To describe our world, supersymmetry must be a broken symmetry, and the compactification must fail to preserve supersymmetry. How many solutions then exist? The lack of exact supersymmetry makes the calculations harder and more vulnerable to error, but as the understanding of string theory grew, so did confidence in these calculations.
It should be emphasised now that there remains a minority who think that these calculations miss something deep and are not reliable. However, if the calculations are trustworthy, one can ask how many ways there are of going from ten dimensions to four dimensions. A few years after the millennium, a crude estimate was made for this, resulting in a number of 10500. This number stuck, and went viral. This large number of apparent solutions of string theory is now often called the string theory landscape.
Whatever its merits, let us stick with this number. This number would say that, whatever the status of string theory in ten dimensions, there are roughly 10500 consistent ways of curling up six of the dimensions to turn the theory into a four-dimensional one. This is a vast – almost inconceivably vast – number. Does this mean that string theory is dead, or at least useless, as a theory of nature? If there are 10500 ways of going from ten to four dimensions, does this render it useless as a part of science?
While the number 10500 sounds gargantuan, reflection shows that this argument is a little silly. Its silliness may be seen through the more familiar issue of genetics and human reproduction. Every one of us has a genetic code inherited, neglecting occasional mutations, in part from our mother and in part from our father. The number of combinations that could have made up our DNA is vast. Personally, I have two sons, and at the time of writing the younger is two and the older is four. There were far more than 10500 possibilities for how our younger son could have turned out. With such a landscape of options, who could ever possibly have predicted the true fact that he looks similar but not identical to his big brother at the same age?
Large numbers may intimidate, but they should not scare. What they should provoke is not flight, but instead reflection on what is the right question to ask. As we have seen, the possible genetic variation in humans is vast beyond count. It is impossible to guess someone’s genetic code, either before or after meeting them in person. Nonetheless, despite the enormous number of possible genetic combinations, you are likely to do quite well by predicting that they have ten fingers, do not have blue skin, and have an adult height somewhere between four and a half and seven feet.
In the context of string theory, what are the right and wrong questions? A good example of a wrong question is to ask what the precise geometry of the extra dimensions is. It is not needed and not relevant. You need not know every mutation on someone’s genome to know they have brown eyes. A more productive line of enquiry is to ask about the general features that arise and the observational consequences they lead to.
We will explore this direction at greater length in chapter 10. To give one example here, compactifications almost always involve large numbers – hundreds or even thousands – of scalar particles called moduli, which have gravitational-strength interactions. It is sensible to search for the consequences of having large numbers of such particles. For example, one can ask: would the presence of a thousand moduli, as opposed to five, lead to observational effects in string theory models of particle physics and cosmology?
This is a type of question that is productively asked. In asking such questions, there is no claim to be identifying the one true theory of the world. However, it takes string theory seriously, treats it as a framework and asks what the typical properties are that this framework leads to. These properties may be testable in the conventional way, and the fact that this does not uniquely reveal the extra-dimensional geometry should be no more disappointing than the fact that long black hair does not tell us someone’s genome.
This is illustrative of the general trend in applications of string theory to particle physics and cosmology. It is no longer expected that string theory will lead to one unique solution for physics in four dimensions. Rather, string theory is more understood as a framework for constructing models, analysing ideas and testing conjectures. If one wants to ask whether some interesting phenomenon, with clear observational consequences, can be realised within a theory of quantum gravity, string theory can be used as a testing ground. It may provide either a proof-of-principle example confirming this, or alternatively an argument suggesting the phenomenon is impossible. At the time of writing, one such example concerns the possible existence of what are called tensor modes within the cosmic microwave background.2
String theory appears to have many solutions. This is neither unusual nor unprecedented in science. The existence of many solutions makes it important to think clearly about the questions it is profitable to ask. However, this existence of a landscape of solutions is behind one particular topic where speculation and conjecture have far out-run reasonable argument. This is the topic of the observed vacuum energy of the universe, often called the cosmological constant, and the theoretical explanation for this phenomenon.
The most striking experimental discovery over the last twenty years has been the discovery that the universe is accelerating. The universe is not only expanding, but the rate of expansion is increasing. Seventy per cent of the energy of the universe lies in the form of a mysterious dark energy’ or ‘cosmological constant’. This dark energy is associated to space itself – it is present wherever space is, and the amount of it is proportional to the volume of space. What makes this energy so mysterious is not its existence – pretty much any theory of particle physics seems to produce such an energy. The mystery comes from its size. The energy is measured at around sixty orders of magnitude smaller than any reasonable estimate; that is, around 0.000000000000000000000000000000000000000000000000000000000001 of what sensible estimates suggest. While the smallness of the cosmological constant had been known for a long time, prior to this discovery it was hoped that there might exist some unknown deep principle that caused the vacuum energy to vanish. No such principle was known, but it was not difficult to conjecture that one might exist. With this possibility removed, the question has to be faced of why the cosmological constant is both non-zero and also so much smaller than all reasonable estimates.
