String Theory

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STRING THEORY

String theory is a proposed unified theory of fundamental physics, incorporating both particle physics and gravity. It is based on the idea that the basic building blocks of nature are strings, one-dimensional objects of zero thickness, which form either closed loops or open curves (Figure 1). This theory has not yet been experimentally tested, but it has attracted the attention of theoretical physicists from a wide range of fields because it unifies many of the central concepts of physics and resolves a number of longstanding theoretical problems.

FIGURE 1

A Brief History

Two of the central questions in physics are the nature of matter and the nature of gravity. In the twentieth century, very successful theories of each were discovered: the Standard Model of matter and the general theory of relativity. However, both theories are incomplete. The Standard Model is based on a complicated pattern of particles and forces, similar to the Periodic Table of the elements, and this pattern must be explained. General relativity, when combined with quantum mechanics, suffers from several problems and paradoxes when applied to very short distances or to black holes. Further, ultimately matter and gravity should not be described by two unrelated theories but should be understood in a unified way. String theory is believed to solve all of these problems.

The idea of building blocks that are one-dimensional, rather than zero-dimensional points, is rather novel, and it has had an odd history. It was first developed between 1968 and 1973 as a theory of the strong interaction: mesons such as the pion behave in some respects like open strings. This idea was superceded by the 1973 discovery of the true theory of the strong interaction, quantum chromodynamics, but a small handful of theorists regarded string theory as a compelling idea and continued to develop it. In the following years it was discovered that string theory is actually a theory of gravity, that it implies a symmetry between bosons and fermions (which was named supersymmetry), and that it is free of the unphysical infinities that plagued all previous theories of quantum gravity.

In 1984 a discovery by Michael Green and John Schwarz, known as anomaly cancellation, showed that string theory could also describe quarks, leptons, and gauge interactions. This led to a tremendous wave of research activity, often called the first superstring revolution, as theorists who had been pursuing other approaches to unification began to develop string theory. The discovery of Calabi-Yau compactification by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten and of heterotic string by David Gross, Jeff Harvey, Lance Dixon, and Ryan Rohm strengthened the evidence for string theory. However, work at this time was limited to a certain approximation, known as perturbation theory, which applies only to small numbers of strings interacting weakly. In 1995 Witten, extending results of Chris Hull, Paul Townsend, and others, identified the principle of string duality, which governs the behavior of strongly interacting strings. The ensuing period, known as the second superstring revolution, has produced many further discoveries. One has been a new understanding of the quantum mechanics of black holes, resolving some long-standing puzzles. Another has been the understanding that string theory contains extended structures known as branes, which has led to new ideas for realistic models and for experimental and cosmological tests.

String theory is still an incomplete theory. It is widely believed that the description of the theory in terms of one-dimensional building blocks is not ultimately the simplest or most complete but rather is a stepping-stone toward a more fundamental principle, which is being actively sought by string theorists today. String theory is equivalently referred to as superstring theory, reflecting the central role of supersymmetry. Since 1995 the term M theory has also been used, reflecting the fact that the string picture is believed to be just a stepping-stone to a final theory. What the "M" stands for is deliberately left unspecified, reflecting the unknown nature of the final theory; "magic," "mother," "mystery," "membrane," and "matrix" have all been suggested.

String Theory and Particle Physics

Experimentally, the electron, quarks, photon, and other Standard Model particles are all points: no experiment has revealed any substructure, down to the distance scale of 10^-16 cm. The idea of string theory is that under sufficient magnification the Standard Model particles will be seen to be loops or segments of string. The magnification needed is very large, as the size of the string is expected to be of order 10^-32 cm: this is the Planck length, the distance scale where gravity and quantum mechanics come together. This scale is far beyond the reach of particle accelerators, so that experimental tests of the theory will have to be indirect. (Recent ideas, to be discussed below, raise the possibility that the strings are larger and so more accessible to experiment.)

The string of string theory is much like a violin string: it vibrates, and this vibration can be decomposed into a sum of notes or harmonics. Depending upon which harmonics are excited, and to what degree, the string will behave like different kinds of particles, and all the particles found in the Standard Model can be obtained as different states of vibration of this one building block. In particular, one of the states of vibration of the closed string is a massless particle of spin two. This was a problem when string theory was supposed to describe the strong interaction, as there is no such hadron, but this particle has precisely the properties of the graviton, the particle associated with the gravitational field, and this is why string theory must incorporate gravity. In addition to vibrating, strings can break in two and join together (Figure 2). All of the basic processes of nature, such as an electron emitting a photon or a graviton, or a Z boson decaying into a quark-antiquark pair, arise from this one basic string process.

