String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything (TOE), a self-contained mathematical model
Theory Of Quantum Mechanics
Stephen Hawking, Edward Witten, Juan Maldacena and Leonard Susskind
Supersymmetry- Supergravity- Quantum gravity
String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but made up of 1-dimensional strings. These strings can oscillate, giving the observed particles their flavor, charge, mass and spin. Among the modes of oscillation of the string is a massless, spin-two state -- a graviton. The existence of this graviton state and the fact that the equations describing string theory include Einstein's equations for general relativity mean that string theory is a quantum theory of gravity. Since string theory is widely believed to be mathematically consistent, many hope that it fully describes our universe, making it a theory of everything. String theory is known to contain configurations that describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields. Other configurations have different values of the cosmological constant, and are metastable but long-lived. This leads many to believe that there is at least one metastable solution 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 how much freedom the theory allows to choose the details.
String theories also include objects other than strings, called branes. The word brane, derived from "membrane", refers to a variety of interrelated objects, such as D-branes, black p-branes and Neveu–Schwarz 5-branes. These are extended objects that are charged sources for differential form generalizations of the vector potential electromagnetic field. These objects are related to one another by a variety of dualities. Black hole-like black p-branes are identified with D-branes, which are endpoints for strings, and this identification is called Gauge-gravity duality. Research on this equivalence has led to new insights on quantum chromodynamics, the fundamental theory of the strong nuclear force. The strings make closed loops unless they encounter D-branes, where they can open up into 1-dimensional lines. The endpoints of the string cannot break off the D-brane, but they can slide around on it.
The full theory does not yet have a satisfactory definition in all circumstances, since the scattering of strings is most straightforwardly defined by a perturbation theory. The complete quantum mechanics of high dimensional branes is not easily defined, and the behavior of string theory in cosmological settings (time-dependent backgrounds) is not fully worked out. 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).
String theory can be formulated in terms of an action principle, either the Nambu-Goto actionor the Polyakov action, which describe how strings propagate through space and time. In the absence of external interactions, string dynamics are governed by tension and kinetic energy, which combine to produce oscillations. The quantum mechanics of strings implies these oscillations exist in discrete vibrational modes, the spectrum of the theory.
On distance scales larger than the string radius, each oscillation mode behaves as a different species of particle, with its mass, spin and charge 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 but distinct musical notes. In the analogy, different notes correspond to different particles. One difference is the guitar string exists in 3 dimensions, so that there are only two dimensions transverse to the string. Fundamental strings exist in 9 dimensions and the strings can vibrate in any direction, meaning that the spectrum of vibrational modes is much richer.
String theory includes both open strings, which have two distinct endpoints, and closedstrings making a complete loop. The two types of string behave in slightly different ways, yielding two different spectra. For example, in most string theories one of the closed string modes is the graviton, and one of the open string modes is the photon. Because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings.
The earliest string model, the bosonic string, incorporated only bosonic degrees of freedom. This model describes, in low enough energies, a quantum gravity theory, which also includes (if open strings are incorporated as well) gauge fields such as the photon (or, in more general terms, any gauge theory). 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. In broad terms, bosons are the constituents of radiation, but not of matter, which is made of fermions. Investigating how a string theory may include fermions in its spectrum 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 different kinds have been described, but all are now thought to be different limits of M-theory.
Some qualitative properties of quantum strings can be understood in a fairly simple fashion. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg'suncertainty principle. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". As a consequence, the minimum size of a string is related to the string tension.
A point-like particle's motion may be described by drawing a graph of its position (in one or two dimensions of space) against time. The resulting picture depicts the worldline of the particle (its 'history') in spacetime. By analogy, a similar graph depicting the progress of a stringas time passes by can be obtained; the string (a one-dimensional object — a small line — by itself) will trace out a surface (a two-dimensional manifold), known as the worldsheet. The different string modes (representing different particles, such as photon or graviton) are surface waves on this manifold.
A closed string looks like a small loop, so its worldsheet will look like a pipe or, in more general terms, a Riemann surface (a two-dimensional oriented manifold) with no boundaries (i.e., no edge). An open string looks like a short line, so its worldsheet will look like a strip or, in more general terms, a Riemann surface with a boundary.
