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Quantum

Quantum

Superstrings

01 Nov 2003

String theory is either a theory of everything - which automatically unites gravity with the other three forces in nature - or a theory of nothing, but finding the correct form of the theory is like searching for a needle in a stupendous haystack

Figure 1

As I sit down to write this article I feel that I have taken on a task rather like trying to summarize the history of the world in 10 pages. It is just too large a subject, with too many lines of thought and too many threads to weave together. In the 34 years since it began, string theory has developed into an enormous body of knowledge that touches on every aspect of theoretical physics.

String theory is a theory of composite hadrons, an aspiring theory of elementary particles, a quantum theory of gravity, and a framework for understanding black holes. It is also a powerful technical tool for taming strongly interacting quantum field theories and, perhaps, a basis for formulating a fundamental theory of the universe. It even touches on problems in condensed-matter physics, and has also provided a whole new world of mathematical problems and tools.

All I can do with this gargantuan collection of material is to make my own guess about which aspects of string theory are most likely to form the core of a future physical theory, perhaps 100 years from now. It will come as no surprise to my friends that my choice revolves around those things that have most interested me in the last several years. No doubt many of them will disagree with my judgement. Let them write their own articles.

String theory is considered to be a branch of high-energy or elementary particle physics. However, a high-energy theorist from the 1950s, 1960s or 1970s would be surprised to read a recent string-theory paper and find not a single Feynman diagram, cross-section or particle decay rate. Nor would there be any mention of protons, neutrinos or Higgs bosons in the majority of current literature. What the reader would find are black-hole metrics, Einstein equations, Kaluza-Klein theories and plenty of fancy geometry and topology. The energy scales of interest are not MeV, GeV or even TeV, but energies at the Planck scale – the scale at which the classical concepts of space and time break down.

The Planck energy is equal to h-bar5/G, where h-bar is Planck’s constant divided by 2π, c is the speed of light and G is the gravitational constant, and it corresponds to masses that are some 19 orders of magnitude larger than the proton mass. This is the energy of the universe when it was just 10-43s old, and it will probably be forever out of range of any particle accelerator. To understand physics at the Planck scale we need a quantum theory of gravity.

In the days when my career was beginning, a typical colloquium on high-energy physics would often begin by stating that there are four forces in nature – electromagnetic, weak, strong and gravitational – followed by a statement that the gravitational force is much too weak to be of any importance in particle physics so we will ignore it from now on. That has all changed.

Today the other three forces are described by the gauge theories of quantum chromodynamics (QCD) and quantum electrodynamics (QED), which together make up the Standard Model of particle physics. These quantum field theories describe the fundamental forces between particles as being due to the exchange of field quanta: the photon for the electromagnetic force, the W and Z bosons for the weak force, and the gluon for the strong force. In the string-theory community, however, the electromagnetic, strong and weak forces are generally considered to be manifestations of certain “compactifications” of space from 10 or 11 dimensions to the four familiar dimensions of space-time. But before I report on the status of string theory, I want to tell you how it came about that so many otherwise sensible high-energy theorists became interested in quantum gravity.

Why quantum gravity?

Elementary particles have far too many properties – such as spin, charge, colour, parity and hypercharge – to be truly elementary. Particles obviously have some kind of internal machinery at some scale. Protons and mesons reveal their “parts” at the modestly small distance of about 10-15 m, but quarks, leptons and photons hide their structure much more effectively. Indeed, no experiment has ever seen direct evidence of size or structure for any of these particles.

The first indication that the true scale of elementary particles might be somewhere in the neighbourhood of the Planck scale came in the 1970s. Howard Georgi and Sheldon Glashow, then at Harvard University, showed that the very successful, but somewhat contrived, Standard Model could be elegantly unified into a single theory by enlarging its symmetry group. The new construction was astonishingly compact and most particle theorists assumed that there must be some truth to it. But its predictions for the coupling constants – the constants that describe the strengths of the strong, weak and electromagnetic interactions – were wrong.

