# String theory

Template:String-theory
**String theory** is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics. For this reason, string theories are able to avoid problems associated with the presence of pointlike particles in a physical theory. Study of string theories has revealed that they require not just strings but other objects, variously including points, membranes, and higher-dimensional objects.

Interest in string theory is driven largely by the hope that it will prove to be a theory of everything. It is a possible solution of the quantum gravity problem, and in addition to gravity it can naturally describe interactions similar to electromagnetism and the other forces of nature. Superstring theories include fermions, the building blocks of matter. It is not yet known whether string theory is able to describe a universe with the precise collection of forces and matter that we observe, nor how much freedom to choose those details the theory will allow. No string theory has yet made falsifiable predictions that would allow it to be experimentally tested.

Work on string theory has led to advances in mathematics, mainly in algebraic geometry. String theory has also led to insight into supersymmetric gauge theories, which will be tested at the new Large Hadron Collider experiment.

## Contents

## History

String theory was originally invented to explain peculiarities of hadron behavior. In particle-accelerator experiments, physicists observed that the angular momentum of a hadron is exactly proportional to the square of its energy. No simple model of the hadron, such as picturing it as a set of smaller particles held together by spring-like forces, was able to explain these relationships. In order to account for these "Regge trajectories," physicists turned to a model where each hadron was in fact a rotating string, moving in accordance with Einstein's special theory of relativity. This led to the development of bosonic string theory, which is still the version first taught to many students. (The original need for a viable theory of hadrons has been fulfilled by quantum chromodynamics, the theory of quarks and their interactions. It is now hoped that string theory or some descendant of it will provide a fundamental understanding of the quarks themselves.)

Bosonic string theory is formulated in terms of the Nambu-Goto action, a mathematical quantity which can be used to predict how strings move through space and time. By applying the ideas of quantum mechanics to the Nambu-Goto action—a procedure known as quantization—one can deduce that each string can vibrate in many different ways, and that each vibrational state appears to be a different particle. The mass the particle has, and the fashion with which it can interact, are determined by the way the string vibrates—in essence, by the "note" which the string sounds. The scale of notes, each corresponding to a different kind of particle, is termed the "spectrum" of the theory.

These early models included both *open* strings, which have two distinct endpoints, and *closed* strings, where the endpoints are joined to make a complete loop. The two types of string behave in slightly different ways, yielding two spectra. Not all modern string theories use both types; some incorporate only the closed variety.

However, the bosonic theory has problems. Most importantly, as the name implies, the spectrum of particles contains only bosons, particles like the photon which obey particular rules of behavior. While bosons are a critical ingredient of the Universe, they are not its only constituents. Investigating how a string theory may include fermions in its spectrum led to supersymmetry, a mathematical relation between bosons and fermions which is now an independent area of study. String theories which include fermionic vibrations are now known as superstring theories; several different kinds have been described.

In the 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of an unknown 11-dimensional theory called M-theory. These discoveries sparked the second superstring revolution. (Several meanings of the "M" have been proposed; physicists joke that the true meaning will only be chosen when the theory is finally understood.)

Many recent developments in the field relate to D-branes, objects which physicists discovered must also be included in any theory which includes open strings of the super string theory.

## Basic properties

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories discovered by adding supersymmetry. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'.

While understanding the details of string and superstring theories requires considerable mathematical sophistication, some qualitative properties of quantum strings can be understood in a fairly intuitive 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's uncertainty 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". Consequently, the minimum size of a string must be related to the string tension.

## Extra dimensions

One intriguing feature of string theory is that it predicts the number of dimensions which the universe should possess. 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 hand". The first person to add a fifth dimension to Einstein's four was the German mathematician Theodor Kaluza in 1919. The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926.

Instead, string theory allows one to compute the number of spacetime dimensions from first principles. Technically, this happens because Lorentz invariance can only be satisfied in a certain number of dimensions. This is roughly like saying that if we measure the distance between two points, then rotate our observer by some angle and measure again, the observed distance only stays the same if the universe has a particular number of dimensions.