Interaction in the subatomic world: world lines of point-likeparticles in the Standard Model or a world sheet swept up by closed strings in string theory
Strings can split and connect. This is reflected by the form of their worldsheet (in more accurate terms, by its topology). For example, if a closed string splits, its worldsheet will look like a single pipe splitting (or connected) to two pipes (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 ingoing string, and the other — the outgoing one). An open string doing the same thing will have its worldsheet looking like a ring connected to two strips.
Note that the process of a string splitting (or strings connecting) is a global process of the worldsheet, not a local one: Locally, the worldsheet looks the same everywhere, and it is not possible to determine a single point on the worldsheet where the splitting occurs. Therefore, these processes are an integral part of the theory, and are described by the same dynamics that controls the string modes.
In some string theories (namely, closed strings in Type I and some versions of the bosonic string), strings can split and reconnect in an opposite orientation (as in a Möbius strip or a Klein bottle). These theories are called unoriented. In formal terms, the worldsheet in these theories is a non-orientable surface.
Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed 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 compactifieddown 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 connected to one another as if they are each a special case of some more fundamental theory (thought to be M-theory). These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
These dualities link quantities that were also thought to be separate. 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 field theory and quantum particle 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.
|Bosonic||26||Only bosons, no fermions, meaning only forces, no matter, with both open and closed strings; major flaw: aparticle with imaginary mass, called the tachyon, representing an instability in the theory.|
|I||10||Supersymmetry between forces and matter, with both open and closed strings; no tachyon; group symmetry isSO(32)|
|IIA||10||Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; masslessfermions are non-chiral|
|IIB||10||Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; masslessfermions are chiral|
|HO||10||Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is SO(32)|
|HE||10||Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is E8×E8|
Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional spacetime (called the bulk), while open strings have their ends attached to D-branes, which are membranes of lower dimensionality (their dimension is odd — 1, 3, 5, 7 or 9 — in type IIA and even — 0, 2, 4, 6 or 8 — in type IIB, including the time direction).
Number of dimensionsEdit
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, in order 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 criteria. 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 not described by a real number, but by a completely different type of mathematical quantity. So 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 "by both hands", 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.
Compact dimensionsEditCalabi–Yau manifold (3D projection)
Two different ways have been proposed to resolve this apparent contradiction. The first is tocompactify 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 a 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-space-time 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 too can be properly localized to a sub-spacetime. In addition, spacetime may be stratified, containing strata of various dimensions, allowing us to inhabit a 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.
D-branesEditMain article: D-brane
Another key feature of string theory is the existence of D-branes. These are membranes of different dimensionality (anywhere from a zero dimensional membrane—which is in fact a point—and up, including 2-dimensional membranes, 3-dimensional volumes, and so on).
D-branes are defined by the fact that worldsheet boundaries are attached to them. D-branes have mass, since they emit and absorb closed strings that describe gravitons, and — in superstring theories —charge as well, since they couple to open strings that describe gauge interactions.
From the point of view of open strings, D-branes are objects to which the ends of open strings are attached. The open strings attached to a D-brane are said to "live" on it, and they give rise to gauge theories "living" on it (since one of the open string modes is a gauge boson such as the photon). In the case of one D-brane there will be one type of a gauge boson and we will have an Abelian gauge theory (with the gauge boson being the photon). If there are multiple parallel D-branes there will be multiple types of gauge bosons, giving rise to a non-Abeliangauge theory.
D-branes are thus gravitational sources, on which a gauge theory "lives". This gauge theory is coupled to gravity (which is said to exist in thebulk), so that normally each of these two different viewpoints is incomplete.
Testability and experimental predictionsEdit
|Is there a string theory vacuum that exactly describes everything in our universe? Is it uniquely determined by low energy data?|
Several major difficulties complicate efforts to test string theory. The most significant is the extremely small size of the Planck length, which is expected to be close to the string length (the characteristic size of a string, where strings become easily distinguishable from particles). Another issue is the huge number of metastable vacua of string theory, which might be sufficiently diverse to accommodate almost any phenomena we might observe at lower energies.
