After two or three thousand years of
Fairy tales for the naive,
Are we mature enough to
The Sistine Chapel,
And seek The Creator
Using the brains we have developed
Through time ?
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory aims to explain all types of observed elementary particles using quantum states of these strings. In addition to the particles postulated by the standard model of particle physics, string theory naturally incorporates gravity, and so is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. Besides this hypothesized role in particle physics, string theory is now widely used as a theoretical tool in physics, and has shed light on many aspects of quantum field theory and quantum gravity.
The earliest version of string theory, called bosonic string theory, incorporated only the class of particles known as bosons, although this theory developed into superstring theory, which posits that a connection (a "supersymmetry") exists between bosons and the class of particles called fermions. String theory requires the existence of extra spatial dimensions for its mathematical consistency. In realistic physical models constructed from string theory, these extra dimensions are typically compactified to extremely small scales.
String theory was first studied in the late 1960s as a theory of the strong nuclear force before being abandoned in favor of the theory of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it an outstanding candidate for a quantum theory of gravity. After five consistent versions of string theory were developed, it was realized in the mid-1990s that these theories could be obtained as different limits of a conjectured 11-dimensional theory called M-theory.
Many theoretical physicists (including Stephen Hawking, Edward Witten, and Juan Maldacena) believe that string theory is a step towards the correct fundamental description of nature. This is because string theory allows for the consistent combination of quantum field theory and general relativity, agrees with general insights in quantum gravity such as the holographic principle and black hole thermodynamics, and has passed many non-trivial checks of its internal consistency. According to Hawking, "M-theory is the only candidate for a complete theory of the universe." Other physicists, such as Richard Feynman, Roger Penrose, and Sheldon Lee Glashow, have criticized string theory for not providing novel experimental predictions at accessible energy scales and say that it is a failure as a theory of everything.
The starting point for string theory is the idea that the point-like particles of elementary particle physics can also be modeled as one-dimensional objects called strings. According to string theory, strings can oscillate in many ways. On distance scales larger than the string radius, each oscillation mode gives rise to a different species of particle, with its mass, charge, and other properties determined by the string's dynamics. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles. An analogy for strings' modes of vibration is a guitar string's production of multiple distinct musical notes. In this analogy, different notes correspond to different particles.
In string theory, one of the modes of oscillation of the string corresponds to a massless, spin-2 particle. Such a particle is called a graviton since it mediates a force which has the properties of gravity. Since string theory is believed to be a mathematically consistent quantum mechanical theory, the existence of this graviton state implies that string theory is a theory of quantum gravity.
String theory includes both open strings, which have two distinct endpoints, and closed strings, which form a complete loop. The two types of string behave in slightly different ways, yielding different particle types. For example, all string theories have closed string graviton modes, but only open strings can correspond to the particles known as photons. Because the two ends of an open string can always meet and connect, forming a closed string, all string theories contain closed strings.
The earliest string model, the bosonic string, incorporated only the class of particles known as bosons. This model describes, at low enough energies, a quantum gravity theory, which also includes (if open strings are incorporated as well) gauge bosons such as the photon. However, this model has problems. What is most significant is that the theory has a fundamental instability, believed to result in the decay (at least partially) of spacetime itself. In addition, as the name implies, the spectrum of particles contains only bosons, particles which, like the photon, obey particular rules of behavior. Roughly speaking, bosons are the constituents of radiation, but not of matter, which is made of fermions. Investigating how a string theory may include fermions led to the invention of supersymmetry, a mathematical relation between bosons and fermions. String theories that include fermionic vibrations are now known as superstring theories; several kinds have been described, but all are now thought to be different limits of a theory called M-theory.
Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it fully describes our universe, making it a theory of everything. One of the goals of current research in string theory is to find a solution of the theory that is quantitatively identical with the standard model, with a small cosmological constant, containing dark matter and a plausible mechanism for cosmic inflation. It is not yet known whether string theory has such a solution, nor is it known how much freedom the theory allows to choose the details.
