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Volume 3, Number 6
PHYSICAL REVIEW LETTERS
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28 November
Unified Interactions: Two Strings to One Bow
Jesse d’Eça
University of Ottawa, Student number 2353839, Ottawa, Ontario J9H5K4, Canada
(Received, revised manuscript received, published 28 November 2003)
In a world with three ordinary dimensions and some additional very small dimensions, particles are strings and
membranes. However, the search of higher-dimensional gravity is presently receiving a lot of attention because of its
connection with string theory and black hole physics. The required theories of Quantum Mechanics and General
Relativity have been fixed for so long that in the past, constructing string theories was complicated. There are several
string theories, which hold similar views by very general properties of the strings. The ultimate goal behind the creation
of string theory is to unite the gravitational force with quantum mechanics and relativity. This procedure takes us one
step closer to the unification of all the interactions that appear to us as being all very different from one another.
Models consisting of matrices are field theories in a diversity of dimensions. The degrees of freedom are matrices.
Historically, these models are developed in similar relationship with string theory. M-theory includes all the theories of
strings. The ‘M’ stands for Matrix, which refers to a new non-commutative relationship. There are problems caused by
our living in three spatial dimensions.
DOI: 00.0000/PhysRevLett.99.999999
PACS numbers: 00.00.Vc, 00.00.Sb, 00.00.Es
A
ll forces occur from the interactions
between different objects. Only a few distinct
kinds of interactions are at work. Gravitational
forces arise between objects because of their
masses. Electromagnetic forces are due to
electric charges at rest or in motion. Nuclear
forces (strong and weak) control the interactions
between subatomic particles if distances less
than about 10-15m separate them. It may be that
even this degree of categorization will prove to
be unnecessarily great; the theoretical physicist’s
dream is to find a unifying idea that would allow
us to understand these interactions as one
continuous governing force (see figure 1).
The study of physics is essentially the attempt
to understand these interactions and all their
consequences. For instance, James Clerk
Maxwell unified electricity and magnetism, thus
the first step towards a complete unification of
all known interactions. String theory attempts to
further it by trying to unify the electromagnetic
force with the gravitational force. Hence, the
second step towards the Grand Unified Theory
(GUT) is described by string theory. To undergo
the attempt, the critical requirement of string
theory is to unify quantum mechanics with
general relativity, and also to explain the
spectrum of particles and forces observed in
nature. [2]
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Figure 1: A Grand Unified Theory will unify the strong,
weak, and electromagnetic interactions. All the known
interactions that are said to be observe are all different
aspects of the same, unified interaction, albeit the
strong and weak and electromagnetic interactions are
so different in strength and effect. Nonetheless, current
data and theory suggests that these varied forces
merge into one force when the particles being affected
are at a high enough energy.
One of the great challenges in physics is to
develop a theory of gravity that obeys the rules
of quantum mechanics. String theory, which
describes all matter in terms of tiny string-like
objects and multidimensional membranes, may
ultimately solve the problem. String theory
proponents have had a series of triumphs
recently as they've calculated black hole
properties, including their ability to radiate, in
agreement with non-string calculations.
Relativistic quantum field theory has worked
very well to describe the observed behaviours
and properties of elementary particles (0-D). The
gravitational force is weak compared to the other
basic interactions. Particle theory only works
© 2003 - The American Physical Society
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Volume 3, Number 6
PHYSICAL REVIEW LETTERS
when we neglect gravity. Hence, particle theory
does not account for gravity.
General relativity has yielded a great deal of
insight into the Universe: the orbits of planets,
the evolution of stars and galaxies, the origin of
the Universe and recently observed black holes
and gravitational lenses. The theory only works
when we pretend that the Universe is purely
classical and that quantum mechanics is not
needed in our description of Nature.
In the 28 November PRL, physicists suggest
that a theory more elementary than particle
theory, known as string theory, is believed to
relate quantum mechanics with general relativity.
S
tring theory is important in modern physics
in which the fundamental particles are thought of
as excitation modes of elementary strings
(fundamental harmonics). Think of a guitar
string that has been tuned by stretching the string
under tension across the guitar [5]. Depending on
how the string is plucked and how much tension
is in the string, different musical notes (i.e.
excitation modes) will be created by the string.
