M-theory

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Note: M-Theory is a highly technical subject. Non-technical readers might wish to read M-theory Simplified first.

M-Theory, developed by Edward Witten of the Institute for Advanced Study in Princeton, N.J., unites five different superstring theories, by placing them in the same arena, like 5 islands in a large sea. It is one of the leading candidates for what has been colloquially called the theory of everything; however it has yet to produce testable predictions. Dr. Witten is quoted as saying, "String theory is 21st century physics which fell into the 20th century by accident." The theory makes heavy use of the principle of "duality" (detailed below). Two theories are "dual" to each other if they can be shown to be equivalent under a certain interchange. The five superstring theories enclosed by M-Theory are:

type I, type IIA, type IIB, E8 X E8 heterotic ( or HEt), and SO(32) heterotic (or HOt).

  • The type II theories have two supersymmetries in the ten-dimensional sense, the rest just one.
  • The type I theory is special in that it is based on unoriented open and closed strings.
  • The other four are based on oriented closed strings.
  • The IIA theory is special because it is non-chiral (parity conserving).
  • The other four are chiral (parity violating).

In each of these cases there is an 11th dimension that becomes large at strong coupling. In the IIA case the 11th dimension is a circle. In the HE case it is a line interval , which makes eleven-dimensional space-time display two ten-dimensional boundaries. The strong coupling limit of either theory produces an 11-dimensional space-time. This eleven-dimensional description of the underlying theory is called "M- theory". A string's space-time history can be viewed mathematically by functions like

Xμ(σ,τ)

that describe how the string's two-dimensional sheet coordinates (σ,τ) map into space-time Xμ

There are other functions on the two-dimensional sheet that describe other degrees of freedom, for instance those associated with supersymmetry and gauge symmetries. Classical string theory dynamics are denoted by an invariance that conforms with 2D quantum field theory. This conformal invariance is symmetry under a change of length scale. One-dimensional strings differ from higher dimensional analogs due to the fact that the 2D theory is renormalizable (contains no glitches of short-distance infinities).
Objects with p dimensions, i.e, "p-branes," have a (p+1)-dimensional world volume theory. For p > 1, these theories are non-renormalizable. This feature gives strings a special status, even though higher dimensional p-branes do occur in superstring theory.

Insight into non-perturbative properties of superstring theory apparently stems from the study of a special class of p-branes called Dirichlet p-branes( D-branes). This name results from the boundary conditions assigned to the ends of open strings. Normal open strings of the type I theory satisfy the Neumann boundary condition which states " no momentum flows on or of the end of a string." On the other hand, T duality infers the existence of dual open strings with specified positions known as Dirichlet boundary conditions in the dimensions that are T-transformed. Generally, in type II theories, we can imagine open strings with specific positions for the end-points in some of the dimensions. This lends an inference that they must end on a preferred surface. Superficially, this notion seems to break the relativistic invariance of the theory, possibly paradoxical. The resolution here of this paradox is that strings end on a p-dimensional dynamic object, the D-brane. D-branes have been studied for a number of years, their significance explained by Polchinski just recently. Why are we mentioning them ?

The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many of the previously identified p-branes are D-branes ! Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many useful applications, the most remarkable being the study of black holes. Strominger and Vafa have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case first explored was static extremal charged black holes in five dimensions. Strominger and Vafa proved for large values of the charges the entropy S = log N, where N is equal to the number of quantum states that system can be in, agrees with the [[Bekenstein-Hawking]] prediction (1/4 the area of the event horizon). This result has been generalized to black holes in 4D as well as to ones that are near extremal (and radiate correctly) or rotating, a remarkable advance. It has not yet been proven that there is any problematic breakdown of quantum mechanics due to black holes.



Further Reading:

M-theory Simplified for a non-technical easier explanation.


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