On the other hand, all string theory models are quantum mechanical, Lorentz invariant, unitary, and contain Einstein's General Relativityas a low energy limit. Therefore, to falsify string theory, it would suffice to falsify quantum mechanics, fundamental Lorentz invariance,or general relativity. Other potential falsifications of string theory would include the confirmation of a model from theswamplandor observations of positive curvature in cosmology. However, these falsifications do not necessarily correspond to predictions which are unique to string theory, and finding a way to experimentally verify string theory via unique predictions remains a major challenge.
One such unique prediction of string theory is the existence of string harmonics: at sufficiently high energies, the string-like nature of particles would become obvious. There should be heavier copies of all particles, corresponding to higher vibrational harmonics of the string. But it is not clear how high these energies are. In most conventional string models they would be not far below the Planck energy, around 1014 times higher than the energies accessible in the newest particle accelerator, the LHC, making this prediction impossible to test with any particle accelerator in the foreseeable future. However, in models with large extra dimensions they could be potentially be produced at the LHC or at energies not far above its reach.
String theory as currently understood makes a series of predictions for the structure of the universe at the largest scales. Many phases in string theory have very large, positive vacuum energy . Regions of the universe that are in such a phase will inflate exponentially rapidly in a process known as eternal inflation. As such, the theory predicts that most of the universe is very rapidly expanding. However, these expanding phases are not stable, and can decay via the nucleation of bubbles of lower vacuum energy. Since our local region of the universe is not very rapidly expanding, string theory predicts we are inside such a bubble. The spatial curvature of the "universe" inside the bubbles that form by this process is negative, a testable prediction . Moreover, other bubbles will eventually form in the parent vacuum outside the bubble and collide with it. These collisions lead to potentially observable imprints on cosmology . However, it is possible that neither of these will be observed if the spatial curvature is too small and the collisions are too rare.
Under certain circumstances, fundamental strings produced at or near the end of inflation can be "stretched" to astronomical proportions. These cosmic strings could be observed in various ways, for instance by their gravitational lensing effects. However, certain field theories also predict cosmic strings arising from topological defects in the field configuration.
Strength of gravityEdit
Theories with extra dimensions predict that the strength of gravity increases much more rapidly at small distances than is the case in 3 dimensions (where it increase as r-2). Depending on the size of the dimensions, this could lead to phenomena such as the production of micro-black holes at the LHC, or be detected in microgravity experiments.
String theory was originally proposed as a theory of hadrons, and its study has led to new insights on quantum chromodynamics, a gauge theory, which is the fundamental theory of the strong nuclear force. To this end, it is hoped that a gravitational theory dual to quantum chromodynamics will be found.
A mathematical technique from string theory (the AdS/CFT correspondence) has been used to describe qualitative features of quark–gluon plasma behavior in relativistic heavy-ion collisions;the physics, however, is strictly that of standard quantum chromodynamics, which has been quantitatively modeled by lattice QCD methods with good results.
The discovery of supersymmetry could also be considered evidence, since it was discovered in the context of string theory, and all consistent string theories are supersymmetric. However, the absence of supersymmetric particles at energies accessible to the LHC would not necessarily disprove string theory, since the energy scale at which supersymmetry is broken could be well above the accelerator's range.
A central problem for applications is that the best-understood backgrounds of string theory preserve much of the supersymmetry of the underlying theory, which results in time-invariant spacetimes: At present, string theory cannot deal well with time-dependent, cosmological backgrounds. However, several models have been proposed to predict supersymmetry breaking, the most notable one being the KKLT model, which incorporates branes and fluxes to make a metastable compactification.
AdS/CFT relates string theory to gauge theory, and allows contact with low energy experiments in quantum chromodynamics. This type of string theory, which describes only the strong interactions, is much less controversial today than string theories of everything (although two decades ago, it was the other way around).
Coupling constant unificationEdit
Grand unification natural in string theories of everything requires that the coupling constants of the four forces meet at one point under renormalization group rescaling. This is also a falsifiable statement, but it is not restricted to string theory, but is shared by grand unified theories. The LHC will be used both for testing AdS/CFT, and to check if the electroweakstrong unification does happen as predicted.
