by Lee Smolin
The first step in the assessment of any theory is a comparison with observation and experiment. This was discussed in the last chapter. We learned that even after all the work that has been put into string theory, there is no realistic possibility for a definitive confirmation or falsification of a unique prediction from it by a currently doable experiment.
Some scientists would take that as reason enough to give up, but string theory was invented to solve certain theoretical puzzles. Even absent experimental test, we might be willing to support a theory that provided convincing solutions to outstanding problems. In the first chapter, I described the five major problems facing theoretical physics. The theory that will close Einstein’s revolution should resolve all of them. It is thus fair to assess string theory by asking how well it does this.
Let’s begin by recapping exactly what we know about string theory.
There is, first of all, no complete formulation of it. There is no accepted proposal for what the basic principles of string theory are, or for what the main equations of the theory should be. Nor is there proof that such a complete formulation exists. What we know of string theory consists mostly of approximate results and conjectures that concern the following four classes of theories.
The best-understood theories feature strings moving in simple backgrounds, such as flat ten-dimensional spacetime, where the geometry of the background is unchanging in time and the cosmological constant is zero. There are also many cases where some of the nine spatial dimensions are curled up, while the rest remain flat. These are the theories we understand best, because detailed calculations can be done of strings and branes moving and interacting in these backgrounds.
In these theories, we describe the motion and interaction of the strings on the background spaces in terms of an approximation procedure called perturbation theory. What has been proved is that these theories are well defined and give finite and consistent predictions up to a second order in that approximation scheme. Other results support, but so far do not prove, the consistency of these theories. In addition, a large number of results and conjectures describe a network of duality relations among these theories.
However, every one of these theories disagrees with established facts about our world. Most of them have unbroken supersymmetry, which is not observed in the real world. The few that don’t have unbroken supersymmetry predict that fermions and bosons have superpartners of equal mass, which is also not observed, and they also predict the existence of infinite-range forces in addition to gravity and electromagnetism, which again are unobserved.
In the case of a world with a negative cosmological constant, there is an argument for the existence of a class of string theories based on the Maldacena conjecture. This relates string theory on certain spaces with a negative cosmological constant to certain supersymmetric gauge theories. So far, these string theories cannot be explicitly constructed and studied except for certain very special, highly symmetric extremal cases. The weaker versions of the Maldacena conjecture are supported by a lot of evidence, but it is unknown exactly which version of the conjecture is true. If the strongest version is true, then string theory is equivalent to gauge theory, and this relation provides an exact description of string theories with a negative cosmological constant. However, these theories also cannot describe our universe, because we know the cosmological constant is positive.
An infinite number of other theories are conjectured to exist that correspond to strings moving on more complicated backgrounds, in which the cosmological constant is not zero, in which the spacetime background geometry is evolving in time, or in which the background involves branes and other fields. This includes a vast number of cases where the cosmological constant is positive, in agreement with observation. It has so far been impossible to precisely define these string theories or do explicit calculations to draw predictions from them. The evidence for their existence is based on the satisfaction of certain necessary but far from sufficient conditions.
In twenty-six spacetime dimensions, there is a theory, without fermions or supersymmetry, called the bosonic string. This theory has tachyons, which lead to infinite expressions, rendering the theory inconsistent.
It has been suggested that all the conjectured and constructed theories are unified in a deeper theory, called M-theory. The basic idea is that all the theories we understand will correspond to solutions of this deeper theory. There is evidence for its existence in the many duality relationships that are conjectured or demonstrated to hold among the various string theories, but so far no one has been able to formulate its basic principles or write down its basic laws.
From this summary, we can see why any evaluation of string theory will necessarily be controversial. If we restrict our attention to the theories that are known to exist—those that allow us to do actual calculations and make predictions—we must conclude that string theory has nothing to do with nature, because every single one of these disagrees with experimental data. So the hope that string theory may describe our world rests wholly on a belief in string theories whose existence is only conjectured.
Nevertheless, many working string theorists believe that the conjectured theories exist. This belief seems to be based on indirect reasoning, as follows:
They conjecture that a general formulation of string theory exists and is defined by unknown principles and unknown equations. This unknown theory is conjectured to have many solutions, each of which provides a consistent theory of strings propagating on some background spacetime.
They then write down equations that are conjectured to approximate the true equations of the unknown theory. It is then conjectured that these approximate equations give necessary but not sufficient conditions for a background to have consistent string theories. These equations are versions of Kaluza-Klein theory, in that they involve general relativity extended to higher dimensions.
For every such solution to these approximate equations, they conjecture the existence of a string theory, even if they cannot write it down explicitly.
