String Theory

INDEX

· What is String Theory?

· So, what the String Theory then?

· A brief theory of String Theory

· How many String Theories are there?

· Basic Properties

· Extra Dimensions

· D-Branes

· What is black Hole Entropy?

· Problems and Controversies

· Conclusion

· References

What is String Theory?

We live in a wonderfully complex universe, and we are curious about it by nature. Time and again we have wondered--- why are we here? Where did we and the world come from? What is the world made of? It is our privilege to live in a time when enormous progress has been made towards finding some of the answers. String theory is our most recent attempt to answer the last (and part of the second) question.

So, what is the world made of? Ordinary matter is made of atoms, which are in turn made of just three basic components: electrons whirling around a nucleus composed of neutrons and protons. The electron is a truly fundamental particle (it is one of a family of particles known as leptons), but neutrons and protons are made of smaller particles, known as quarks. Quarks are, as far as we know, truly elementary.

Our current knowledge about the subatomic composition of the universe is summarized in what is known as the Standard Model of particle physics. It describes both the fundamental building blocks out of which the world is made, and the forces through which these blocks interact. There are twelve basic building blocks. Six of these are quarks--- they go by the interesting names of up, down, charm, strange, bottom and top. (A proton, for instance, is made of two up quarks and one down quark.) The other six are leptons--- these include the electron and its two heavier siblings, the muon and the tauon, as well as three neutrinos.

There are four fundamental forces in the universe: gravity, electromagnetism, and the weak and strong nuclear forces. Each of these is produced by fundamental particles that act as carriers of the force. The most familiar of these is the photon, a particle of light, which is the mediator of electromagnetic forces. (This means that, for instance, a magnet attracts a nail because both objects exchange photons.) The graviton is the particle associated with gravity. The strong force is carried by eight particles known as gluons. Finally, the weak force is transmitted by three particles, the W+, the W- , and the Z.

The behavior of all of these particles and forces is described with impeccable precision by the Standard Model, with one notable exception: gravity. For technical reasons, the gravitational force, the most familiar in our every day lives, has proven very difficult to describe microscopically. This has been for many years one of the most important problems in theoretical physics-- to formulate a quantum theory of gravity.

In the last few decades, string theory has emerged as the most promising candidate for a microscopic theory of gravity. And it is infinitely more ambitious than that: it attempts to provide a complete, unified, and consistent description of the fundamental structure of our universe. (For this reason it is sometimes, quite arrogantly, called a 'Theory of Everything').

The essential idea behind string theory is this: all of the different 'fundamental ' particles of the Standard Model are really just different manifestations of one basic object: a string. How can that be? Well, we would ordinarily picture an electron, for instance, as a point with no internal structure. A point cannot do anything but move. But, if string theory is correct, then under an extremely powerful 'microscope' we would realize that the electron is not really a point, but a tiny loop of string. A string can do something aside from moving--- it can oscillate in different ways. If it oscillates a certain way, then from a distance, unable to tell it is really a string, we see an electron. But if it oscillates some other way, well, then we call it a photon, or a quark, or a ... you get the idea. So, if string theory is correct, the entire world is made of strings!

Perhaps the most remarkable thing about string theory is that such a simple idea works--- it is possible to derive (an extension of) the Standard Model (which has been verified experimentally with incredible precision) from a theory of strings. But it should also be said that, to date, there is no direct experimental evidence that string theory itself is the correct description of Nature. This is mostly due to the fact that string theory is still under development. We know bits and pieces of it, but we do not yet see the whole picture, and we are therefore unable to make definite predictions. In recent years many exciting developments have taken place, radically improving our understanding of what the theory is.

So, what the String Theory then?

Think of a guitar string that has been tuned by stretching the string under tension across the guitar. Depending on how the string is plucked and how much tension is in the string, different musical notes will be created by the string. These musical notes could be said to be excitation modes of that guitar string under tension.

In a similar manner, in string theory, the elementary particles we observe in particle accelerators could be thought of as the "musical notes" or excitation modes of elementary strings.

In string theory, as in guitar playing, the string must be stretched under tension in order to become excited. However, the strings in string theory are floating in spacetime, they aren't tied down to a guitar. Nonetheless, they have tension.

String theories are classified according to whether or not the strings are required to be closed loops, and whether or not the particle spectrum includes fermions. In order to include fermions in string theory, there must be a special kind of symmetry called supersymmetry, which means for every boson (particle that transmits a force) there is a corresponding fermion (particle that makes up matter). So supersymmetry relates the particles that transmit forces to the particles that make up matter.

Fig: -There are two basic types of string theories: those with closed string loops that can break into open strings, shown above, and those with closed string loops that can't break into open strings.

A Brief History of String Theory

Here is a very brief outline of the development of string theory, the details of which will eventually fill many large volumes written by many people directly and indirectly involved in this rich and fascinating story.

