A new picture of string theory
At one time, string theorists believed there
were five distinct superstring theories: type
I, types IIA and IIB,
and the two heterotic string theories.
The thinking was that out of these five candidate theories, only one
was the actual correct Theory of Everything,
and that theory was the theory whose low energy limit, with ten dimensions
spacetime compactified down to four, matched the physics observed in
our world today. The other theories would be nothing more than rejected
string theories, mathematical constructs not blessed by Nature with
But now it is known that this naive picture
was wrong, and that the the five superstring
theories are connected to one another as if they are each a special
case of some more fundamental theory, of which there is only one. These
theories are related by transformations that are called dualities.
If two theories are related by a duality transformation, it means that
the first theory can be transformed in some way so that it ends up looking
just like the second theory. The two theories are then said to be dual
to one another under that kind of transformation.
These dualities link quantities that were also
thought to be separate. Large and small distance scales, strong and
weak coupling strengths -- these quantities have always marked very
distinct limits of behavior of a physical system, in both classical
field theory and quantum particle physics. But strings can obscure the
difference between large and small, strong and weak, and this is how
these five very different theories end up being related.
Large and small distance
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 of extra space dimensions
in a ten dimensional superstring theory.
Suppose we're in ten spacetime dimensions, which
means we have nine space and one time. Take one of those nine space
dimensions and make it a circle of radius R, so that traveling in that
direction for a distance L=2pR
takes you around the circle and brings you back to where you started.
A particle traveling around this circle will
have a quantized momentum around the circle, and this will contribute
to the total energy of the particle. But a string is very different,
because in addition to traveling around the circle, the string can wrap
around the circle. The number of times the string winds around the circle
is called the winding number, and that is also quantized.
Now the weird thing about string theory is that
these momentum modes and the winding modes can be interchanged, as long
as we also interchange the radius R of the circle with the quantity
Lst2/R, where Lst is the string length.
If R is very much smaller than the string length,
then the quantity Lst2/R is going to be very large.
So exchanging momentum and winding modes
of the string exchanges a large distance scale
with a small distance scale.
This type of duality is called T-duality.
T-duality relates Type IIA superstring theory
to Type IIB superstring theory. That means if we take Type IIA
and Type IIB theory and compactify them both on a circle, then switching
the momentum and winding modes, and switching the distance scale, changes
one theory into the other! The same is also true for the two
So T-duality obscures
the difference between large and small distances. What looks
like a very large distance to a momentum mode of a string looks, looks
to a winding mode of a string like a very small distance. This is very
counter to how physics has always worked since the days of Kepler and
Strong and weak coupling
What is a coupling constant? This is some number
that tells us how strong an interaction is. Newton's constant is the
coupling constant for the gravitational force, for example. If Newton's
constant were twice the size it is measured to be now, then we would
feel twice as much gravitational force from the Earth, and the Earth
would feel twice as much from the Moon and the Sun, and so on. A larger
coupling constant means a stronger force, and a smaller coupling constant
means a weaker force.
Every force has a coupling constant. For electromagnetism,
the coupling constant is proportional to the square of the electric
charge. When physicists study the quantum behavior of electromagnetism,
they can't solve the whole theory exactly, so they break it down to
little pieces, and each little piece that they can solve has a different
power of the coupling constant in front of it. At normal energies in
electromagnetism, the coupling constant is small, and so the first few
little pieces make a good approximation to the real answer. But if the
coupling constant gets large, that method of calculation breaks down,
and the little pieces become worthless as an approximation to the real
This also can happen in string theory. String
theories have a coupling constant. But unlike in particle theories,
the string coupling constant is not just a number, but depends on one
of the oscillation modes of the string, called the dilaton.
Exchanging the dilaton field with minus itself exchanges a very large
coupling constant with a very small one.
This symmetry is called S-duality.
If two string theories are related by S-duality, then one theory with
a strong coupling constant is the same as the other theory with weak
coupling constant. Notice that the theory with strong coupling cannot
be understood by means of expanding in a series, but the theory with
weak coupling can. So if the two theories are related by S-duality,
then we just need to understand the weak theory,
and that is equivalent to understanding the
strong theory. For a physicist, that is the proverbial two-for-one
Superstring theories related by S-duality
are: Type I superstring theory with heterotic
SO(32) superstring theory, and Type
IIB theory with itself.
What does it mean?
T-duality is something unique to string physics.
It's something particles cannot do, because a particle cannot get wrapped
around a circle like a string. If string theory is a correct theory
of Nature, then this implies that one some deep level, the separation
between large vs. small distance scales in physics is not a fixed separation
but a fluid one, dependent upon the type of probe we use to measure
distance, and how we count the states of the probe.
The same goes for S-duality, which teaches us
that the strong coupling limit of one string theory can describe the
weak coupling limit of a different string theory.
This sounds like it goes against all traditional
physics, but this is indeed 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 objects and
magnitudes of interactions are measured in curved spacetime.