KaluzaKlein in string theory
Superstring theory is a possible unified theory of
all fundamental forces, but superstring theory requires a 10 dimensional
spacetime, or else bad quantum states called ghosts with unphysical
negative probabilities become part of the spectrum and spoil Lorentz
invariance. Fermions are very complicated to work with in higher dimensions,
so for the sake of simplicity let's consider bosonic string theory,
which is Lorentz invariant and ghostfree (albeit tachyonic) in d=26.
A particle trajectory only has one parameter: the proper time along
the path of the particle. Going from particles to strings adds a new
parameter: the distance along the string
and that's what makes the outcome of KaluzaKlein compactification
far more interesting in string theory than it is in particle theory.
If we compactify x^{25} on a circle of radius
R, we get the usual KaluzaKlein quantized momentum in that direction
We want gravity in the theory, so we need to look at closed strings.
A closed string can do something that a particle cannot do: get wrapped
around the circle in the compact dimension.
A closed string can be wrapped around the circle once,
twice, or any number of times, and the number of times the string is
wrapped around the circle is called the winding number w. The string
oscillator sum in the x^{25} direction changes by a constant
piece in a way that is consistent with the periodicity of the closed
string and the compact dimension
The string tension T_{string} is the energy
per unit length of the string. If the string is wound w times around
a circular dimension with radius R, then the energy E_{w} stored
in the tension of the wound string is
The mass of an excited string depends on the number
of oscillator modes N and Ñ excited in the two directions of
propagation around the closed string, minus the constant vacuum energy.
KaluzaKlein compactification adds the quantized momentum in the compact
dimensions, and the tension energy from the string being wrapped w times
around the compact dimension, so that the total squared mass becomes
A very crucial feature of this mass equation is the
symmetry under
This is what makes string theory so different from particle theory.
The theory doesn't really distinguish between the quantized momentum
modes, and the winding modes of the string in the compact dimension.
This creates a symmetry between small and large
distances that is not present in KaluzaKlein compactification
of a particle theory.
This symmetry is called Tduality. Tduality is a
symmetry that relates different string theories that everyone thought
were completely unrelated before Tduality was understood. Tduality
preceded the Second Superstring Revolution.
The theory gains extra massless particles when the
radius R of the compact dimension takes the minimum value possible given
the above symmetry of Tduality, which is just the string scale itself
This is another purely stringy effect, not occurring with particles.
The KaluzaKlein compactification of strings can
be done on more than one dimension at once. When n dimensions are compactified
into circles, then this is called toroidal
compactification, because the product of n copies of a circle
is an ntorus, or T^{n} for short.
When fermions are added to make superstrings, the
mathematics becomes more complicated but the structures and symmetries
become more rich. The most studied superstring compactification is heterotic
string theory compactified on a CalabiYau
space in sixdimensions (or three complex dimensions).
These general models all have in common that the
spacetime is a direct product
where M_{4} is the fourdimensional noncompact spacetime, and
X_{6} is some sixdimensional compact internal space. This means
that the metric on M_{4} doesn't depend at all on the coordinates
in the internal space. In this case, the gravitational coupling constant
that we measure as Newton's constant G_{N} is related to the
gravitational coupling G_{10} of the full tendimensional superstring
theory by
where V_{X} is the volume of X6.
In terms of the Planck mass M_{Planck}, which
is the quantum gravity mass scale determined by the gravitational coupling
G_{N}, this relationship becomes
where the mass M_{S} is the fundamental mass scale of the full
tendimensional theory_{.}
Braneworlds
In the KaluzaKlein picture, the extra dimensions
are envisioned as being rolled up in compact space with a very small
volume, with massive excited states called KaluzaKlein modes whose
mass makes them too heavy to be observed in current or future accelerators.
The braneworld scenario for having extra dimensions
while hiding them from easy detection relies on allowing the extra dimensions
to be noncompact, but with a warped
metric that depends on the extra dimensions and so is not a direct product
space. A simple model in five spacetime dimensions is the RandallSundrum
model, with metric
In this scenario, the threedimensional space that we experience is
a threedimensional subspace, called a 3brane, located at f=0,
with another 3brane located at f=p,
or y=pr_{c}. The full
fourdimensional space, or fivedimensional spacetime, is referred to
as the bulk. The warping or curving of the bulk gives rise to a cosmological
constant, which is proportional to the parameter k.
Since the extra space dimension is noncompact, we
would expect the force law of gravity to change. However in this picture,
the warping of the brane causes the the graviton to become bound to
our brane, so that the graviton wave function falls away very rapidly
away in the direction of the extra dimension.
This spacetime also has oscillations in the extra
dimension that are the KaluzaKlein modes, but in this case there is
a continuous spectrum of modes. This would seem to rule the model out,
except that the KaluzaKlein modes here are so weakly coupled that they
can't be detected on the brane.
Why would this model be preferable to having compact
extra dimensions? In KaluzaKlein compactification, the Planck mass
in the full tendimensional superstring
The parameter M is the fundamental mass scale in the full theory in
the bulk, and k is about the same size as M. So for kr_{c}>>1,
the Planck scale measured on our brane would be about the same size
as the Planck scale as measured in the full theory. This avoids the
situation in the KaluzaKlein compactification where the Planck mass
in four spacetime dimensions depends on the volume of the compactified
space, which is hard to control dynamically.
How could they be observed?
One problem with theoretical models of gravity and
particle physics is that before they can make unique testable predictions
of new physics, they have to be worked on so that they don't contradict
any existing theoretical or experimental knowledge. That can be a long
process, and it's not really over for superstring theories or for braneworld
models, especially not braneworld models derived from superstring theories.
In superstring theory with KaluzaKlein compactification,
there are several different energy scales that come into play in going
from a string theory to a low energy effective particle theory that
is consistent with observed particle physics and cosmology.
The attribute of superstring theory that looks the most
promising for experimental detection is supersymmetry.
Supersymmetry breaking and compactification of higher dimensions have
to work together to give the low energy physics we observe in accelerator
detectors.
Braneworld models in general are very different from superstring
KaluzaKlein compactification models because they don't require there
to be so many steps between the Planck scale and the electroweak scale.
The huge difference between the Planck scale and the electroweak scale
is called the gauge hierarchy problem.
Supersymmetry is interesting to particle physicists
because it can address this problem. But some braneworld models need
supersymmetry for the brane geometry to be stable.
If supersymmetry is detected at nextgeneration particle
physics experiments, then the details of the supersymmetric physics
will have something to say, hopefully, about any underlying superstring
model and whether there is KaluzaKlein compactification of extra space
dimensions into some tiny rolled up internal space, or whether we are
all living as the four dimensional equivalent of flies stuck on the
wall of a higher dimensional Universe.