Georgi, along with Helen Quinn and Steven Weinberg, also at Harvard, soon solved this problem when they realized that the coupling constants are not really constants at all – they vary with energy. If the known couplings are extrapolated they all intersect the predictions of the unified theory at roughly the same scale. Moreover, this scale is close to the Planck scale. The implication of this was clear: the scale of the internal machinery of elementary particles is the Planck scale. And since the gravitational constant, G, appears in the definition of the Planck energy, to many of us this inevitably meant that gravitation must play an essential role in determining the properties of particles.

The earliest attempts to reconcile gravity and quantum mechanics – notably by Richard Feynman, Paul Dirac and Bryce DeWitt, who is now at the University of Texas at Austin – were based on trying to fit Einstein’s general theory of relativity into a quantum field theory like the hugely successful QED. The goal was to find a set of rules for calculating scattering amplitudes in which the photons of QED are replaced by the quanta of the gravitational field: gravitons. But gravitational forces become increasingly strong as the energy of the participating quanta increases, and the theory proved to be wildly out of control. Attempting to treat the graviton as a point particle simply gave rise to far too many degrees of freedom at short distances.

In a sense the failure of this “quantum gravity” theory was a good sign. The theory itself gave no insight into the internal machinery of elementary particles, and it offered no explanation for the other forces of nature. At best it was more of the same: an effective (but not very) description of gravitation with no deeper insight into the origin of particle properties. At worst, it was mathematical nonsense.

Strings as hadrons

We all know that science is full of surprising twists, but the discovery of string theory was particularly serendipitous. The theory grew out of attempts in the 1960s to describe the interactions of hadrons – particles that contain quarks, such as the proton and neutron. This was a problem that had nothing to do with gravity. Gabriele Veneziano, now at CERN, and others had written down a simple mathematical expression for scattering amplitudes that had certain properties that were fashionable at that time. It was soon discovered by Yoichiro Nambu of the University of Chicago and myself, and in a slightly different form by Holger Bech Nielsen at the Niels Bohr Institute, that these amplitudes were the solution of a definite physical system that consists of extended 1D elastic strings.

For the two years that followed, string theory was the theory of hadrons. One of the spectacular discoveries made in this early period was that the mathematical infinities that occur in quantum field theory are completely absent in string theory. However, from the very beginning there were big problems in interpreting hadrons as strings. For example, the earliest version of the theory could only accommodate bosons, whereas many hadrons – including the proton and neutron – are fermions.

The distinction between bosons and fermions is one of the most important in physics. Bosons are particles that have integer spins, such as 0, h-bar and 2h-bar, whereas fermions have half-integer spins of h-bar/2, 3h-bar/2 and so on. All fundamental matter particles, such as quarks and leptons, are fermions, while the particles that carry fundamental forces – the photon, W and Z, and so on – are all bosons.

Fermionic versions of string theory were soon discovered and, moreover, they turned out to have a surprising symmetry called supersymmetry that is now totally pervasive in high-energy physics. In supersymmetric theories all bosons have a fermionic superpartner and vice versa. The early development of “superstring” theory was due to pioneering work by John Schwarz of Caltech, Andrei Neveu of the University of Montpellier II, Michael Green of Cambridge and Pierre Ramond of the University of Florida, and much of the subsequent technical development was carried out in a famous series of papers by Green and Schwarz in the 1980s.

Another apparently serious problem with the string theory of hadrons concerned dimensions. Although the original assumptions in string theory were simple enough, the mathematics proved internally inconsistent, at least if the number of dimensions of space-time was four. The source of this problem was quite deep, but, strangely, if space-time has 10 dimensions it contrives to cancel out. The reasons were not at all easy to understand, but the extraordinary mathematical consistency of superstring theory in 10 dimensions was compelling. However, so was the obvious fact that space-time has four dimensions, not 10.