The problem with this reasoning is that the first step is a conjecture. We do not know that the theory or the equations that would define it really exist. That makes the second step a conjecture as well. Also, we do not know that the conjectured approximate equations give us sufficient, as opposed to necessary, conditions for a string theory to exist.
There is a danger with this kind of reasoning—with assuming what needs to be proved. If you believe in the assumptions of the argument, then the theories whose existences are implied can be studied as examples of string theories. But it must be remembered that they are not string theories, nor theories of any kind, but rather solutions of classical equations. Their significance depends entirely on the existence of theories that no one has been able to formulate and conjectures that no one has been able to prove. Given this, there appear to be no compelling reasons to believe that any string theory that has not been explicitly constructed exists.
What conclusions can be drawn from all this? First, given the incomplete state of knowledge of string theory, there is a wide range of possible futures. Based on what we know now, a theory may well emerge that fulfills the original hopes. It is also possible that there is no real theory there and that all there ever will be is a large set of approximate results about special cases that hold only because they are constrained by special symmetries.
The inevitable conclusion seems to be that string theory itself—that is, the theory of strings moving on background spacetimes—is not going to be a fundamental theory. If string theory is to be relevant at all for physics, it is because it provides evidence for the existence of a more fundamental theory. This is generally recognized, and the fundamental theory has a name—M-theory—even if it has not yet been invented.
This may not be as bad as it seems. For example, most quantum field theories are not known to exist at a rigorous level. The quantum field theories that particle physicists study—including quantum electrodynamics, quantu
m chromodynamics, and the standard model—share with string theory the fact that they are defined only in terms of an approximation procedure. (Although these theories have been proven to give finite and consistent results to all orders of approximation.) Still, there is good reason to believe that the standard model does not exist as a rigorously defined mathematical theory. This is not disturbing, as long as we believe that the standard model is only a step toward a deeper theory.
String theory was at first thought to be that deeper theory. On the present evidence, we must admit that it is not. Like the quantum field theories, string theory seems to be an approximate construction that (to the extent it is relevant for nature) points to the existence of a more fundamental theory. This does not necessarily make string theory irrelevant, but to prove its worth it must do at least as well as the standard model. It must predict something new that turns out to be true and it must explain phenomena that have been observed. We have seen that so far it does not do the first. Does it do the second?
We can answer this by assessing how well string theory answers the five key problems outlined in chapter 1.
Let’s start with the good news. String theory was originally motivated by the third problem, the problem of unification of the particles and forces. How has it stood up as such a unifying theory?
Quite well. On the backgrounds where consistent string theories are defined, the vibrations of a string include states that correspond to all the known kinds of matter and forces. The graviton, the particle that carries the gravitational force, comes out of the vibrations of loops (i.e., closed strings). The photon, carrier of the electromagnetic force, also emerges from the vibrations of a string. The more complicated gauge fields, in terms of our understanding of the strong and weak nuclear forces, also come out automatically; that is, string theory predicts generally that there are gauge fields similar to these, although it does not predict the particular mix of forces we see in nature.
Thus—at least on the level of the bosons, or force-carrying particles, on a background spacetime—string theory unifies gravity with the other forces. All four fundamental forces arise as vibrations of one fundamental kind of object, a string.
What about unifying the bosons with the particles that make up matter, like quarks and electrons and neutrinos? It turns out that these also arise as states of vibration of strings, when supersymmetry is added. Thus, supersymmetric string theories unify all the different kinds of particles with one another.
Moreover, string theory does all this with a simple law: that the strings propagate through spacetime so as to take up the least amount of area. Nor is there any need to have separate laws describing how particles interact; the laws by which strings interact follow directly from the simple law that describes how they propagate. And since the various forces and particles all are just vibrations of strings, the laws that describe them follow as well. Indeed, the whole set of equations describing the propagation and interactions of the forces and particles has been derived from the simple condition that a string propagates so as to take up the least area in space-time. The beautiful simplicity of this is what excited us originally and what has kept many people so excited: a single kind of entity, satisfying a single simple law.
What about the first problem in chapter 1, the problem of quantum gravity? Here the situation is mixed. The good news is that the particles carrying the gravitational force come out of the vibrations of strings, as does the fact that the gravitational force exerted by a particle is proportional to its mass. Does this lead to a consistent unification of gravity with quantum theory? As I stressed in chapters 1 and 6, Einstein’s general theory of relativity is a background-independent theory. This means that the whole geometry of space and time is dynamical; nothing is fixed. A quantum theory of gravity should also be background-independent. Space and time should arise from it, not serve as a backdrop for the actions of strings.
String theory is not currently formulated as a background-independent theory. This is its chief weakness as a candidate for a quantum theory of gravity. We understand string theory in terms of strings and other objects moving on fixed classical background geometries of space that don’t evolve in time. So Einstein’s discovery that the geometry of space and time is dynamical has not been incorporated into string theory.