Before the first superstring revolution, many other ideas were explored for unifying the Standard Model and explaining its patterns. Three ideas of particular note are grand unification, supersymmetry, and extra dimensions. These can be thought of as new symmetry principles, meaning that Standard Model particles that appear to be different are really the same kind of particle but with, in some sense, a different orientation. Each of these ideas had some successes, and one of the attractive features of string theory is that it automatically incorporates all three of these enlarged symmetries.

Extra Dimensions

The idea that space-time has more than the four visible dimensions is almost as old as general relativity. Gravity and electromagnetism are similar in that both forces fall off as the inverse square of the distance; they differ in that gravity couples to energy and momentum, and electromagnetism

FIGURE 2

couples to charge. Theodore Kaluza in 1919 and Oscar Klein in 1926 put forward the idea that elec tromagnetism would actually originate from grav ity if space-time were five-dimensional, with the fifth dimension too small to be seen directly (Figure 3). The gravitional field has a polarization (spin). If this polarization is fully aligned along the large dimensions, the five-dimensional graviton be haves like a four-dimensional graviton and pro duces the gravitational force. If it is partly aligned along the small dimension, then it behaves like a four-dimensional photon and produces the elec tromagnetic force. What is seen as electric charge is actually momentum that is directed along the small dimension. This elegant unification of the two then-known forces fascinated some of the greatest physicists of the early twentieth century. Albert Einstein and Wolfgang Pauli each spent sub stantial periods trying to develop it further.

String theory requires that space-time have ten dimensions (nine space and one time). Ultimately this originates from the mathematical structure of the supersymmetry algebra. There are actually five consistent string theories, known as types I, IIA, IIB, heterotic SO(32), and heterotic E₈ × E₈. These differ primarily in the way that the supersymmetry acts on the states of the string; also, type I theory has both open and closed strings, while the other four have only closed strings.

FIGURE 3

In Kaluza-Klein theory, the single extra dimension forms a circle. The six extra dimensions of string theory can have a much more complex topology and geometry. Although we do not see these dimensions directly, their shape determines the physics that we do see—the spectrum of particles, and their masses and couplings. A relatively simple set of spaces known as Calabi-Yau manifolds, when combined with the E₈ × E₈ heterotic string, give a result very much like the grand unified supersymmetric Standard Model. One of the central problems in string theory is that there are many different Calabi-Yau manifolds, as well as other possible spaces, and to account for the precise details of the Standard Model requires knowing the precise shape of the extra dimensions.

The simplest estimate of the size of the extra dimensions in string theory gives the Planck length, which would put them far beyond direct experimental detection. As will be discussed below, more recent ideas raise the possibility that they are much larger and might have a variety of observable effects.

String Theory and Quantum Gravity

General relativity and quantum mechanics are two of the central principles in physics, and each has been verified experimentally in great detail. General relativity is important at astronomical scales, but its effects are negligible in the microscopic regime of atomic and particle physics. Quantum mechanics is essential to microscopic physics, but its effects are negligible at astronomical scales. Thus, in ordinary circumstances one does not encounter general relativistic and quantum effects together.

However, general relativity and quantum mechanics conflict with one another, and this conflict will appear in certain extreme situations. At very short distances one encounters the problem of space-time foam. General relativity states that space-time is curved and that the effect of this curvature is gravity. Quantum mechanics states, roughly speaking, that nothing sits still (the uncertainty principle). Taken together, these imply that space-time does not sit still, its shape is constantly fluctuating. These fluctuations are totally negligible on astronomical scales and even on the scales of particle physics, but as one goes to very short distances they become more evident. At the Planck length of 10^-32 cm they become so severe that shorter distances do not make sense at all—space-time, in a sense, tears itself to pieces. In the language of particle physics, this means that quantum gravity is not renormalizable: when the effects of virtual gravitons are included in quantum mechanical amplitudes, the result is infinite (Figure 4).

The problem of renormalization arose for the three particle interactions as well and in each case was an important clue to the correct theory. In the case of gravity, string theory removes the problem by changing the theory at distances below the Planck scale.

Other approaches to this problem are still under study, but to date string theory is the only known finite theory of quantum gravity. The problem of quantum foam means that distances smaller than the Planck scale cannot make sense, and so the historic progression toward ever-smaller constituents must end. In string theory, the size of a string represents a minimum length, the shortest distance that can be probed.

Modern String Theory

String Duality and D-branes

As physics has progressed toward more unified theories based on more fundamental principles, there has been a growing expectation that there is

FIGURE 4

a unique theory that incorporates all of the laws of physics. The existence of five different string theories was therefore a puzzle (though the situation is much better than quantum field theory, where there is an infinite number of theories that are characterized by different symmetries, particles, masses, and couplings). In 1995 it was understood, through string duality, that these are all part of a single theory. Essentially, they are different phases (Figure 5), related to each other much like the liquid, solid, and gas phases of water. In the case of water, one varies the pressure and temperature to change one phase into another. In the case of string theory, one varies the shape and size of the extra dimensions, and in certain regimes the theory behaves like one or the other of the string theories. There are also new phases, most notably a phase known as D = 11 supergravity where a new space-time dimension, the eleventh, appears.