Thus by about 1972 theorists were beginning to question the relevance of string theory for hadrons. In fact, there were other serious physical shortcomings in addition to the bizarre need for 10 dimensions. A mathematical string can vibrate in many patterns, which represent a different type of particle, and among these are certain patterns that represent massless particles. But most dangerous of all were massless particles with two units of spin angular momentum (“spin-two”). There are certainly spin-two hadrons, but none that have anything like zero mass. Despite all efforts, the massless spin-two particle could not be removed or made massive.

Eventually, mathematical string theory gave way to QCD as a theory of hadrons, which had its own explanation of the string-like behaviour of these particles without the bad side effects. For most high-energy theorists, string theory had lost its reason for existence. But a few bold souls saw opportunity in the debacle. A massless spin-two field might not be good for hadronic physics, but it is just what was needed for quantum gravity, albeit in 10D. This is because just as the photon is the quantum of the electromagnetic field, the graviton is the quantum of the gravitational field. But the gravitational field is a symmetric tensor rather than a vector, and this means the graviton is spin-two, rather than spin-one like the photon. This difference in spin is the principal reason why early attempts to quantize gravity based on QED did not work.

A theory of everything

The massless spin-two graviton led to a radical shift in perspective among theorists. The focus of mainstream high-energy physics at the time was on energy scales anywhere from the hadronic scale of a few GeV to the weak interaction scale of a few hundred GeV. But to explore the idea that string theory governs gravity, the energy scale of string excitations has to jump from the hadronic scale to the Planck scale. In other words, with barely a blink of the eye, string theorists would leapfrog 19 orders of magnitude, and therefore completely abandon the idea that progress in physics proceeds incrementally. Heady stuff, but also the source of much irritation in the rest of the physics community.

Another reason for annoyance was somebody’s idea to start referring to string theory as a “theory of everything”. Even string theorists found this irritating, but there is actually a technical sense in which string theory can either be a theory of everything or a theory of nothing. One of the problems in describing hadrons with strings was that it proved impossible to allow for the hadrons to interact with other fields, such as electromagnetic fields, as they clearly do experimentally. This was a deadly flaw for a theory of hadrons, but not for a theory in which all matter, including photons, are strings. In other words, either all matter is strings, or string theory is wrong. This is one of the most exciting features of the theory.

But what about the problem of dimensions? Here again, a sow’s ear was turned into a silk purse. The basic idea goes back to Theodor Kaluza in 1919, who tried to unify Einstein’s gravitational theory with electrodynamics by introducing a compact space-like fifth dimension. Kaluza discovered the beautiful fact that the extra components of the gravitational field tensor in 5 dimensions behaved exactly like the electromagnetic field plus one additional scalar field. Somewhat later, in 1938, Oskar Klein and then Wolfgang Pauli generalized Kaluza’s work so that the single compact dimension was replaced by a 2D space. If the 2D space is the surface of a sphere then a remarkable thing happens when Kaluza’s procedure is followed. Instead of electrodynamics, Klein and Pauli discovered the first “non-Abelian” gauge theory, which was later rediscovered by Chen Ning Yang and Robert Mills. This is exactly the same class of theories that is so successful in describing the strong and electromagnetic interactions in the Standard Model.

One may ask whether particles move in the extra dimensions. For example, can a particle that appears to be standing still in our usual 3D space have velocity or momentum components in the compact dimensions? The answer is yes, and the corresponding components of momentum define new conserved quantities (figure 1). What is more, these quantities are quantized in discrete units. In short, they are “charges” similar to electric charge, isospin and all the other internal quantum numbers of elementary particles. The answer to the problem of dimensions in string theory is obvious: six of the 10 dimensions should be wrapped up into some very small compact space, and the corresponding quantized components of momenta become part of the internal machinery of elementary particles that determines their quantum numbers.

