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M-Theory
M-theory
M-theory is 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.
The following article is somewhat technical in nature; see
M-theory simplified for a less technical article.
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
T-duality, as are the two Heterotic theories. Type I and Heterotic SO(32)
are related by the
S-duality. Type IIB is also S-dual with itself.
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The type II theories have two supersymmetries in the
ten-dimensional sense, the rest just one.
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The type I theory is special in that it is based on
unoriented open and closed strings.
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The other four are based on oriented closed strings.
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The IIA theory is special because it is non-chiral
(parity conserving).
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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
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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 contant 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.
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,
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 of this
paradox is 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 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:
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Michael J. Duff, The Theory Formerly Known as Strings,
Scientific American, February 1998
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John Gribbin, The Search for Superstrings, Symmetry,
and the Theory of Everything,
ISBN 0316329754 , Little, Brown & Company, 1ST BACK B Edition, August
2000, specifically pages 177-180.
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Brian Greene, The Elegant Universe: Superstrings,
Hidden Dimensions, and the Quest for the Ultimate Theory,
ISBN 0393046885 , W.W. Norton & Company, February 1999
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