The disagreement between theory and experiment here is so vast that it is almost an insult to the subject. It is an experimental result that openly mocks any claims of theorists to understand the world. Manifestly clear and important, the problem represents a plum target for ambitious theorists who want to prove themselves. It has attracted many, but no solutions have been proposed that have been found satisfactory or generally acceptable.
It is here, it is claimed, that the presence of many solutions of string theory rides to the rescue. The string theory landscape, it is said, can explain why the cosmological constant is small – and the explanation is the anthropic principle.
The anthropic principle is essentially the statement that the answer to some questions is conditioned by the fact that we are here to ask them. If I ask why the density of atoms around me is so vastly greater than almost anywhere else in the universe, a reasonable response is the fact that life cannot exist in empty space. As humans, we cannot live in a vacuum. We can only ask the question on earth, as that is the only place where we can be alive. Our answer then lies in the simple fact that we are here to ask the question in the first place.
Few would find this response unreasonable. However, while the anthropic principle is not vacuous, it can be seductive. It offers the dangers of the open cookie jar at Fat Camp – the soft route of easy temptation. It also encourages a solipsistic attitude to science. For example, I could ask why the Cuban missile crisis did not end in mutual assured destruction, with a nuclear conflagration that destroyed the world. If I felt sufficiently brazen, I could respond that the answer is the anthropic principle. If nuclear war had occurred in 1962, my parents are unlikely to have met a decade later, and I would never have been born in
1981. However, many would feel that the fact that I, Joseph Conlon, am right now contemplating the marvel of my own existence is not a satisfactory explanation for why Kennedy and Khrushchev managed to avoid taking their respective nations to war.
In the context of the cosmological constant, the structure of the anthropic argument is as follows. String theory has many solutions – perhaps 10500 or greater. Each leads to a different value of the cosmological constant. Just as our galaxy is a tiny part of a much greater universe, our universe is also a tiny part of a much greater multiverse, in which all these solutions are actually realised. Across this multiverse of 10500, 101000 or 101500 universes, these different solutions will realise all possible values of the cosmological constant. Somewhere in the multiverse, there is a universe with a cosmological constant as small as in our universe, and with the same laws of physics as in our universe.
What, the argument continues, are the conditions for life to form? Life requires many billions of years of evolution. Billions of years of stable conditions can only arise in a universe that does not expand too rapidly, which requires a very small cosmological constant. Therefore, a precondition for the existence of intelligent beings who ask the question, ‘Why is the cosmological constant so small?’ is a small cosmological constant. As all possible values of the cosmological constant are sampled in the multiverse, the anthropic principle then – it is claimed – explains why the cosmological constant is so small. The answer is us and our existence. We only exist to ask the question in universes with a small cosmological constant.
What is wrong with this argument? It is the sheer utter extravagance of the speculation, uncoupled from either rigorous calculation or experimental test. The argument requires the physical existence of 10500 additional universes, none of which we can probe experimentally. The argument also lacks the redeeming precision of cut-and-dried mathematical argument.
It has all the flaws of accounting for the entirety of human history by my own existence. We are all of us the product of many ancestors. If history had taken a different turn, we as individuals would not exist – but it would be the height of egotistic solipsism to assert that my existence represents a decent answer to the question of why Brutus killed Caesar. If we look at the entire observable universe, the anthropic explanation of the cosmological constant requires that this has been replicated 10500 times beyond our view, every time with different laws of physics and histories. None of these other universes are observable – and all to explain one single number whose empirical existence was established barely a decade ago!
The history of science shows that certain problems are often not mature for solution at the time they are posed. Instead, they have to be left alone for decades or even centuries before they can be sensibly tackled. The idea of atoms had to wait over two thousand years to move from speculation to understanding. Chemists understood the existence of different elements in the 18th and 19th centuries, and that they had different properties, but it would require the advent of quantum mechanics in the 1920s to understand where the elements came from. Until new tools had appeared, these problems were inaccessible.
The existence of a cosmological constant became generally accepted around the turn of the millennium. There is no reason that it needs to be understood in the lifetimes of our grandchildren, let alone within a few years. The ideas needed may be centuries away. The most serious problem with the anthropic landscape is that it provides a cheap and lazy explanation that does not come from hard calculation and also has no clear experimental test. It sounds exciting, but does not offer lasting sustenance, and may even act as a deterrent against necessary hard work developing new calculational tools.
Of course, this does not mean that the anthropic approach is necessarily wrong. However, the triumph of science has been not because it contains ideas that are not necessarily wrong, but because it contains ideas that are, in some important sense, known to be true: ideas which have either passed experimental test or are glued together by calculation. The anthropic landscape is neither of these. It represents incontinence of speculation joined to constipation of experiment.
Sometimes, the only thing that can be done with an intractable scientific problem is to wait – not for new universes, but for another, doubtless very different, set of tricks. The theory of the cosmological constant may well be one such case.
