The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next

by Lee Smolin

  How can this be? Haven’t I pointed out that string theory is based on a very simple law? The problem is that the law is simple only when applied to the original theory in twenty-six dimensions. When supersymmetry is added, it becomes quite a bit more complicated.

  There are additional results, which show for every term that certain possible infinite expressions that might have occurred do not in fact appear. A powerful proof of this sort was published in 1992 by Stanley Mandelstam. Recently a great deal of progress has been made by Nathan Berkovits, an American physicist who chooses happily to work in São Paulo. Berkovits has invented a new formulation of superstring theory. He achieves a proof good for each term in the perturbation theory, subject only to a couple of additional assumptions. It is too early to tell if those additional assumptions will be easy to lift. Still, this is substantial progress toward a proof. The issue of finiteness is not one that gets much attention from most string theorists, and I have enormous respect for the few that are still working hard on the problem.

  There is one more worrying issue surrounding finiteness. In the end, even if each term in a calculation turns out to be finite, the exact answers to the calculation are derived by summing up all the terms. Since there are an infinite number of terms to be added, the result could again be infinite. While this summing up has not yet been done, there is evidence (too technical to describe here) that the result will be infinite. Another way to say this is that the approximation procedure comes only so close to the real predictions and then diverges from them. This is a common feature of quantum theories. It means that perturbation theory, while a useful tool, cannot be used to define the theory.

  On present evidence, without a proof or a counterexample in hand, it is simply impossible to know whether string theory is finite. The evidence can be read either way. After a lot of hard work (albeit by just a handful of people), there are several partial proofs. This can be read either as clear evidence that the conjecture is true or that something is amiss. If such gifted physicists have tried and failed, and if every attempt remains incomplete, it may be because the conjecture they are trying to prove is false. The reason that mathematics invented the idea of proof and made it the criterion for belief is that human intuition has so often proved faulty. Widely believed conjectures sometimes turn out to be false. This is not an issue of mathematical rigor. Physicists don’t typically aspire to the same level of rigor demanded by their mathematician cousins. There are a number of interesting and widely accepted theoretical results that don’t have mathematical proof. But this is not one of those cases. There is no proof of string theory’s finiteness even at a physicist’s level of rigor.

  Given this, I don’t have a view on whether supersymmetric string theory will turn out to be finite or not. But if something so central to the claims of a theory is thought to be true, work should be expended to turn that intuition into proof. To be sure, there are many cases where a popular conjecture remains unproved for generations, but this is usually because key insights are missing. Even if the end result proves what everyone believes anyway, the effort is usually repaid by our gaining a much deeper insight into the area of mathematics that gave birth to the conjecture in the first place.

  We will come back to why the finiteness of string theory is such a controversial issue. For now, we should simply note that it’s not an isolated example. Several of the key conjectures that drove the two string revolutions remain unproved. These include the strong-weak duality and the Maldacena duality. In both cases, there is a lot of evidence that some form of relation between different theories is true. Even if the strict equivalences claimed in the conjectures are false, these are important ideas and results. But in any hard-nosed accounting, we have to distinguish between conjecture, evidence, and proof.

  Some claim that the Maldacena conjecture offers independent proof that string theory yields a good quantum theory of gravity, at least in the case of certain geometries. They assert that string theory is in some cases precisely equivalent to an ordinary gauge theory in three spatial dimensions, giving a good quantum theory of gravity reliable to any order of approximation.

  The problem with this assertion is that, as noted, the strong form of the Maldacena conjecture remains unproved. There is impressive evidence for some relationship between Maldacena’s ten-dimensional supersymmetric string theory and the maximally super gauge theory, but what we currently have is not yet a proof of the full conjecture. The evidence is just as easily explained by there being only a partial correspondence between the two theories, neither of which is precisely defined. (Recently there has been progress on approaching the gauge theory through a second approximation scheme, called lattice gauge theory.) The present evidence is consistent with Maldacena’s conjecture of complete equivalence being false, either because the two theories are in fact different or because one or both of them do not, strictly speaking, exist. On the other hand, if the strong form of the Maldacena conjecture turns out to be true—which is also consistent with the present evidence—then string theory provides good quantum theories of gravity, in the special case of backgrounds with a negative cosmological constant. Moreover, those theories would be partly background-independent, in that a nine-dimensional space is generated from physics in a three-dimensional space.

  There is other evidence that string theory can provide a unification of gravity with quantum theory. The strongest results involve branes and black holes. These results are extraordinary, but, as we discussed in chapter 9, they do not go far enough. They are so far limited to very special black holes, and there appears to be little hope that the exact results will soon be extended to black holes in general, including the kinds believed to exist in nature; the results may be due to the extra symmetry these special black holes have. Finally, the string theory results do not include an actual description of the quantum geometry of the special black holes; they are limited to the study of model brane systems that share many properties with black holes but exist in an ordinary flat spacetime, and they are studied in an approximation in which the gravitational force is turned off.