All properties of elementary particles (charge,
mass, spin, etc) come from the vibration of the
strings. Let’s take for example mass: the more
frequent the vibration, the more energy. Given
that mass is the same as energy, increased mass
comes from higher vibration. In string theory,
the string must be stretched under tension to
become excited, but the strings in string theory
are floating in space-time. The string tension is
denoted by the quantity 1/(2*π*l2), where l2 is
equal to the square of the string length scale [6].
This is equivalent to a loading of about 1039 tons.
To correspond with quantum gravity, the average
size of a string should be the length scale of
quantum gravity, the Planck length (≈ 10 -33 cm),
but have no thickness (so, only 1-D) [1].
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 [2]. In order to
include fermions in string theory, there must be
supersymmetry, which means for every boson
(particle that transmits a force) there is a
corresponding fermion (particle that makes up
matter). A supersymmetric string theory is called
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a superstring theory [5]. Thus supersymmetry
relates the particles that transmit forces to the
particles that make up matter. It is known that
supersymmetric particles are too massive to be
detected at current particle accelerators. Highenergy supersymmetry is to be expected from
string theory.
Included in the excitations of a string is a
particle with zero mass and two units of spin.
The particle that carries the gravitational force,
known as the graviton, would correspond to this
excitation. [3]
If a graviton is added to quantum field theory,
the calculations, implied in Einstein’s theory of
gravity, that are supposed to describe Nature
become erroneous. This is due to particle
interactions occurring at one point of space-time,
analogous to zero distance between those
particles. Conversely, the calculations do make
sense in string theory given that the strings
collide over a small-finite distance (see Figure 2,
below). The nonzero distance behaviour is such
that we can combine quantum mechanics and
gravity. This is the consequence of a string
excitation that carries the gravitational force.
Figure 2: Particle physics interactions can occur at
zero distance -- but Einstein's theory of gravity makes
no sense at zero distance.
String interactions don't
occur at one point but are spread out in a way that
leads to more sensible quantum behaviour.
There are five kinds of superstring theories.
For them to be able to unify all the known
interactions, the Universe must have nine spatial
dimensions and one time dimension. This idea of
an extra-dimensional Universe was established in
the Kaluza-Klein theory (1921) where only one
extra dimension is needed for electromagnetism
to be derived from gravity in a unified theory.
That fourth dimension is curled into a tiny circle,
which will be examined anon [4]. See figure 3,
next page:
© 2003 - The American Physical Society
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PHYSICAL REVIEW LETTERS
Figure 3: Sketches of how a compactified dimension is
formed (Kaluza-Klein space).
We’re all familiar with time and three of the
space dimensions: the other six together are
known as Calabi-Yau spaces (see figure 4). Six
exact extra dimensions are required in
superstring theory; otherwise, bad quantum
states called ghosts with unphysical negative
probabilities become a part of the spectrum. To
get from ten space-time dimensions to four
space-time dimensions, the number of string
theories grows since there are so many ways to
make six dimensions smaller than the four
known dimensions observed in our universe.
Figure 4: A representation of a 6-dimensional
Calabi-Yau space.
String theorists are realizing that these five
types of string theories are one unique theory
understood differently (this period in time was
called the second string revolution). The
combination of those theories is called M-theory,
which is understood as Matrix theory. For their
combination to exist, they have to be related to
one another, each one a special case of some
more fundamental theory, of which there is only
one.
These
theories
are
related
by
transformations that are called dualities. This
means that any one theory can be converted in
some way so that it ends up like another theory.
The two theories are said to be dual to one
another under that particular transformation.
These dualities connect quantities that were
thought to be separate, such as large and small
distance scales, strong and weak strength
coupling. These quantities are distinct limits of
behaviour of a physical system, both classical
and quantum.
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The duality symmetry that obscures our ability
to distinguish between large and small distance
scales is called T-duality, and comes about from
the compactification (see reference [1]) of extra
space dimensions in a ten-dimensional
superstring theory. A particle has the property of
momentum, whereas a string can wrap around a
circle. The number of times that the string winds
around the circle is called the winding number
(see figure 5). Both properties are quantized.
Accordingly, exchanging momentum and
winding modes of the string exchanges a large
distance scale with a small distance scale. For
example, if we compactify two theories on a
circle, then switching the momentum and
winding modes, and switching the distance
scales, changes one theory into the other. A very
large distance scale to a momentum mode of a
string looks to a winding mode of a string like a
very small distance.