The only problem is that when the calculation is done, the universe's dimensionality is not four as one may expect (three axes of space and one of time), but twenty-six. More precisely, bosonic string theories are 26-dimensional, while superstring and M-theories turn out to involve 10 or 11 dimensions. In Rambu's Theory, the 26 dimensions comes from a famous equation; [1-(D-2)/24]=0

However, these models appear to contradict observed phenomena. Physicists usually solve this problem in one of two different ways. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable in our phenomenal experience. We achieve the 6-dimensional model's resolution with Calabi-Yau spaces. In 7 dimensions, they are termed G_{2} manifolds. Essentially these extra dimensions are "compactified" by causing them to loop back upon themselves.

A standard analogy for this is to consider multidimensional space as a garden hose. If we view the hose from a sufficient distance, it appears to have only one dimension, its length. This is akin to the 4 macroscopic dimensions we are accustomed to dealing with every day. If, however, one approaches the hose, one discovers that it contains a second dimension, its circumference. This "extra dimension" is only visible within a relatively close range to the hose, just as the extra dimensions of the Calabi-Yau space are only visible at extremely small distances, and thus are not easily detected.

(Of course, everyday garden hoses exist in three spatial dimensions, but for the purpose of the analogy, we neglect its thickness and consider only motion on the *surface* of the hose. A point on the hose's surface can be specified by two numbers, a distance along the hose and a distance along the circumference, just as points on the Earth's surface can be uniquely specified by latitude and longitude. In either case, we say that the object has two spatial dimensions. Like the Earth, garden hoses have an interior, a region that requires an extra dimension; however, unlike the Earth, a Calabi-Yau space has no interior.)

Another possibility is that we are stuck in a 3+1 dimensional subspace of the full universe, where the "3+1" reminds us that time is a different kind of dimension than space. Because it involves mathematical objects called D-branes, this is known as a braneworld theory.

In either case, gravity acting in the hidden dimensions produces other non-gravitational forces such as electromagnetism. 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.

## Problems

String theory remains to be verified. No version of string theory has yet made a prediction which differs from those made by other theories—at least, not in a way that could be checked by a currently feasible experiment. In this sense, string theory is still in a "larval stage": it possesses many features of mathematical interest, and it may yet become supremely important in our understanding of the Universe, but it requires further developments before it is accepted or falsified. Since string theory may not be tested in the foreseeable future, some scientists have asked if it even deserves to be called a scientific theory: it is not yet a falsifiable theory in the sense of Popper.

It is by no means the only theory currently being developed which suffers from this difficulty; any new development can pass through a stage of uncertainty before it becomes conclusively accepted or rejected. As Richard Feynman noted in *The Character of Physical Law,* the key test of a scientific theory is whether its consequences agree with the measurements we take in experiments. It does not matter who invented the theory, "what his name is", or even how aesthetically appealing the theory may be—"if it disagrees with experiment, it's wrong." (Of course, there are subsidiary issues: something may have gone wrong with the experiment, or perhaps the person computing the consequences of the theory made a mistake. All these possibilities must be checked, which may take a considerable time.) These developments may be in the theory itself, such as new methods of performing calculations and deriving predictions, or they may be advances in experimental science, which make formerly ungraspable quantities measurable.

Human beings do not have the technology to observe strings (which are said to be roughly of Planck length, about 10^{-35} meters across). Eventually, we may be able to observe strings in a meaningful way, or at least to gain substantial insight by observing cosmological phenomena which may elucidate string physics.

In the early 2000s, string theorists revived interest in an older concept, the cosmic string. Originally discussed in the 1980s, cosmic strings are a different type of object than the entities of superstring theories. For several years, *cosmic* strings were a popular model for explaining various cosmological phenomena, such as the way galaxies formed in the early Universe. However, further experiments — and in particular the detailed measurements of the cosmic microwave background — failed to support the cosmic-string model's predictions, and the cosmic string fell out of vogue. If such objects did exist, they must be few and far between. Several years later, it was pointed out that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size. Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. Furthermore, modern superstring theories offer other objects which could feasibly resemble cosmic strings, such as highly elongated one-dimensional D-branes (known as "D-strings"). As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". Older proposals for detecting cosmic strings could now be used to investigate superstring theory. For example, astronomers have also detected a few cases of what might be string-induced gravitational lensing.

Superstrings, D-strings or other stringy objects stretched to intergalactic scales would radiate gravitational waves, which could presumably be detected using experiments like LIGO. They might also cause slight irregularities in the cosmic microwave background, too subtle to have been detected yet but possibly within the realm of future observability.