  Some argue that these extremal brane systems will become black holes when the gravitational force is turned back on. But string theory cannot follow up that argument with a detailed description of how a black hole is formed. To do this, you would need a string theory that worked in a spacetime that was evolving in time, and we have seen that this does not now exist.

  Since the original results on black holes, there have been a number of imaginative proposals for how to describe real black holes in string theory. But they all suffer from a general problem, which is that whenever they stray from the very special black holes where we can use supersymmetry to do calculations, they fail to lead to precise results. Once we study ordinary black holes, or when we try to go inside to ask what happens to the singularity, we are unavoidably in the regime where the spacetime geometry evolves in time. Supersymmetry cannot work here, and neither do all the beautiful calculational tools that depend on it. So we are left with the same dilemma that afflicts so much research in string theory: We get marvelous results for very special cases, and we are unable to decide whether the results extend to the whole theory or are true only of the special cases where we can do the calculations.

  Given these limitations, can it be claimed that string theory resolves the puzzles of black-hole entropy, temperature, and information loss implied by the discoveries of Jacob Bekenstein and Stephen Hawking? The answer is that while there are suggestive results, string theory cannot yet claim to have solved these problems. For the extremal and near-extremal black holes, calculations using the model systems of branes do reproduce all the details of the formulas that describe the thermodynamics of the corresponding black holes. But these are not black holes, they are just systems constrained by the requirements of having a large amount of supersymmetry to have the thermal properties of black holes. The results do not provide a description of the actual quantum geometry of black holes. Hence t
hey do not explain Bekenstein and Hawking’s results in terms of a microscopic description of black holes. Moreover, as noted, the results apply only to a very special class of black holes and not to those of real physical interest.

  To summarize: On the basis of current results, we cannot confidently assert that string theory solves the problem of quantum gravity. The evidence is mixed. To a certain approximation, string theory seems to consistently unify quantum theory and gravity and give sensible and finite answers. But it is hard to decide if this is true of the whole theory. There is evidence to support something like the Maldacena conjecture, but no proof of the full conjecture itself, and only the full conjecture will allow us to assert the existence of a good quantum theory of gravity. The black-hole picture is impressive but only for the atypical black holes that string theory is able to model. Beyond these, there is the ever-present problem that string theory is not background-independent and even within that limitation cannot so far describe anything other than static backgrounds where the geometry does not evolve in time.

  What we can say is that within these limitations there is some evidence that string theory points to the existence of a consistent unification of gravity and quantum theory. But is string theory itself that consistent unification? In the absence of a solution to these problems, it seems unlikely.

  Let’s turn to the other problems on chapter l’s list. The fourth problem is to explain the values of the parameters of the standard model of particle physics. It is clear that string theory has failed so far to do this, and there is no reason to believe it can. Instead, as we discussed in chapter 10, the evidence suggests that there are such a vast number of consistent string theories that the theory will make few if any predictions on this point.

  The fifth problem is to explain what the dark matter and dark energy are, and to explain the values of the constants in cosmology. Here the situation is also not good. String theories, since they typically include many more particles and forces than have been observed, do offer a number of candidates for the dark matter and energy. Some of the extra particles could be the dark matter. Some of the extra forces could be the dark energy. But string theory offers no specific predictions as to which of the many possible candidates are the dark matter or the dark energy.

  For example, among the possible dark-matter candidates is a particle called an axion (the word refers to certain properties, which I won’t go into). Many, but not all, string theories contain axions, so at first this seems good. But most string theories that contain axions predict that they have properties that disagree with the standard cosmological model. So this seems bad. But then there are so many string theories that some may well contain axions consistent with the cosmological model. It is also possible that the cosmological model is wrong on this point. So it is reasonable to say that if axions are the dark matter, this is consistent with string theory. But this is very far from saying that string theory either predicts that the dark matter is an axion or makes any additional predictions by which observations of the dark matter could falsify string theory.

  The remaining problem on our list is no. 2: the foundational problems of quantum mechanics. Does string theory offer any solution to those problems? The answer is no. String theory so far says nothing directly about the problems in the foundations of quantum theory.

  So here is the accounting. Out of the five key problems, string theory potentially solves one of them completely, the problem of the unification of particles and forces. This is the problem that motivated the invention of string theory and it is still its most impressive success.

  There is evidence that string theory points to a solution to the problem of quantum gravity, but at best it points to the existence of a deeper theory that solves the problem of quantum gravity rather than being itself the solution.

  At the present time, string theory does not solve any of the three remaining problems. It appears incapable of explaining the parameters of the standard models of physics and cosmology. It provides a list of possible candidates for the dark matter and energy but doesn’t uniquely predict or explain anything about them. And so far, string theory has nothing to say about the greatest mystery of all, which is the meaning of quantum theory.