One of the challenges of string theory is that the full theory does not yet have a satisfactory definition in all circumstances. The scattering of strings is most straightforwardly defined using the techniques of perturbation theory, but it is not known in general how to define string theory nonperturbatively. It is also not clear as to whether there is any principle by which string theory selects its vacuum state, the spacetime configuration that determines the properties of our universe (see string theory landscape).
The motion of a point-like particle can be described by drawing a graph of its position with respect to time. The resulting picture depicts the worldline of the particle in spacetime. In an analogous way, one can draw a graph depicting the progress of a string as time passes. The string, which looks like a small line by itself, will sweep out a two-dimensional surface known as the worldsheet. The different string modes (giving rise to different particles, such as the photon or graviton) appear as waves on this surface.
A closed string looks like a small loop, so its worldsheet will look like a pipe. An open string looks like a segment with two endpoints, so its worldsheet will look like a strip. In a more mathematical language, these are both Riemann surfaces, the strip having a boundary and the pipe none.
Strings can join and split. This is reflected by the form of their worldsheet, or more precisely, by its topology. For example, if a closed string splits, its worldsheet will look like a single pipe splitting into two pipes. This topology is often referred to as a pair of pants (see drawing at right). If a closed string splits and its two parts later reconnect, its worldsheet will look like a single pipe splitting to two and then reconnecting, which also looks like a torus connected to two pipes (one representing the incoming string, and the other representing the outgoing one). An open string doing the same thing will have a worldsheet that looks like an annulus connected to two strips.
In quantum mechanics, one computes the probability for a point particle to propagate from one point to another by summing certain quantities called probability amplitudes. Each amplitude is associated with a different worldline of the particle. This process of summing amplitudes over all possible worldlines is called path integration. In string theory, one computes probabilities in a similar way, by summing quantities associated with the worldsheets joining an initial string configuration to a final configuration. It is in this sense that string theory extends quantum field theory, replacing point particles by strings. As in quantum field theory, the classical behavior of fields is determined by an action functional, which in string theory can be either the Nambu–Goto action or the Polyakov action.
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.
Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A p-brane sweeps out a (p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane.
In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to the fact that we impose a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory.
Branes are also frequently studied from a purely mathematical point of view since they are related to subjects such as homological mirror symmetry and noncommutative geometry. Mathematically, branes may be represented as objects of certain categories, such as the derived category of coherent sheaves on a Calabi–Yau manifold, or the Fukaya category.
In physics, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
In addition to providing a candidate for a theory of everything, string theory provides many examples of dualities between different physical theories and can therefore be used as a tool for understanding the relationships between these theories.
S-, T-, and U-duality
These are dualities between string theories which relate seemingly different quantities. Large and small distance scales, as well as strong and weak coupling strengths, are quantities that have always marked very distinct limits of behavior of a physical system in both classical and quantum physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related. T-duality relates the large and small distance scales between string theories, whereas S-duality relates strong and weak coupling strengths between string theories. U-duality links T-duality and S-duality.
Before the 1990s, string theorists believed there were five distinct superstring theories: type I, type IIA, type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8). The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low energy limit, with ten spacetime dimensions compactified down to four, matched the physics observed in our world today. It is now believed that this picture was incorrect and that the five superstring theories are related to one another by the dualities described above. The existence of these dualities suggests that the five string theories are in fact special cases of a more fundamental theory called M-theory.
An intriguing feature of string theory is that it predicts extra dimensions. In classical string theory the number of dimensions is not fixed by any consistency criterion. However, to make a consistent quantum theory, string theory is required to live in a spacetime of the so-called "critical dimension": we must have 26 spacetime dimensions for the bosonic string and 10 for the superstring. This is necessary to ensure the vanishing of the conformal anomaly of the worldsheet conformal field theory. Modern understanding indicates that there exist less trivial ways of satisfying this criterion. Cosmological solutions exist in a wider variety of dimensionalities, and these different dimensions are related by dynamical transitions. The dimensions are more precisely different values of the "effective central charge", a count of degrees of freedom that reduces to dimensionality in weakly curved regimes.
