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Why 3+1 = 11 for small values of 7
A brief overview of (Super)String Theory
by
Allison Gaul
Differential Geometry
In the early 1900's Max Planck put forth his hypothesis of energy quantization, stating that
energy travels along a wave path but in small packets of "quanta" rather than simply in a continuous
wave. Though he did not understand it at the time his hypothesis contained a somewhat mysterious but
necessary constant which has had far reaching applications, and has come to be known today as
"Planck's constant". This constant eventually lead to the determining of the "Planck Scale" at which the
theory of Quantum Mechanics takes over where General Relativity can not go.
While Einstein attempted to explain the curvature of space and time on large scales, Quantum
Mechanics attempted to explain the interaction between particles on a very small scale. For years,
physicists were stumped by the inherent properties of each theory that caused the other to be
contradictory and in some cases paradoxical. Much of the problem lay in the inability of any theory to
describe all of the four natural forces, strong, weak, electromagnetic, and gravity. Out of the attempt to
find a unified theory that agreed with experimental observations of the four forces was born "Quantum
Field Theory".
Many important advances came about as a result of the madcap dash to describe the universe,
including the unification of the electromagnetic force and the weak nuclear force. The "Electroweak"
force, as it is called, is thought to have existed at the time of the big bang and as the universe expanded
outward it began to thin out and dissipate until it's effects were noticeable as being two almost separate
forces. This revolutionary advancement in QFT produced renewed fervor in the search for a theory that
would bring the forces together. Try as they might, the theoretical physics community could not seem
to find a home for gravity, the seemingly most mundane and simple of the forces refused to conform to
any theory put to it. Defeat was reluctantly admitted by many of the unification zealots and more
fruitful projects were taken up.
It seemed that all hope was lost for gravity, while the other forces had their time in the spotlight,
it remained homeless and abandoned, until years later when hope presented itself, in the form of
minuscule undulating pieces of string. In 1970 three physicists by the names of Yoichio Nambu,
Holger Nielsen, and Leonard Susskind, although not together, find that the same theory which
describes the particle spectrum, entitled "dual resonance", also accurately describes oscillating string
material. This seemingly trivial discovery marked the beginning of what is now known as "String
Theory",
The concept of fundamental strings, rather than fundamental particles is a little hard to grasp for
people raised on the ideas of classical particle physics, if not just downright absurd. Yet those little
"wibbling" (as I like to call it) bits of string have held up to every test thrown at them so far, and the
physics community is rapidly beginning to accept that not only are they here to stay but they just might
be correct. This begs the question: "Just what exactly are strings?" The answer unfortunately is some
what frustrating since strings are well, strings. Revert back to your somewhat more classical notions of
quantum mechanics for a moment if you will, and consider the quark. What is a quark? A quark is a
quark, there is an up, down, top, bottom, charmed, and strange, but they are all just quarks. Now carry
that idea over to strings, there are different classifications, but they are fundamentally just strings. This
may still be a little difficult to swallow due to the macroscopic association of strings being made out of
some material, but this idea doesn't translate into elementary composites of the universe.
There are four known types of strings, and in the grand tradition of unimaginative physics
names, they are open, closed, oriented, and non-oriented, all of which can be described by their
quantum numbers of mass, spin, and dimension. Open strings can be thought of as small piece of thread
oscillating in some unseen breeze. These 1-dimensional string models have boundary conditions
(Neumann boundary) that prevent momentum from flowing on or off the end of the strings which is
important when trying to combine multiple string theories. Closed strings are 2-dimensional and also
commonly thought of as small undulating rubber bands, which form closed loops, with periodic
boundary conditions. To make matters more confusing, neither string type is bound to it's state,! There
are in fact five different ways strings can interact. Open strings have the ability to split, creating two
more open strings, closed strings can also perform splitting which results in two like-form strings. Two
open strings may collide, twist and break apart into two new strings , and an open strings also possesses
the ability to twist itself into a loop then disjoin, creating one closed string and one open. Strangely
enough, an open string can close upon itself but a closed string can not open. Due to these specific
interactions, closed string theories can often be described as transforms of open string theories but not
vice versa.
Oriented strings have the property that direction of oscillation along the string can be
determined, whereas non-oriented strings do not. On top of this, closed, oriented, strings haves two
possible modes, and are classified as being either right-movers, or left-movers depending on the
direction of their oscillations. Modes are simply the possible states of excitation for strings, similar to
the integral length necessary for harmonic oscillations of macroscopic strings. It is this limited (dare I
say quantized?) nature of string modes which proves so useful in developing different string theories.
As strings move through space they trace out surfaces referred to as world sheets. These sheets
can be thought of as manifold surfaces that the string's movement is bound to. For open strings there
are two separate sets of boundary conditions Neumann and Dirichlet boundary conditions. Neumann
conditions allow the ends of the string to translate freely but momentum may not travel off the
worldsheet. With Dirichlet boundary conditions the endpoint is fixed to move only on some manifold.
