8 Beyond the Standard Model
501
Thus the KK ideas provide an explanation of the quantization of electric charge.
(Note also that charge conjugation is the parity tranformation y → −y in the fifth
dimesion.) If one indeed identifies the fundamental unit of charge e =
√
2km with
the charge on the electron, then one is forced to take m to be very large: the Planck
mass 10 19 GeV. Such extra scalar particles seem to be present abundantly in string
theories as well. The experimental question was mainly to set bounds on the value
of the radius R(= 1/m) of the fifth dimension. The limits are obtained from precision
electroweak experiments and require m > 7TeV. This is for models with one extra
dimension, in which the gauge bosons propagate in the bulk but the fermions and
Higgs are confined to four dimensions. More on this shortly. There are also bounds
from astrophysics they are not model independent bounds. They are important for
large dimensions in which only the graviton propagates, and they depend on number
of large dimensions. The strongest bounds arise from supernova emissions, giving
1/m < 10 −4 mm for the case of two large extra dimensions. Although researchers
such as Einstein and Pauli had studied such theories for dozens of years this path has
been abandoned. Extra dimensions resurfaced when string theory was formulated.
It was found out that strings are much fussier than particles, (super) strings can
propagate quantum mechanically only in a limited number of dimensions. In fact,
not only was the number of allowed possible dimensions dramatically reduced, the
allowed values of the number of space-time dimensions did not include the value
4. These dimensions are usually required to be 10 or 26 (The origin of this basic
number is, at this stage, disappointingly technical.) In fact while the numbers 10
and 26 result from the theory they need not always be related directly to extra
dimensions. When the extra dimensions emerged in string theory it was suggested
in the spirit of KK that any extra dimensions are very small. As interest was diverted
from string theory back to field theory the ideas of KK were revised allowing one
to take into account the extra interactions discovered since the original work of
KK. It turned out that if one which to trade all the known standard model gauge
interactions, i.e. the all SU(3) ∗ SU(2) ∗ U(1) interactions, for a theory of gravity
alone one was led to consider eleven dimensional gravity. Such theories raised
interest also for other reasons. The return of string theory brought back with it an
intense study of extra dimensions. In particular extra dimensions in the form of
compact Calabi-Yau manifolds which allowed to have four dimensional effective
theories with N = 1 SUSY. The topology of the extra dimensions determined in
some cases the number of zero modes on them, that is the spectrum of massless
particles. The number of generations of particles for example could be correlated
thus to the topology of those extra dimensions offering a solution to the origin of
the repetitive structure of the elementary particles. A vast amount of research on the
possible extra dimensions is ongoing. There have also been attempts to understand
dynamically the origin of the difference between the four large extra dimensions and
the rest. Some efforts were directed to explain how a spontaneous breaking of space
time symmetries could lead to such asymmetry in the properties of the different
dimensions. Solid state systems such as liquid crystals also exhibit such differences.
Other efforts focused on determining why an expanding universe would expand
asymmetrically after a while only in four directions. Another development occurred
501
Thus the KK ideas provide an explanation of the quantization of electric charge.
(Note also that charge conjugation is the parity tranformation y → −y in the fifth
dimesion.) If one indeed identifies the fundamental unit of charge e =
√
2km with
the charge on the electron, then one is forced to take m to be very large: the Planck
mass 10 19 GeV. Such extra scalar particles seem to be present abundantly in string
theories as well. The experimental question was mainly to set bounds on the value
of the radius R(= 1/m) of the fifth dimension. The limits are obtained from precision
electroweak experiments and require m > 7TeV. This is for models with one extra
dimension, in which the gauge bosons propagate in the bulk but the fermions and
Higgs are confined to four dimensions. More on this shortly. There are also bounds
from astrophysics they are not model independent bounds. They are important for
large dimensions in which only the graviton propagates, and they depend on number
of large dimensions. The strongest bounds arise from supernova emissions, giving
1/m < 10 −4 mm for the case of two large extra dimensions. Although researchers
such as Einstein and Pauli had studied such theories for dozens of years this path has
been abandoned. Extra dimensions resurfaced when string theory was formulated.
It was found out that strings are much fussier than particles, (super) strings can
propagate quantum mechanically only in a limited number of dimensions. In fact,
not only was the number of allowed possible dimensions dramatically reduced, the
allowed values of the number of space-time dimensions did not include the value
4. These dimensions are usually required to be 10 or 26 (The origin of this basic
number is, at this stage, disappointingly technical.) In fact while the numbers 10
and 26 result from the theory they need not always be related directly to extra
dimensions. When the extra dimensions emerged in string theory it was suggested
in the spirit of KK that any extra dimensions are very small. As interest was diverted
from string theory back to field theory the ideas of KK were revised allowing one
to take into account the extra interactions discovered since the original work of
KK. It turned out that if one which to trade all the known standard model gauge
interactions, i.e. the all SU(3) ∗ SU(2) ∗ U(1) interactions, for a theory of gravity
alone one was led to consider eleven dimensional gravity. Such theories raised
interest also for other reasons. The return of string theory brought back with it an
intense study of extra dimensions. In particular extra dimensions in the form of
compact Calabi-Yau manifolds which allowed to have four dimensional effective
theories with N = 1 SUSY. The topology of the extra dimensions determined in
some cases the number of zero modes on them, that is the spectrum of massless
particles. The number of generations of particles for example could be correlated
thus to the topology of those extra dimensions offering a solution to the origin of
the repetitive structure of the elementary particles. A vast amount of research on the
possible extra dimensions is ongoing. There have also been attempts to understand
dynamically the origin of the difference between the four large extra dimensions and
the rest. Some efforts were directed to explain how a spontaneous breaking of space
time symmetries could lead to such asymmetry in the properties of the different
dimensions. Solid state systems such as liquid crystals also exhibit such differences.
Other efforts focused on determining why an expanding universe would expand
asymmetrically after a while only in four directions. Another development occurred