M-Theory

M-Theory was a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. According to Dr. Edward Witten, who proposed the theory, mathematical tools which have yet to be invented are needed in order to fully understand it. Edward Witten, PhD (b. 1951 CE d. 2062 CE) was a Twenty-first Century mathematical physicist who introduced M-theory in 1995 CE. He proposed a unification of five superstring theories and the quantum mechanical theory of gravity – 11-dimensional supergravity. Superstring/M-theory was a promising candidate to be the unifying theory for the four fundamental physical interactions (gravity, electromagnetism, strong and weak nuclear) and provides a fundamental link between quantum mechanics and general relativity, previously thought to be incompatible. M-Theory was the most significant precursor to Manifold Theory.

Witten was heralded as the foremost mathematical theoretical physicist of his time and for his contributions to physics and mathematics; he obtained many awards, with the Fields Medal in 1990 (mathematics equivalent to Nobel Prize) being one of them. He inspired his colleagues and sparked what was at the time called the second revolution in superstring theory.

M-theory's relation to superstrings and supergravity
M-Theory in various geometric backgrounds is associated with the different superstring theories (in different geometric backgrounds), and these limits are related to each other by the principle of duality. Two physical theories are dual to each other if they have identical physics after a certain mathematical transformation.

Type IIA and IIB are related by the T-duality, as are the two Heterotic theories. T-duality is a symmetry of string theory, relating type IIA and type IIB string theory, and the two heterotic string theories. T-duality transformations act on spaces in which at least one direction has the topology of a circle. Under the transformation, the radius R of that direction will be changed to 1/R, and "wrapped" string states will be exchanged with high-momentum string states in the dual theory. For example, one might begin with a IIA string wrapped once around the direction in question. Under T-duality, it will be mapped to a IIB string which has momentum in that direction. A IIA string with a winding number of two (i.e., wrapped twice) will be mapped to a IIB string with two units of momentum, and so on.

Type I and Heterotic SO(32) are related by the S-duality. Type IIB is also S-dual with itself. In theoretical physics, S-duality (also a strong-weak duality) is an equivalence of two quantum field theories, or string theories. An S-duality transformation maps the states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1 / g in the dual theory. This has permitted the use of perturbation theory, normally useful only for "weakly coupled" theories with g less than 1, to also describe the "strongly coupled" (g greater than 1) regimes of string theory, by mapping them onto dual, weakly coupled regimes.

In the case of four-dimensional quantum field theories, S-duality was understood to exchange the electric and magnetic fields (and the electrically charged particles with magnetic monopoles) 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.

One interpretation of this result is that the 11th dimension was always present but invisible because the radius of the 11th dimension is proportional to the string coupling constant and the traditional perturbative string theory presumes it to be infinitesimal. Another interpretation is that dimension is not a fundamental concept of M-theory at all.

Characteristics of M-theory
M-theory contains much more than just strings. It contains both higher and lower dimensional objects. These objects are called p-branes where p denotes their dimensionality (thus, 1-brane for a string and 2-brane for a membrane). Higher dimensional objects were always present in superstring theory but could never be studied before the Second Superstring Revolution because of their non-perturbative nature.

Insights into non-perturbative properties of p-branes stem from a special class of p-branes called Dirichlet p-branes (Dp-branes). This name results from the boundary conditions assigned to the ends of open strings in type I superstrings. Open strings of the type I theory can have endpoints which satisfy the Neumann boundary condition. Under this condition, the endpoints of strings are free to move about but no momentum can flow into or out of the end of a string. The T duality infers the existence of open strings with positions fixed in the dimensions that are T-transformed. Generally, in type II theories, open strings can be imagine 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 was thought to break the relativistic invariance of the theory, possibly leading to a paradox. The resolution of this paradox was that strings end on a p-dimensional dynamic object, the Dp-brane.

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 found many useful applications, the most remarkable being the study of black holes. It was shown that D-brane techniques could 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. It was proven 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, agreed with the Bekenstein-Hawking prediction of 1/4 the area of the event horizon.

This result was generalized to black holes in 4D as well as to ones that were near extremal (and radiated correctly) or were rotating.