1921
	Electromagnetism can be derived from gravity in a unified theory if there are four space dimensions instead of three, and the fourth is curled into a tiny circle. Kaluza and Klein made this discovery independently of each other.
1970
	Three particle theorists independently realize that the dual theories developed in 1968 to describe the particle spectrum also describe the quantum mechanics of oscillating strings. This marks the official birth of string theory.
1971
	Supersymmetry is invented in two contexts at once: in ordinary particle field theory and as a consequence of introducing fermions into string theory. It holds the promise of resolving many problems in particle theory, but requires equal numbers of fermions and bosons, so it cannot be an exact symmetry of Nature.
1974
	String theory using closed strings fails to describe hadronic physics because the spin 2 excitation has zero mass. Oops, that makes it an ideal candidate for the missing theory of quantum gravity!! This marks the advent of string theory as a proposed unified theory of all four observed forces in Nature.
1976
	Supersymmetry is added to gravity, making supergravity. This progress is especially important to string theory, where gravity can't be separated from the spectrum of excitations.
1980
	String theory plus supersymmetry yields an excitation spectrum that has equal numbers of fermions and bosons, showing that string theory can be made totally supersymmetric. The resulting objects are called superstrings.
1984
	This was the year for string theory! Deadly anomalies that threatened to make the theory senseless were discovered to cancel each other when the underlying symmetries in the theory belong two special groups. Finally string theory is accepted by the mainstream physics community as an actual candidate theory uniting quantum mechanics, particle physics and gravity.
1991- 1995
	Interesting work on stringy black holes in higher dimensions leads to a revolution in understanding how different versions of string theory are related through duality transformations. This unlocks a surge of progress towards a deeper nonperturbative picture of string theory.
1996
	Using Einstein relativity and Hawking radiation, there were hints in the past that black holes have thermodynamic properties that need to be understood microscopically. A microscopic origin for black hole thermodynamics is finally achieved in string theory. String theory sheds amazing light on the entire perplexing subject of black hole quantum mechanics.

How many String Theories are there?

There are several ways theorists can build string theories. Start with the elementary ingredient: a wiggling tiny string. Next decide: should it be an open string or a closed string? Then ask: will I settle for only bosons ( particles that transmit forces) or will I ask for fermions, too (particles that make up matter)?

If the answer to the last question is "Bosons only, please!" then one gets bosonic string theory. If the answer is "No, I demand that matter exist!" then we wind up needing supersymmetry, which means an equal matching between bosons (particles that transmit forces) and fermions (particles that make up matter). A supersymmetric string theory is called a superstring theory. There are five kinds of superstring theories, shown in the table below: -

A Brief Table of String Theories
Type	Spacetime Dimensions	Details
Bosonic	26	Only bosons, no fermions means only forces, no matter, with both open and closed strings. Major flaw: a particle with imaginary mass, called the tachyon
I	10	Supersymmetry between forces and matter, with both open and closed strings, no tachyon, group symmetry is SO(32)
IIA	10	Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions spin both ways (nonchiral)
IIB	10	Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions only spin one way (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 x E8

Basic properties

String theory is formulated in terms of an action principle, either the Nambu-Goto action or the Polyakov action, which describes how strings move through space and time. Like springs, the strings want to contract to minimize their potential energy, but conservation of energy prevents them from disappearing, and instead they oscillate. By applying the ideas of quantum mechanics to strings it is possible to deduce the different vibrational modes of strings, and that each vibrational state appears to be a different particle. The mass of each particle, and the fashion with which it can interact, are determined by the way the string vibrates — the string can vibrate in many different modes, just like a guitar string can produce different notes. The different modes, each corresponding to a different kind of particle, make up the "spectrum" of the theory. Strings can split and combine, which would appear as particles emitting and absorbing other particles, presumably giving rise to the known interactions between particles.

String theory includes 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 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, which incorporated only bosons, 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, more generally, any Yang-Mills theory). However, this model has problems. Most importantly, the theory has a fundamental instability, believed to result in the decay (at least partially) of space-time itself. Additionally, 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 in its spectrum led to the invention of supersymmetry, a mathematical relation between bosons and fermions’

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 general relativity was 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 (see Kaluza–Klein theory).

Unlike general relativity, string theory allows one to compute the number of spacetime dimensions from first principles. Technically, this happens because, for a different number of dimensions, the theory has a gauge anomaly. This can be understood by noting that in a consistent theory which includes a photon (technically, a particle carrying a force related to an unbroken gauge symmetry), it must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy includes a contribution from 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 will be massless — and the theory consistent — only for a particular number of dimensions.^[4]

When the calculation is done, the universe's dimensionality is not four as one may expect (three axes of space and one of time). Bosonic string theories are 26-dimensional, while superstring and M-theories turn out to involve 10 or 11 dimensions. In bosonic string theories, the 26 dimensions come from the Polyakov equation.^[5] However, these results appear to contradict the observed four dimensional space-time.

