. I found a heady atmosphere, not unlike the atmosphere that would develop later at centers of string theory. Penrose was deeply admired, as Edward Witten would be later. I encountered extremely talented young physicists and mathematicians who passionately believed in twistor theory. Several have gone on to prominence as mathematicians.
Twistor theory certainly led to important advances in mathematics. It gave us a deeper understanding of several of the major equations of physics, including the main equations of Yang-Mills theory, which is the basis of the standard model of particle physics. Twistor theory also gave us a deep and stunningly beautiful understanding of a certain set of solutions to Einstein’s general theory of relativity. These insights have figured significantly in several different developments, including loop quantum gravity.
But twistor theory has so far not blossomed into a viable approach to quantum gravity—chiefly because it hasn’t found a way to incorporate most of general relativity. Still, Penrose and a few colleagues haven’t given up. And a few string theorists, led by Witten, have recently begun working on it, bringing to twistor space some new methods that have moved things forward rapidly. This approach does not so far appear to help twistor theory develop into a quantum theory of gravity, but it is revolutionizing the study of gauge theories—evidence, if any were needed, that it was wrong to neglect twistor theory for so long.
Roger Penrose is not the only first-rate mathematician to invent his own approach to quantum gravity. Perhaps the greatest living mathematician—and certainly the funniest—is Alain Connes, who is the son of a chief of detectives from Marseille and has worked for most of his life in Paris. I love to talk to Alain. I don’t always understand everything he says, but I go away giddy, both from the profundity of his ideas and the absurdity of his jokes. (These tend to be R-rated, even when they are about black holes or pesky Calabi-Yau manifolds.) Once he broke up a conference talk on quantum cosmology by insisting that to show respect, we should all stand up each time the universe was mentioned. But if I don’t always understand Alain, he always understands me; he is one of those people who think so fast that they finish your sentences for you and inevitably improve on what you were about to say. Yet he is so relaxed and confident in himself and his ideas that he is not the least bit competitive, and he is genuinely curious about the ideas of others.
Alain’s approach to quantum gravity has been to go back to the foundations and invent a new mathematics that perfectly unifies the mathematical structures of geometry and quantum theory. This is the math I alluded to in chapter 14, called noncommutative geometry. “Noncommutative” refers to the fact that quantities in quantum theory are represented by objects that do not commute: That is, AB is not equal to BA. The noncommutivity of quantum theory is closely tied to the fact that you can’t measure a particle’s position and momentum at once. When two quantities don’t commute, you can’t know their values simultaneously. Now, this seems counter to the essence of geometry, which starts with a visual image of a su
rface. The very ability to form a visual image implies complete definition and complete knowledge. To make a version of something like geometry built on things that cannot be simultaneously known was a profound step indeed. What is compelling about it is that it offers a new unification of several areas of mathematics, while putting itself forward as the proper math for the next step in physics.
Noncommutative geometry has turned up in several approaches to quantum gravity, including string theory, DSR, and loop quantum gravity. But none of these capture the depth of Connes’s original conception, which he and a few mathematicians, mostly in France, continue to develop.5 The various versions of it that appear in other programs are based on superficial ideas, such as making the coordinates of space and time into noncommuting quantities. Connes’s idea is much deeper; it is a unification at the foundations of algebra and geometry. It could only be the invention of someone who does not just exploit mathematics but thinks strategically and creatively about the structure of mathematical knowledge and its future.
Like the old twistor theorists, the few followers Connes has acquired are committed. For a conference at Penn State University on different approaches to quantum gravity, Alain recommended a famous elder French physicist named Daniel Kastler. The gentleman broke his leg in a bicycle accident a week before the conference, but he clambered out of the hospital and got himself to the Marseille airport, arriving just in time to open the proceedings with the following proclamation: “There is one true Alain, and I am his messenger.” String theorists aren’t the only ones who have their true believers, but the noncommutative geometers surely have a better sense of humor.
One success of noncommutative geometry is that it leads directly to the standard model of particle physics. As Alain and his colleagues discovered, if you take Maxwell’s theory of electromagnetism and write it on the simplest possible noncommutative geometry, out pops the Weinberg-Salam model unifying electromagnetism with the weak nuclear force. In other words, the weak interactions, together with the Higgs fields, show up automatically and correctly.
Recall from chapter 2 that one way to tell whether a particular unification is successful is that there is immediately a sense that the idea agrees with nature. The fact that the correct unification of the weak and electromagnetic forces falls out from the simplest version of Connes’s idea is compelling. It is the kind of thing that might have happened with string theory but didn’t.
There is another set of approaches focusing on how classical spacetime and particle physics could emerge from an underlying discrete structure. These are models developed by condensed-matter physicists, such as Robert Laughlin, of Stanford; Grigori Volovik, of the Helsinki University of Technology; and Xiao-Gang Wen, of MIT. Recently these approaches have been taken up by young people in quantum gravity, such as Olaf Dreyer. These models are primitive, but they do show that aspects of special relativity, such as the universality of an upper speed limit, can emerge from certain kinds of discrete quantum systems. One provocative assertion of Volovik and Dreyer is that the problem of the cosmological constant is solved—because it was never actually a problem in the first place. They claim that the idea that there was a problem was a mistake, a consequence of taking background-dependent theories too seriously. The mistake, they argue, comes from splitting asunder the basic variables of a theory and treating some of them as frozen background and the others as quantum fields.6 If they are right about this, it’s the most important result to have come out of quantum gravity in many years.