It is interesting to reflect that apart from a few special one-dimensional theories, no rigorous background-dependent quantum field theory exists. All are defined only in terms of approximation procedures. Perhaps string theory shares this property because it is background-dependent. It is tempting to suggest that any consistent quantum field theory must be background-independent. If true, this would imply that unification of quantum theory with general relativity is not optional, it is forced.
There are claims that general relativity can, in a certain sense, be derived from string theory. This is a significant claim, and it is important to understand the sense in which it is true, for how can a background-independent theory be derived from a background-dependent theory? How can a theory in which the geometry of space-time is dynamical be derived from a theory that requires a fixed geometry?
The argument for this is as follows: Consider a spacetime geometry and ask if there is a consistent quantum-mechanical description of strings moving and interacting in that geometry. When you investigate this proposition, you find that a necessary condition for the string theory to be consistent is that, to a certain approximation, the spacetime geometry is a solution to the equations of a higher-dimensional version of general relativity. So there is a sense in which the equations of general relativity emerge from the conditions for a string to move consistently. This is the basis for the claims that string theorists make for general relativity’s derivation from string theory.
There’s a catch, though. What I have just described is the situation in the original twenty-six-dimensional bosonic string. But, as noted, this theory has an instability, the tachyon, so it is not really a viable theory. To make the theory stable, one can make it supersymmetric. And supersymmetry gives rise to additional necessary conditions that the background geometry must satisfy. Currently the only supersymmetric string theories known in detail to be consistent live on background spacetimes that do not evolve in time.1 So in these cases it cannot be asserted that all of general relativity is recovered as an approximation in supersymmetric string theory. It is true that many solutions to general relativity are recovered, including all the solutions in which some dimensions are flat and others are curled up. But these are very special; the generic solution of general relativity describes a world whose spacetime geometry changes in time. This captures Einstein’s essential insight that the geometry of spacetime is dynamical and evolving. You cannot recover only those solutions with no time dependence and still say that general relativity is derived from string theory. Nor can you claim to have a theory of gravity, since many gravitational phenomena involving time dependence have been observed.
In response, some string theorists conjecture that there are consistent string theories on spacetime backgrounds that vary in time but these are just much more difficult to study. They cannot be supersymmetric, and to my knowledge, there is no explicit general construction of such theories. The evidence for them is of two kinds. First, there is an argument that at least small amounts of time dependence can be introduced without disturbing the conditions required to eliminate the tachyon and make the theory consistent. This argument is plausible, but with no detailed construction it’s hard to judge. Second, some special cases have been worked out in detail; however, the most successful of these have a hidden symmetry in time, so they don’t fit. Others have possible problems with instabilities, or are worked out only at the level of classical equations that don’t go far enough to show whether or not they really exist. Still others have a very fast time dependence, governed by the scale of the string theory itself.
In the absence of an explicit construction of a string theory on a general time-dependent spacetim
e, or a compelling argument for its existence without assuming the existence of a meta-theory, it cannot now be asserted that all of general relativity can be derived from string theory. This is another issue that remains open, to be decided by future work.
One can still ask whether string theory gives a consistent theory that includes gravity and quantum theory in those cases where the theory can be constructed explicitly. That is, can we at least describe gravitational waves and forces so weak that they can be seen as barely rippling the geometry of space? And can we do this completely consistently with quantum theory?
This can be done to a certain approximation. So far, the attempts to prove it beyond that level of approximation have not fully succeeded, although a lot of positive evidence has been gathered and no counterexample has emerged. Certainly, it’s widely believed by string theorists to be true. At the same time, the obstacles to proving it seem substantial. The approximation method, perturbation theory, gives answers to any physical question by a sum of an infinite number of terms. For the first several terms, each one is smaller than the one before, so you get an approximation just by calculating a few terms. This is what is usually done in string theory and quantum field theory. To prove the theory finite, then, you have to prove that for any calculation you might do to answer a physical question, each of the infinite number of terms is finite.
Here is where things stand now. The first term is obviously finite, but that corresponds to classical physics, so there is no quantum mechanics in it. The second term, the first that could possibly be infinite, can easily be shown to be finite. It took until 2001 for a complete proof of the finiteness of the third term. It was heroic work, carried out over many years by Eric D’Hoker at UCLA and his collaborator Duong H. Phong, at Columbia University.2 They have since then been working on the fourth term. They understand a great deal about this term but have no proof so far that it is finite. Whether or not they succeed in proving all the infinite terms finite remains to be seen. Part of the problem they face is that the algorithm for writing down the theory becomes ambiguous past the second term, so they need to first find the right definition for the theory before they can try proving that it gives finite answers.