Two different 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 in our phenomenal experience. In order to retain the supersymmetric properties of string theory, these spaces must be very special. The 6-dimensional model's resolution is achieved with Calabi-Yau spaces. In 7 dimensions, they are termed G₂ manifolds. 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 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 wave lengths (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 (i.e. three spatial dimensions plus the time dimension) subspace of the full universe. This subspace is supposed to be a D-brane, hence this is known as a braneworld theory. Many people believe that some combination of the two ideas – compactification and branes – will ultimately yield the most realistic theory.

In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza and Klein's early work demonstrated that general relativity with five large dimensions and one small dimension 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-Branes

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. Thus D-branes can emit and absorb closed strings; therefore they have mass (since they emit gravitons) and - in superstring theories - charge as well (since they emit closed strings which are gauge bosons).

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.

Thus D-branes are gravitational sources, on which a gauge theory "lives". This gauge theory is coupled to gravity (which is said to exist in the bulk), so that normally each of these two different viewpoints is incomplete.

Fig: - The particle view of nature is a description that works exceedingly well to describe three of the four observed forces of nature

What is black hole entropy?

Two important thermodynamic quantities are temperature and entropy. Temperature we all know from our fevers, weather reports and ovens. Entropy however is foreign to everyday life for most people.

Suppose we have a box filled with gas of some type of molecule called M. The temperature of that gas in that box tells us the average kinetic energy of those vibrating molecules of gas. Each molecule as a quantum particle has quantized energy states, and if we understand the quantum theory of those molecules, theorists can count up the available quantum microstates of those molecules and get some number. The entropy is the logarithm of that number.

When it was discovered that black holes can decay by quantum processes, it was also discovered that black holes seem to have the thermodynamic properties of temperature and entropy. The temperature of the black hole is inversely proportional to its mass, so the black hole gets hotter and hotter as it decays.

The entropy of a black hole is one fourth of the area of the event horizon, so the entropy gets smaller and smaller as the black hole decays and the event horizon area becomes smaller and smaller.

But until string theory there was not a clear relation between quantum microstates of a quantum theory and this supposed black hole entropy.

Problems and controversy

String theory remains to be verified. No version of string theory has yet made an experimentally verifiable prediction that differs from those made by other theories. In this sense, string theory is still in a "larval stage": it is not a proper physical theory. It possesses many features of mathematical interest and may yet become important in our understanding of the universe, but it requires further developments before it is accepted or discarded. 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 falsifiable in the sense of Popper.

For example, while supersymmetry is now seen as a vital ingredient of string theory, supersymmetric models with no obvious connection to string theory are also studied. Therefore, if supersymmetry were detected at the Large Hadron Collider it would not be seen as a direct confirmation of the theory. More importantly, if supersymmetry were not detected, there are vacua in string theory in which supersymmetry would only be seen at much higher energies, so its absence would not falsify string theory. By contrast, if observing the Sun during a solar eclipse had not shown that the Sun's gravity deflected light by the predicted amount, Einstein's general relativity theory would have been proven wrong.

One hope for testing string theory is that a better understanding of how string theory deals with singularities and time-dependent backgrounds would allow physicists to understand the predictions of string theory for the big bang, and see how cosmic inflation can be incorporated into the theory. This has led to some deep theoretical progress, and some early models of string cosmology, such as brane inflation, trans-Planckian effects, string gas cosmology and the ekpyrotic universe, but fundamental progress must be made before it is understood what, if any, distinctive predictions the theory makes for cosmology. A recent popular suggestion is that brane inflation may produce cosmic strings which could be observed through their gravitational radiation, or lensing of distant galaxies or the cosmic microwave background.

On a more mathematical level, another problem is that, like many quantum field theories, 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 non-perturbative definition of the theory is still lacking.

Philosophically, string theory cannot be truly fundamental in its present formulation because it is background-dependent: each string theory is built on a fixed spacetime background. Since a dynamic spacetime is the central tenet of general relativity, the hope is that M-theory will turn out to be background-independent, giving as solutions the many different versions of string theories, but no one yet knows how such a fundamental theory can be constructed. A related problem is that the best understood backgrounds of string theory preserve much of the supersymmetry of the underlying theory, and thus are time-invariant: string theory cannot yet deal well with time-dependent, cosmological backgrounds.

Another problem is that the vacuum structure of the theory, called the string theory landscape, is not well understood. As string theory is presently understood, it appears to contain a large number of distinct vacua, perhaps 10⁵⁰⁰ or more. Each of these corresponds to a different universe, with a different collection of particles and forces. What principle, if any, can be used to select among these vacua is an open issue. While there are no known continuous parameters in the theory, there is a very large discretuum (coined in contradistinction to continuum) of possible universes, which may be radically different from each other. Some physicists believe this is a benefit of the theory, as it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant. However, such explanations are not usually regarded as scientific in the Popperian sense.

Conclusion

This period in string history has been given the name the second string revolution.

And now the biggest rush in string research is to collapse the table above into one theory, which some people want to call M theory, for it is the Mother of all theories.