Gauge-gravity duality is a conjectured duality between a quantum theory of gravity in certain cases and gauge theory in a lower number of dimensions. This means that each predicted phenomenon and quantity in one theory has an analogue in the other theory, with a "dictionary" translating from one theory to the other.
Description of the dualityEdit
In certain cases the gauge theory on the D-branes is decoupled from the gravity living in the bulk; thus open strings attached to the D-branes are not interacting with closed strings. Such a situation is termed a decoupling limit.
In those cases, the D-branes have two independent alternative descriptions. As discussed above, from the point of view of closed strings, the D-branes are gravitational sources, and thus we have a gravitational theory on spacetime with some background fields. From the point of view of open strings, the physics of the D-branes is described by the appropriate gauge theory. Therefore in such cases it is often conjectured that the gravitational theory on spacetime with the appropriate background fields is dual (i.e. physically equivalent) to the gauge theory on the boundary of this spacetime (since the subspace filled by the D-branes is the boundary of this spacetime). So far, this duality has not been proven in any cases, so there is also disagreement among string theorists regarding how strong the duality applies to various models.
Examples and intuitionEdit
The best known example and the first one to be studied is the duality between Type IIB superstring on AdS5 ×S5 (a product space of a five-dimensional Anti de Sitter space and a five-sphere) on one hand, and N = 4 supersymmetric Yang–Mills theory on the four-dimensional boundary of the Anti de Sitter space (either a flat four-dimensional spacetime R3,1 or a three-sphere with time S3 ×R). This is known as theAdS/CFT correspondence, a name often used for Gauge / gravity duality in general.
This duality can be thought of as follows: suppose there is a spacetime with a gravitational source, for example an extremal black hole.When particles are far away from this source, they are described by closed strings (i.e., a gravitational theory, or usually supergravity). As the particles approach the gravitational source, they can still be described by closed strings; also, they can be described by objects similar to QCD strings,which are made of gauge bosons (gluons) and other gauge theory degrees of freedom.So if one is able (in adecoupling limit) to describe the gravitational system as two separate regions — one (the bulk) far away from the source, and the other close to the source — then the latter region can also be described by a gauge theory on D-branes. This latter region (close to the source) is termed the near-horizon limit, since usually there is an event horizon around (or at) the gravitational source.
In the gravitational theory, one of the directions in spacetime is the radial direction, going from the gravitational source and away (toward the bulk). The gauge theory lives only on the D-brane itself, so it does not include the radial direction: it lives in a spacetime with one less dimension compared to the gravitational theory (in fact, it lives on a spacetime identical to the boundary of the near-horizon gravitational theory). Let us understand how the two theories are still equivalent:
The physics of the near-horizon gravitational theory involves only on-shell states (as usual in string theory), while the field theory includes alsooff-shell correlation function. The on-shell states in the near-horizon gravitational theory can be thought of as describing only particles arriving from the bulk to the near-horizon region and interacting there between themselves. In the gauge theory, these are "projected" onto the boundary, so that particles that arrive at the source from different directions will be seen in the gauge theory as (off-shell) quantum fluctuations far apart from each other, while particles arriving at the source from almost the same direction in space will be seen in the gauge theory as (off-shell) quantum fluctuations close to each other. Thus the angle between the arriving particles in the gravitational theory translates to the distance scale between quantum fluctuations in the gauge theory. The angle between arriving particles in the gravitational theory is related to the radial distance from the gravitational source at which the particles interact: The larger the angle the closer the particles have to get to the source in order to interact with each other. On the other hand, the scale of the distance between quantum fluctuations in a quantum field theory is related (inversely) to the energy scale in this theory, so small radius in the gravitational theory translates to low energy scale in the gauge theory (i.e., the IR regime of the field theory), while large radius in the gravitational theory translates to high energy scale in the gauge theory (i.e., the UV regime of the field theory).
A simple example to this principle is that if in the gravitational theory there is a setup in which the dilaton field (which determines the strength of the coupling) is decreasing with the radius, then its dual field theory will be asymptotically free, i.e. its coupling will grow weaker in high energies.