This section uses units where (Planck's constant)/2p
and the speed of light = 1. This choice of units is called natural
units. With this choice, mass has units of inverse length,
and vice versa. The conversion factor is 2x10^{7} eV = 1/meter.
Unification and group theory
The success of spontaneous symmetry breaking in explaining
electroweak physics led physicis to wonder whether the three particle
theories of the SU(3)xSU(2)xU(1) model
could be the spontaneously broken version of a higher unified theory
at some higher energy scale, a single theory with only one gauge group
and one coupling constant. This type of theory is called a Grand
Unified Theory, or GUT for short.
The quantum behavior of the known particle coupling
constants supports the idea of Grand Unification. Because of renormalization,
the electromagnetic coupling constant grows larger
at high energies, whereas the coupling constants for the weak and strong
nuclear interactions grow smaller at
higher energies. At the mass scale
the three coupling constants become equal. Therefore, this ought to
be the mass scale where the single gauge symmetry of a Grand Unified
Theory would become spontaneously broken into the three distinct gauge
symmetries of the SU(3)xSU(2)xU(1) model.
The single gauge group of a GUT has to be mathematically
capable of containing the group product SU(3)xSU(2)xU(1)
of the three gauge groups relevant to low energy particle physcs. The
best candidate for such a theory is unitary group SU(5),
which would give 24 gauge bosons mediating the single unified force,
but there are also other GUT models based on other groups, such as the
orthogonal group SO(10), which would
give 45 gauge bosons and contain the SU(5) theory as a subgroup.
The problem with Grand Unification is that the unified
gauge bosons allow quarks to couple to leptons in such a way that two
quarks can be converted into an antiquark and an antilepton. For example,
two up quarks would be allowed to turn into a positron and and a down
antiquark.
A proton consists of two up quarks and and down quark.
A neutral pion consists of a down quark and a down antiquark. Therefore
the unified gauge boson in a GUT could mediate proton decay by the interaction
and other related decays.
The proton lifetime predicted in a GUT is about
whereas the current best measurement of the proton lifetime is
It's important to note here that proton decay can happen through radiative
corrections even in the Standard Model, so we don't expect the proton
lifetime to be infinite.
However, it seems that the proton doesn't decay as
quickly as predicted by a GUT. This situtation is improved when supersymmetry
is added to the GUT, and this will be explained in next section.
What about gravity?
Einstein's elegant and experimentally tested theory
of gravity called General
Relativity is not a normal gauge theory like electromagnetism.
The symmetry is not a unitary group symmetry like U(1)
or SU(3), but instead a symmetry under
general coordinate transformations in
four spacetime dimensions. This does not lead to a renormalizable quantum
field theory, and so gravity cannot be unified with the other three
known physical forces in the context of a Grand(er) Unified Theory.
But string theory claims to be a unified theory encompassing
all known forces including gravity. How can that be? The main symmetry
apparent in string theory is conformal invariance, or superconformal
invariance, on the string world sheet. This symmetry dicates the spectrum
of allowed mass and spin states in the theory. The spin two graviton
and the spin one gauge bosons exist within this framework naturally
as part of the tensor structure of the quantized string spectrum.
This is another reason why physicists have become
so impressed by string theory. There
exists a completely novel way of putting gravity and the other known
forces together in the context of a single symmetry, that is much more
powerful than the ordinary quantum gauge theory of particles. But the
question is  is this really the way that nature does it? The answer
to that may take a long time to sort out.
Symmetry breaking in string theory
The two string theories that have shown the most
promise for yielding a pattern of symmetry breaking that is like Grand
Unification plus gravity are the heterotic superstring theories based
on the groups SO(32) and E_{8}xE_{8}. However, these
are supersymmetric theories in ten spacetime dimensions, so the symmetry
breaking scheme also has to be involved with breaking the supersymmetry
(because fermions and bosons don't come in pairs in the real world)
and dealing with the extra six space dimensions in some manner. So the
possibilities, and the possible complications, are much wider in string
theory than in ordinary quantum gauge field theories.
Forgetting these complications for a moment, focus on the
group theory of the E_{8}xE_{8}
model. The group E_{8} is an
exceptional group with interesting properties
too complex to explain here. The common suppostion is that one of the
E_{8} groups remains unbroken,
and decouples from physical observation as a kind of shadow matter.
The other E_{8} has the right
mathematical structure to break down to an SU(5)
GUT via E_{8} > E_{6} >
SO(10) > SU(5).
The symmetry breaking scale would presumably start somewhere
near the Planck scale
and end up at the GUT scale of about 10^{14} GeV. The spontaneous
symmetry breaking mechanism would presumably be scalar field potentials
of the form shown above, where a subset of the scalar fields with normal
modes like the radial mode become massive, and the remaining massless
scalar fields become longitudinal modes of massive gauge bosons to break
the gauge symmetry down to the next level.
But  in string theory, at the level of perturbation theory where the
physics is most understood  the scalar potentials seem to be flat
in all directions and hence the scalar fields all remain massless.
The solution to symmetry breaking in string theory has to be nonperturbative
and is still regarded as an unsolved problem.