Life in six dimensions

Much of the development of string theory is therefore concerned with 6D spaces. These spaces, which can be thought of as generalized Kaluza-Klein compactification spaces, were originally studied by mathematicians and are known as Calabi-Yau spaces. They are tremendously complicated and are not completely understood. But in the process of studying how strings move on them, physicists have created an unexpected revolution in the study of Calabi-Yau spaces.

In particular, it was discovered that a compactification radius of size R is completely equivalent to a space with size 1/R from the point of view of string theory. This connection, which is known as T-duality, has a mathematically profound generalization called mirror symmetry, which states that there is an equivalence between small and large spaces (see box above). Mirror symmetry of Calabi-Yau spaces – which are not only of different sizes but have completely different topologies – was completely unsuspected before physicists began studying quantum strings moving on them.

I wish it was possible to draw a Calabi-Yau space but they are tremendously complicated. They are six-dimensional, which is three more than I can visualize, and they have very complicated topologies, including holes, tunnels and handles. Furthermore, there are thousands of them, each with a different topology. And even when their topology is fixed there are hundreds of parameters called moduli that determine the shape and size of the various dimensions. Indeed, it is the complexity of Calabi-Yau geometry that makes string theory so intimidating to an outsider. However, we can abstract a few useful things from the mathematics, one of them being the idea of moduli.

The simplest example of a modulus is just the compactification radius, R, when there is only a single compact dimension. In more complicated cases, the moduli determine the sizes and shapes of the various features of the geometry. The moduli are not constants but depend on the geometry of the space itself, in the same way that the radius of the universe changes with time in a manner that is controlled by dynamical equations of motion. Since the compact dimensions are too small to see, the moduli can simply be thought of as fields in space that determine the local conditions. Electric and magnetic fields are examples of such fields but the moduli are even simpler: they are scalar fields (i.e. they have only one component), rather than vector fields. String theory always has lots of scalar-field moduli and these can potentially play important roles in particle physics and cosmology.

All of this raises an interesting question: what determines the compactification moduli in the real world of experience? Is there some principle that selects a special value of the moduli of a particular Calabi-Yau space and therefore determines the parameters of the theory, such as the masses of particles, the coupling constants of the forces, and so on? The answer seems to be no: all values of the moduli apparently give rise to mathematically consistent theories. Whether or not this is a good thing, it is certainly surprising.

Ordinarily we might expect the vacuum or ground state of the world to be the state of lowest energy. Furthermore, in the absence of very special symmetries, the energy of a region of space will depend non-trivially on the values of the fields in that region. Finding the true vacuum is then merely an exercise in computing the energy for a given field configuration and minimizing it. This is, to be sure, a difficult task, but it is possible in principle. In string theory, however, we know from the beginning that the potential energy stored in a given configuration has no dependence on the moduli fields.

The reason that the field potential is exactly zero for every value of the moduli is that string theory is supersymmetric. Supersymmetry has both desirable and undesirable consequences. Its most obvious drawback is the requirement that for every fermion there is a boson with exactly the same mass, which is clearly not a property of our world.

A more subtle difficulty involves the aforementioned fact that the vacuum energy is independent of the moduli. As well as telling us that we cannot determine the moduli by minimizing the energy, supersymmetry also tells us that the quanta of the moduli fields are exactly massless. No such massless fields are known in nature and, furthermore, such fields are very dangerous. Indeed, massless moduli would probably lead to long-range forces that would compete with gravity and violate the equivalence principle – the cornerstone of general relativity – at an observable level.

On the plus side, the vanishing vacuum energy that is implied by supersymmetry ensures that the cosmological constant vanishes. If it were not for supersymmetry, the vacuum would have a huge zero-point energy density that would make the radius of curvature of space-time not much bigger than the Planck scale – a most undesirable situation. Supersymmetry also stabilizes the vacuum against various hypothetical instabilities, and it allows us to make exact mathematical conclusions. Indeed, T-duality and mirror symmetry are examples of those exact consequences.

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