6.3 FIFTY YEARS ON
The above topics are characteristic of the post-millennial view of string theory, and similar features are seen in other applications of string theory – for example, techniques from string theory have also recently been used to improve the methods for calculating scattering amplitudes for the strong force. Unlike in its earliest years, here string theory is not being proposed as the fundamental theory of the strong force – but it is being used to develop more efficient calculational methods within quantum chromodynamics.
While the details in each case may be different, what is common is a view of string theory as a framework, an assortment of ideas, or a set of tools and concepts – but not as something providing the one true answer to a single question about nature.
In 2015 then, what is string theory? The shortest definition is the consistent theory of quantum relativistic strings – and everything thereby included. Despite all changes, there have been no logical additions to the original ideas. The many discoveries made over the last thirty years were already there, implicitly, in the first string theories written down in the early 1970s, even though no one suspected it at the time. The connections between different string theories, the duality relations between strong and weak coupling, the existence of a hidden eleventh dimension in the theory, the AdS/CFT correspondence, the idea of M-theory – none of these were supernumerary accessories to the original equations. They have not been added in. They were all already present in the equations of the 1970s, veiled and awaiting discovery.
String theory’ now also includes all the useful mathematics, ideas and applications found along the way. The AdS/CFT duality relating gauge and gravitational theories – that is string theory. The mathematics of Calabi-Yau geometries – that is string theory. Techniques for doing computations in quantum field theory – these are string theory. Collections of models for the early universe – these are string theory. The term has become a loose one, encompassing a range of topics that are only tenuously connected to vibrating one-dimensional objects.
It is not that the old ideas have gone away – it is just that string theory is now both a theory and a set of techniques. String theory has become an umbrella term both for a variety of communities with different motivations, interests and methods, and also for the problems they study. The name nowadays is as much that of a social grouping as a description of what these people actually do.
Considered narrowly, string theory remains a candidate theory of quantum gravity – a claim about the laws of nature at the smallest possible distance scales. Considered broadly, string theory is now also an amorphous blob of results, techniques, outlooks, ideas, methodologies and calculations. While these draw on the vision of string theory as a theory of quantum gravity, they do not rely on it.
At the time of writing, the most recent Strings conference occurred in Princeton in July 2014. As part of that conference, several grandees of the subject gave vision talks on matters past, or passing, or to come. One of these talks was given by the recently retired Cambridge professor Michael Green, who had started work on the subject over forty years ago when it was a still viewed as a candidate theory of the strong interactions. One of Green’s slides gave his outlook on string theory:
As time goes by and String Theory evolves, it is more and more apparent that it is not just a
Theory of String-like Elementary Particles’,
but it is a
‘Magnificent theoretical framework that interrelates a very wide range of topics in physics and mathematics’.
The unpredictable trajectory of String Theory since its inception is part of what makes ou
r subject so exciting and so challenging.
If in 2015 – as the Islamic State of Iraq and the Levant claims to reestablish the Caliphate, the 200-1 outsider Jeremy Corbyn is elected leader of Britain’s Labour Party, England dismiss Australia for 60 runs and regain the Ashes, and the United States Supreme Court announces a constitutional right to same-sex marriage – you wish to be a card-carrying string theorist in good standing, there is no requirement to work on quantum gravity – and not even any requirement to work on strings.
In the third part of the book, we shall now look in more detail at the reasons why people do choose to work on string theory, starting with the direct experimental evidence for it.
1From a more technical point of view, the reason these solutions are exact is because they preserve N = 2 supersymmetry. This additional supersymmetry controls corrections to such an extent that the results are guaranteed to be exact. The heterotic string on a Calabi-Yau preserves N = 1 supersymmetry – which, while useful, is not enough to guarantee an exact solution.
2At a technical level, these occur when fields undergo what are called trans-Planckian excursions during an epoch of inflation. It is currently a topic of very active debate whether such excursions are allowed by quantum gravity, or whether a no-go result may exist.
III
What For?
CHAPTER 7
Direct Experimental Evidence for String Theory
There is no direct experimental evidence for string theory.
CHAPTER 8
Why Strings? Quantum Field Theory
Let us try again. Why do people work on string theory?
‘Quantum field theory’ is three small words: but three small words that have spawned a bookcase full of textbooks. We have seen in earlier chapters that the Standard Model of particle physics is an example of a quantum field theory. Quantum field theories are in a sense just – just! – the quantum mechanics of theories such as electromagnetism. That is, they do not exist as something additional to ordinary quantum mechanics, but are rather particular examples of quantum mechanical theories. They are the quantum mechanical theories of fields (for example, the electromagnetic field). They operate under the same rules and obey the same equations as any other quantum mechanical theory. However, to categorise quantum field theories simply as examples of quantum mechanical theories is as accurate as categorising Hamlet simply as an example of a book. The statement is as wrong as it is possible to be while still being right. Quantum field theories are both highly important and filled with subtleties. As outlined in the introductory chapters, there are indeed sufficient subtleties that these theories spent their first twenty years being viewed as fundamentally flawed, and their second twenty years being regarded as calculational black magic, before a deeper conceptual understanding finally arose.
Why String Theory? Page 16