While intriguing, these cosmological proposals fall short in one respect: testing a theory requires that the test be capable, at least in principle, of *falsifying* the theory. For example, if observing the Sun during a solar eclipse had *not* shown that the Sun's gravity deflected light, Einstein's general relativity theory would have been proven wrong. (Barring, of course, the chance of experimental error.) Not finding cosmic strings would not demonstrate that string theory is fundamentally wrong — merely that the particular idea of highly stretched strings acting "cosmic" is in error. While many measurements could in principle be made that would suggest that string theory is on the right track, scientists have not at present devised a stringent "test".

On a more mathematical level, another problem is that, like quantum field theory, much of string theory is still only formulated perturbatively (*i.e.,* as a series of approximations rather than as an exact solution). Although nonperturbative techniques have progressed considerably — including conjectured complete definitions in space-times satisfying certain asymptotics — a full nonperturbative definition of the theory is still lacking.

## See also

- List of string theory topics
- Kaluza-Klein theory
- Superstring theory
- M-theory
- F-theory
- Conformal field theory
- Graviton
- Quantum gravity
- Loop quantum gravity
- Supergravity
- Supersymmetry
- Theory of everything

## References and further reading

### Popular books and articles

- Davies, Paul, and Julian R. Brown.
*Superstrings: A Theory of Everything?*. Cambridge University Press (1988). ISBN 0-521-43775-X. - Greene, Brian,
*The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory*, W.W. Norton & Company; Reissue edition (2003) ISBN 0-393-05858-1. - Gribbin, John,
*The Search for Superstrings, Symmetry, and the Theory of Everything*. London, Great Britain: Little Brown and Company (1998). ISBN 0-316-32975-4. - Kaku, Michio,
*Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension.*New York, Oxford University Press (1994). ISBN 0-195-08514-0. - Penrose, Roger,
*The Road to Reality ; A Complete Guide to the Laws of the Universe*, Jonathan Cape (2004), ISBN 0-224-04447-8. - Witten, Edward.
*The Universe on a String*. Astronomy magazine (June 2002). An easy divulgative article for everybody outside physics wanting to understand the very basics. Written by the expert himself.

### Textbooks

- Green, Michael, John Schwarz and Edward Witten,
*Superstring theory*, Cambridge University Press (1987). The original textbook.- Vol. 1: Introduction, ISBN 0-521-35752-7.
- Vol. 2: Loop amplitudes, anomalies and phenomenology, ISBN 0-521-35753-5.

- Johnson, Clifford,
*D-branes*, Cambridge University Press (2003). ISBN 0-521-80912-6. - Polchinski, Joseph,
*String Theory*, Cambridge University Press (1998). A modern textbook.- Vol. 1: An introduction to the bosonic string, ISBN 0-521-63303-6.
- Vol. 2: Superstring theory and beyond, ISBN 0-521-63304-4.

- Zwiebach, Barton.
*A First Course in String Theory.*Cambridge University Press (2004). ISBN 0-521-83143-1. Errata are available online.

### External links

- Superstringtheory.com - The "Official String Theory Web Site", created by Patricia Schwarz.
- The Elegant Universe - A Three-Hour Miniseries with Brian Greene by NOVA (original PBS Broadcast Dates: October 28th, 8-10 PM and November 4, 8-9 PM, 2003). Various images, texts, videos and animations explaining string theory.
- Beyond String Theory - An ongoing project by Jan Troost, a string physicist working for the French CNRS.
- Superstrings! String Theory Home Page - Online tutorial.
- Superstrings - Michael Green on string theory in a Scientific American article, September 1986.
- The Symphony of Everything - A short interactive introduction to string theory.
- SCI.physics.STRINGS - The home page of a newsgroup dedicated to string theory.
- Resource Letter - A guide to the string theory literature.
- The Nth Dimension - A comprehensive compilation of materials concerning string theory. Created by classmates in the four-month project.
- "Cosmic strings reborn?" - A talk given by Tom Kibble in September 2004.
- Ed Witten's KITP Public Lecture - Slides and audio from an Ed Witten lecture where he introduces string theory and discusses its challenges.
- The Reference Frame A blog supporting string theory
- Not Even Wrong A blog critical of string theory
- Is string theory even wrong? - A criticism of string theory.

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