Fig. 15. A spin network, which is a state of quantum geometry in loop quantum gravity and related theories. There are quanta of volume associated with the nodes and quanta of area associated with the edges.
Fig. 16. Spin networks evolve in time through a series of local changes like these.
All the approaches I’ve been describing are background-independent. Several begin with an assumption that spacetime is composed of discrete building blocks. One would like to do better and show that the discreteness of space and time is a consequence of putting the principles of quantum theory and relativity theory together. This is what loop quantum gravity accomplishes. It did so starting with Ashtekar’s revolutionary reformulation in 1986 of Einstein’s general theory of relativity. What we found was that with no added input, but merely by rewriting Einstein’s theory in terms of a new set of variables, it was possible to derive precisely what a quantum spacetime is.
The key idea behind loop quantum gravity is actually an old one, which we have already discussed in chapter 7. It is the idea of a description of a field, like the electromagnetic field, directly in terms of its field lines. (The word “loop” comes from the fact that, in the absence of matter, the field lines can close on themselves, forming a loop.) This was the vision of Holger Nielsen, Alexander Polyakov, and Kenneth Wilson, and it was one of the ideas that led to string theory. Basically, string theory is the development of this visionary idea in a context of a fixed background of space and time. Loop quantum gravity is the same idea but developed in a completely background-independent theory.
This work was made possible by Ashtekar’s great discovery that general relativity could be expressed in language like that of a gauge field. The metric of spacetime, then, turns out to be something like an electric field. When we tried to treat the corresponding field lines quantum-mechanically, we were forced to treat them without a background because there was none—the field lines already described the geometry of space. Once we made them quantum-mechanical, there was no classical geometry left. So we had to reinvent quantum field theory in order to work without a background metric. To make a long story short, it took the input of many people, with a variety of skills from physics and mathematics, but we succeeded. The result is loop quantum gravity.
The resulting picture is very simple. A quantum geometry is a certain kind of graph (see Fig. 15). A quantum spacetime is a sequence of events in which the graph evolves by local changes in its structure. This is best illustrated by examples, which are shown in Fig. 16.
The theory leads to many successes. It has been proved finite, in three senses:
Quantum geometry is finite, so that areas and volumes come in discrete units.
When you compute the probabilities for the quantum geometries to evolve into different histories, they always come out finite (at least in a certain formulation of the theory called the Barrett-Crane model).
When the theory is coupled to a matter theory, such as the standard model of particle physics, the infinities that ordinarily occur are rendered finite: That is, without gravity, you have to carry out a special procedure to isolate the infinite expressions and render them unobservable; with gravity, there simply are no infinite expressions.
It should be emphasized that there is no uncertainty associated with the foregoing statements. The main results of loop quantum gravity have been proved by rigorous theorems.
The biggest challenge facing loop quantum gravity has from the beginning been to explain how classical spacetime emerges. In the last few years, there has been major progress on this problem, partly thanks to the invention of new approximation procedures. These showed that the theory has quantum states describing universes where the geometry is, to a good approximation, classical. An important step was taken last year by Carlo Rovelli, of the Centre de Physique Théorique in Marseille, and his colleagues, in which they found strong evidence that loop quantum gravity predicts that two masses will attract each other in precisely the way that Newton’s law specifies.7 These results also indicate that at low energies the theory has gravitons, so loop quantum gravity is indeed a theory of gravity.
A lot of effort is now going into applying loop quantum gravity to real-world phenomena. There is a precise description of black-hole horizons that gets the entropy right. These results agree with Bekenstein’s and Hawking’s old predictions that black holes have entropy and temperature (see chapter 6). As I write, one hot topic among graduate students and postdocs is to predict modi
fications of Hawking’s result for the thermodynamics of black holes that, were they to be measured in some future study of a physical black hole, could confirm or falsify loop quantum gravity.
Loop quantum gravity has also been the basis for models that allow the strongly time-varying geometries inside black holes to be studied. Several calculations give evidence that the singularities inside black holes are removed. Thus time can continue beyond the point at which classical general relativity predicted it must end. Where does it go? It seems to go into newly created regions of space-time. The singularity is replaced by what we call a spacetime bounce. Just before the bounce, the matter inside the black hole was contracting. Just after the bounce, it is expanding, but into a new region that did not exist before. This is a very satisfying result, as it confirms an earlier speculation of Bryce DeWitt and John Archibald Wheeler. The same techniques have been used to study what happens in the very early universe. Again, theorists find evidence that the singularity is eliminated, which would mean that the universe existed before the Big Bang.
The elimination of the singularity in black holes provides a natural answer to the black-hole information paradox of Hawking. As noted in chapter 6, the information is not lost; it goes into the new region of spacetime.
The control that loop quantum gravity has given us over the very early universe has made it possible to compute predictions for real observations. Two postdocs at the Perimeter Institute, Stefan Hofmann and Oliver Winkler, were recently able to derive precise predictions for quantum-gravity effects that may be seen in future observations of the cosmic microwave background.8