This manifold is called a D-brane or Dp-brane ('p' is an integer which is the number of spatial
dimensions of the manifold). D-branes can range in dimension states from -1 to whatever the limitation
of their space is. For p= -1 all space and time coordinates are fixed and no movement is allowed, this
state is referred to as a D-instanton. Interestingly at p=0 space coordinates are fixed but the time
coordinate is free, when this occurs we have a D-particle.
In order to make string theories more macroscopic we require “superstrings” and in order to
have superstrings we must exist in a 10-dimensional spacetime. Since there are only 4 dimensions that
we can measure empirically, the other six must be curled tightly around each other and compactified to
an order near the Planck scale. While this may be hard to conceptualize, it was an idea first proposed
around the time of General Relativity by Kaluza and Klein. The concept worked with 5-dimensions and
showed that if you curl the fifth dimension into a circle about the other 4 then you would only have 4
observable dimensions, and a fifth which interacted with the others. The physical representation wold
be similar to that of a 5th dimensional sphere. Strings possess the interesting capability of being able to
wind around dimensions, and this twisting and curling creates “winding modes”. This winding can
occur an integral number of times and contributes to the momentum of the string such that p = Rw,
where R is the radius of the winding. As the radius decreases the momentum begins to appear more
continuous until eventually the string becomes compactified. In this manner the dimensional curling
can be achieved!
To curl the extra dimensions together we need a 6-dimensional compactified manifold, known
as a Calabi-Yau manifold. Specifically these manifolds are invaluable models for comptactification
because they leave much of the original supersymmetry intact. A Calabi-Yau N-fold has holonomy
SU(N) leaves 2^(1-N) of the original supersymmetry unbroken, which is essential for preserving the
characteristics of the curled dimensions. Though originally it seemed that superstrings suggested a
unique 10-dimensional set, the Calabi-Yau configurations are far from unique, numbering up in the
millions. These manifolds can then be linked together by conifold transformations to produce limitless
possibilities of 10-dimensional spaces.
With so many different options, it's no wonder so many different string theories have
developed! However if we wish to find a universally applicable theory we must resolve these
differences. It is no coincidence that many string theories share similar aspects, and often can be
described in terms of one another, when this happens the theories are said to be “dual”. The simplest
form of duality is known as T-duality and relates the geometries of two theories. If one theory predicts
compactification on a circle of R radius, and the other one predicts a circle of radius 1/R, then they are
said to be T-dual (if only it were all this simple). The second form of duality relates coupling between
two theories, thus if one theory is strongly coupled at 10-dimensions, the other must be weakly coupled
there in order for the two theories to be S-dual. While this seems like merely an interesting addendum
to string classifications, it is in fact essential! Due to the existence of dualities, it is possible to relate
all 5 of the major string theories in a sort of daisy chain of relations based on geometries and coupling,
under a single theory known simply as “M-theory”.
This final theory (I promise), implies 11-dimensional space, and is described not in terms of
strings but instead in terms of their manifolds, branes. Where other theories are plentiful, M-theory is
unique! It links the other theories together in a set of dualities, and suggests that they are not entirely
separate theories but rather, they are boundary conditions. The geometry of M-Theory is somewhat
akin to that of an 11-dimensional torus, smooshed into a circle to give 10-dimensions. Though it is a
relatively new theory, it is one that scientists and mathematicians are studying with great zeal, for it
holds at low and high energies, and seems to fully incorporate fully all it's subtheories. Though it may
not be the proverbial holy grail of physics known as the “Grand Unification Theory” it has brought us
significantly closer to finding it, and finally answering the question “just how many dimensions are
there?”
References
–
The Elegant Universe, Brian Greene
I received the bulk of my information from this text, it has an excellent background of the
formation of string theory and though it approaches it from a more physics-y angle than a
strictly
mathematicial one, it does an excellent job of giving easy to understand examples and
scenarios.
–
Hiding In the Mirror, Lawrence Krauss
This book is about the social “obessession” with discovering the nature of higher dimensions,
and how this has effected not just math and physics but history and art as well as other fields. It
goes from Cubist paintings to Calabi-Yau manifolds and was really interesting.
–
www.Superstringtheory.com
I love this website, I really do. Whoever thought that taking a complex mathematical concept
and making a whimsical website colored in lavender and pink was nothing less than brilliant.
The information here is very good and easy to read. They include some good math for Einstein's
cosmology but don't touch on tensor mathematics or any actual equations involved in string
theory. I was sorely tempted to include many of the equations specific to general relativity
(since those are more easily digested than the complex geometries of string theory) but decided
it would be getting off topic
–
www.Wikipedia.com
Used for information and pictures on Calabi-Yau, and pictures for some of the slides. It's pretty
hit or miss as a reference but this time it was helpful.