In addition to the strings themselves, string theory contains a variety of higher-dimensional objects known as branes. These were discussed before 1995, but in the context of string duality it became clear that they play a central role. A particular class known as D-branes (Figure 6) were shown by Joseph Polchinski to have the special property that strings can end on them, and play an important role in understanding many of the phases of the theory.

FIGURE 5

FIGURE 6

Black Hole Entropy and Information

Black holes are among the most extreme objects in physics, and they present another situation where the laws of relativity and quantum physics come together. In the early 1970s it was found that black holes satisfy laws parallel to those of thermodynamics. In particular, Jakob Bekenstein and Stephen Hawking argued that they have an entropy, implying a microscopic structure of states. The nature of these states was mysterious until 1996, when Strominger and Cumrun Vafa showed that they were accounted for by string theory. In particular, for certain charged black holes, D-branes give a precise construction of the microscopic states.

Hawking also discovered that black holes radiate and eventually disappear and that this leads to a paradox. The particles produced in the decay do not depend on what initially forms the black hole, so information is lost in the process of black hole formation and decay; this information loss is inconsistent with the laws of quantum mechanics. Hawking argued the quantum mechanics must therefore be modified. This claim inspired much further work that showed that if quantum mechanics were not modified then the principle that physical processes are local in space-time must break down in a subtle way (there are other alternatives, but these are generally regarded as less likely). This issue is not yet decided, but string duality relates black holes to ordinary systems that do satisfy the laws of quantum mechanics, and this suggests in fact it is the principle of space-time locality, not quantum mechanics, that must be modified.

Braneworlds, Large Extra Dimensions, and Low-energy Strings

The existence of branes of various dimensionalities suggests that we might actually live on a three-brane, a brane with three space dimensions (plus time, of course). String theory still requires nine space dimensions, or ten in the D = 11 supergravity phase, so our brane would be embedded in this higher-dimensional space. The open strings that attach to the D-brane can give rise to all the particles of the Standard Model except the graviton. Thus there are string models in which everything seen in nature except for gravity is attached to a brane, while the gravitational field lines spread out in all the dimensions. This is different from the previous extra-dimensional ideas, where there are no branes, and all particles live in the full set of extra dimensions.

For extra dimensions without branes, dimensional analysis indicates that both the size of the dimensions and the string scale are near the Planck length and so remote from experiment. The situation with branes is more complicated. The size of the dimensions is not fixed by theory, and it could be very much larger. It is then important to consider the experimental limits on the size. In 1998, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali argued that, although particle accelerators probe physics down to 10^-16 cm, the extra dimensions could be much larger than this. If everything but gravity were attached to a brane, the extra dimensions would not be seen readily at accelerators but only in gravitational experiments. They would show up as a change in the gravitational force law, from the inverse square law that comes from the field lines spreading in three space dimensions, to a different behavior. Since the inverse square law is tested only down to 0.1 cm, the extra dimensions could be as large as this.

The large extra dimensions might be seen at a particle accelerator in an indirect way. If two particles moving along the brane collide with enough energy to leave the brane and move into the extra dimensions, they become undetectable, and the result is an event in which energy seems to disappear. This process could also occur in such environments as the core of a supernova, and the observation of neutrinos from supernovae actually gives a more stringent upper limit than the force law experiments, around 10^-4 cm. Arkani-Hamed, Dimopoulos, and Dvali also showed that in theories with large extra dimensions the string size is larger than the Planck length, so that string physics, and gravitational effects like microscopic black holes, might be seen at particle accelerators. There is no definite prediction yet for the size of the extra dimensions, and many theorists still expect that they are too small to be observed, but it is an exciting new possibility that is under theoretical and experimental study.

The Future of Theory and Experiment

The existence of branes of all dimensions raises the issue of whether the one-dimensional strings are truly fundamental. So also does the string-duality phase diagram, which shows that strings exist only in certain phases. These and other arguments have led string theorists to believe that the defining principle of the theory, the analog of the equivalence principle of general relativity and the uncertainty principle of quantum mechanics, has yet to be found. Many ideas are under investigation. A common theme is that physics is expected to be nonlocal, even before the Planck length. Two concepts being considered are the holographic principle, which is connected with the black hole information problem, and noncommutative geometry, which is an extension of the uncertainty principle involving only lengths and not momenta.

Experimentally, string theory does not yet make firm predictions. The part of the theory that is most likely to be accessible to accelerators is supersymmetry; large extra dimensions are a striking but less probable signature. String theory may eventually make distinctive predictions for cosmology. Finally, experience shows that as the theory is understood better, unexpected new possibilities are found.