  Beyond this, are there any successes to speak of? One place we usually look for successes of a theory is in the predictions it makes for new experiments or observations. As we have said, string theory makes absolutely no predictions of this kind. Its strength is that it unifies the kinds of particles and forces we know about. If we didn’t know about gravity, for example, we could predict its existence from string theory. This is not nothing. But it is not a prediction for a new experiment. Furthermore, there is no possibility of falsifying the theory—proving it wrong—by finding that an experiment or observation disagrees with the predictions of the theory.

  If string theory makes no new predictions, then we should at least ask how well it accounts for the data we have. Here the situation is peculiar. Because of the incomplete state of our knowledge, we have to divide the many possible string theories into two groups and investigate each one separately. The first group is made up of those string theories known to exist, and the second contains those that are conjectured to exist but not yet constructed.

  Because of the recent observations that the expansion of the universe is accelerating, we’re forced to focus on string theories in the second class, since they are the only ones that agree with these findings. But we do not know how to compute the probabilities for strings to move and interact in these theories. Nor are we able to show that these theories exist; the evidence we have for them is in their backgrounds satisfying certain necessary but far from sufficient conditions. So even in the very best case, if there is any string theory that describes our universe, new techniques will have to be invented to calculate predictions for experiments that work in these new theories. The known string theories, as noted, all disagree with observed facts about our world: Most have unbroken supersymmetry; the others predict that fermions and bosons come in pairs of equal mass; and they all predict the existence of new (and so far unobserved) infinite-range forces. It is hard to avoid the conclusion that, however well motivated, string theory has failed to realize the hopes so many held for it twenty years ago.

  In the heyday of 1985, one of the most enthusiastic proponents of the new revolutionary theory was Daniel Friedan, then at the University of Chicago’s Enrico Fermi Institute. Here is what he had to say in a recent paper:

  String theory failed as a theory of physics because of the existence of a manifold of possible background spacetimes. . . . The long-standing crisis of string theory is its complete failure to explain or predict any large distance physics. String theory cannot say anything definite about large distance physics. String theory is incapable of determining the dimension, geometry, particle spectrum and coupling constants of macroscopic spacetime. String theory cannot give any definite explanations of existing knowledge of the real world and cannot make any definite predictions. The reliability of string theory cannot be evaluated, much less established. String theory has no credibility as a candidate theory of physics.3

  Still, many string theorists soldier on. But how is it that, in the face of the problems we have been discussing, so many bright people continue to work on string theory?

  One point that string theorists are passionate about is that the theory is beautiful, or “elegant.” This is something of an aesthetic judgment that people may disagree about, so I’m not sure how it should be evaluated. In any case, it has no role in an objective assessment of the accomplishments of the theory. As we saw in Part I, lots of beautiful theories have turned out to have nothing to do with nature.

  Some young string theorists argue that even if string theory does not succeed as the final unification, it has spin-offs that aid our understanding of other theories. They particularly refer to the Maldacena conjecture, discussed in chapter 9, which provides a way to study certain gauge theories from calculatio
ns that are easy to do in the corresponding gravity theory. This certainly works well for theories with supersymmetry, but if it is to be relevant for the standard model, it must work well for the gauge theories that have no supersymmetry. In this case there are other techniques, and the question is how well the Maldacena conjecture compares with those. The jury is still out. A good test case is a simplified version of a gauge theory in which there are only two dimensions of space. It has recently been solved, using a technique that owes nothing to supersymmetry or string theory.4 It can also be studied through a third approach—brute-force calculation by computer. The computer calculations are believed to be reliable, hence they can serve as a benchmark against which to compare the predictions of other approaches. Such a comparison shows that the Maldacena conjecture does not do as well as the other technique.*

  Some theorists also point to potential advances in mathematics as a reason to continue working on strings. One such potential advance involves the geometry of the six-dimensional spaces that string theorists study as possible examples of the compactified dimensions. In some cases, unexpected and striking properties of these six-dimensional geometries have been predicted by using the mathematics of string theory. This is welcome, but we should be clear about what happened. There was no contact with physics. What happened took place purely on a mathematical plane: String theory suggested conjectures that relate different mathematical structures. The string theories suggested that properties of the six-dimensional geometries could be expressed as simpler mathematical structures that could be defined on the two-dimensional surfaces the strings sweep out in time. The name of such a structure is a conformal field. What was suggested is that properties of certain six-dimensional spaces were mirrored in the structures of these conformal field theories. This led to surprising relations between pairs of six-dimensional spaces. This is a wonderful spin-off from string theory. And for it to be useful, we do not have to believe that string theory is a theory of nature. For one thing, conformal field theory plays a role in many different applications, including condensed-matter physics and loop quantum gravity. So it has nothing uniquely to do with string theory.