Another alternative has the extra dimensions
made really big. All the matter and gravity
propagates in a three-dimensional subspace
called three brane (e.g. a sheet of paper is a two
brane of three dimensional space). It is possible
that our world is pinned to a 3-dimensional
region, called a brane that is located in a higher
order space (multi-dimensional universe) [3]. We
live in a universe where our three familiar
dimensions of space are “flat”, but there are
additional dimensions, which are curled-up very
tightly so that they have an extremely small
radius: 10-33cm (roughly the Planck scale).
Figure 5: Winding
compactified space.
modes
of
strings
around
a
We could sense these extra dimensions
through their effect on gravity. While the forces
that hold our world together (electromagnetic,
weak, and strong interactions) are constrained to
the 3+1 plane dimensions, the gravitational
interaction always occupies the entire universe,
thus allowing it to sense the effects of extra
dimensions. Gravity depends on the number of
space-time dimensions. Gravity feels like a
strong force at the macroscopic distance scales
© 2003 - The American Physical Society
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PHYSICAL REVIEW LETTERS
where humans experience it, but gravity is truly a
much weaker force from a microscopic point of
view. In view of the fact that gravity is a very
weak force and the radius of extra small
dimensions is tiny, the effects of gravity are
difficult to differentiate. Providentially, the
theory proposed by Arkani-Hamed, Dimopoulos,
and Dvali, fixed that the gravitational interaction
is greatly enhanced if the colliding particles have
sufficiently high energy [3]. This enhancement is
due to the winding modes of the graviton around
the compactified dimensions many times (see
figure 6). Each time it winds around, it gives rise
to a small gravitational force between the
colliding particles. If the number of revolutions
that the graviton makes around the curled extra
dimensions is large enough, the gravitational
interaction is enhanced.
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S
uperstring theory has the direct prospective
of being the accurate theory for describing the
fundamental nature of our universe. Quantum
physics, bosons, fermions, gauge groups, and
gravity are all implicated in the theories. In the
last several years, there has been vast progress in
understanding the overall structure of the theory
including D-branes and string duality. It is a
reasonable outcome for a quantum theory of
gravity because Einstein’s theory of gravity tells
us that gravity is about how the sizes of the
objects and magnitudes of the interactions are
measured in curved space-time (hence, the tiny
curled-up dimensions of space). Nevertheless,
there is much work yet to be done.
References:
[1]
Green, M. (Michael B.), John H.
Schwarz, and Edward Witten. Superstring
theory. Cambridge, Eng., New York, Cambridge
University Press, 1987. 2 v.
QC794.6.S85G74 1987
Figure 6: The gravitons spiral around the compactified
extra tiny dimensions towards our Universe, thus
creating a gravity effect.
To further explain the weak strength of the
gravitational force, there exists a duality between
strong and weak coupling. A coupling constant
describes how strong an interaction is. A larger
coupling constant means a stronger force; hence,
the coupling constant is of the order of 10 -41 and
it is derived from the masses of the proton and
electron, Planck’s constant, the velocity of light
in empty space, and the Gravitational constant.
String theories have a coupling constant. The
string coupling constant depends on one of the
oscillation modes of the string, called the dilaton
[5]. Exchanging the dilaton field with the
negative value of itself exchanges a very large
coupling constant with a very small one. This
symmetry is called S-duality. Suppose two string
theories are related by S-duality. One theory with
a strong coupling constant is related to the other
with a weak-coupling constant. Therefore, we
just need to understand the weak theory, which is
equivalent to understanding the strong theory.
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[2]
Peat, F. David. Superstrings and the
search for the theory of everything. Chicago,
Contemporary Books, c1988. 362 p.
QC794.6.S85P43 1988
[3]
Arkani-Hamed, Nima, and others. The
universe’s unseen dimensions. Scientific
American, v. 283, Aug. 2000: 62-69.
T1.S5
[4]
Duff, Michael J. The theory formerly
known as strings. Scientific American, v. 278,
Feb. 1998: 64-69.
T1.S5
[5]
The Official String Theory Web Site
Created by physicist Patricia Schwarz, this
website includes a basic introduction to string
theory, an audiovisual lecture, biographies of
theoretical physicists, and a timeline of string
theory. http://superstringtheory.com/
[6]
Physical Review Focus
Physical Review Focus is a free service of the
American Physical Society (APS). Focus stories
explain selected physics research published in
the APS journals Physical Review and Physical
Review Letters (PRL). http://focus.aps.org/
© 2003 - The American Physical Society
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