One such theory is the 11-dimensional M-theory, which requires spacetime to have eleven dimensions, as opposed to the usual three spatial dimensions and the fourth dimension of time. The original string theories from the 1980s describe special cases of M-theory where the eleventh dimension is a very small circle or a line, and if these formulations are considered as fundamental, then string theory requires ten dimensions. But the theory also describes universes like ours, with four observable spacetime dimensions, as well as universes with up to 10 flat space dimensions, and also cases where the position in some of the dimensions is described by a complex number rather than a real number. The notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances.
Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions manually and arbitrarily, and this number is fixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. In technical terms, this happens because a gauge anomaly exists for every separate number of predicted dimensions, and the gauge anomaly can be counteracted by including nontrivial potential energy into equations to solve motion. Furthermore, the absence of potential energy in the "critical dimension" explains why flat spacetime solutions are possible.
This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken gauge symmetry) must be massless. The mass of the photon that is predicted by string theory depends on the energy of the string mode that represents the photon. This energy includes a contribution from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions, since for a larger number of dimensions there are more possible fluctuations in the string position. Therefore, the photon in flat spacetime will be massless—and the theory consistent—only for a particular number of dimensions. When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). The subset of X is equal to the relation of photon fluctuations in a linear dimension. Flat space string theories are 26-dimensional in the bosonic case, while superstring and M-theories turn out to involve 10 or 11 dimensions for flat solutions. In bosonic string theories, the 26 dimensions come from the Polyakov equation. Starting from any dimension greater than four, it is necessary to consider how these are reduced to four-dimensional spacetime.
Two ways have been proposed to resolve this apparent contradiction. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable by present-day experiments.
To retain a high degree of supersymmetry, these compactification spaces must be very special, as reflected in their holonomy. A 6-dimensional manifold must have SU(3) structure, a particular case (torsionless) of this being SU(3) holonomy, making it a Calabi–Yau space, and a 7-dimensional manifold must have G2 structure, with G2 holonomy again being a specific, simple, case. Such spaces have been studied in attempts to relate string theory to the 4-dimensional Standard Model, in part due to the computational simplicity afforded by the assumption of supersymmetry. More recently, progress has been made constructing more realistic compactifications without the degree of symmetry of Calabi–Yau or G2 manifolds.
A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from sufficient distance, it appears to have only one dimension, its length. Indeed, think of a ball just small enough to enter the hose. Throwing such a ball inside the hose, the ball would move more or less in one dimension; in any experiment we make by throwing such balls in the hose, the only important movement will be one-dimensional, that is, along the hose. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions (and a fly flying in it would move in three dimensions). This "extra dimension" is only visible within a relatively close range to the hose, or if one "throws in" small enough objects. Similarly, the extra compact dimensions are only "visible" at extremely small distances, or by experimenting with particles with extremely small wavelengths (of the order of the compact dimension's radius), which in quantum mechanics means very high energies (see wave–particle duality).
Another possibility is that we are "stuck" in a 3+1 dimensional (three spatial dimensions plus one time dimension) subspace of the full universe. Properly localized matter and Yang–Mills gauge fields will typically exist if the sub-spacetime is an exceptional set of the larger universe. These "exceptional sets" are ubiquitous in Calabi–Yau n-folds and may be described as subspaces without local deformations, akin to a crease in a sheet of paper or a crack in a crystal, the neighborhood of which is markedly different from the exceptional subspace itself. However, until the work of Randall and Sundrum, it was not known that gravity can be properly localized to a sub-spacetime. In addition, spacetime may be stratified, containing strata of various dimensions, allowing us to inhabit the 3+1-dimensional stratum—such geometries occur naturally in Calabi–Yau compactifications. Such sub-spacetimes are D-branes, hence such models are known as brane-world scenarios.
In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza's early work demonstrated that general relativity in five dimensions actually predicts the existence of electromagnetism. However, because of the nature of Calabi–Yau manifolds, no new forces appear from the small dimensions, but their shape has a profound effect on how the forces between the strings appear in our four-dimensional universe. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.