8 Beyond the Standard Model
505
always possible to associate this number with the number of dimensions. In fact
the numbers appearing are 15 and 26 and they reflect from one point of view a
conformal anomaly in the theory describing first quantized string theory. In first
quantized field theory of particles can be expressed as a sum over one dimensional
field theories, i.e. quantum mechanics, of a particle moving in space time and
interacting along its space time trajectory. The action of the quantum mechanical
system is geometrical and describes in its simplest form the length of the particle’s
trajectory. The result should not depend on the parameterization of trajectory and
this should be general coordinate invariant. In a more fancy manner the action
describes a particle coupled to gravity evolving in one space time dimension. The
generalization to a description of the motion of a one dimensional object, a string,
follows. As the string is a one dimensional object, its motion in space time spans
a two dimensional manifold. The theory describing its motion is that of a particle,
coupled to gravity, moving in two space-time dimensions. The two dimensions are
called the world sheet and the theory describing them using first quantization is
called the World Sheet theory. The usual counting of quantum degrees of freedom
leads to the conclusion that gravity has minus one degrees of freedom in both one
and two dimensions. In other words the system is over constrained and will require
a higher degree of symmetry to be consistent. In addition the two dimensional
gravitational system has an anomaly which restricts the algebraic structure of the
conformal system describes by the string. A motion of a (super) string in (10) 26
flat dimensions removes the anomaly. There are many other ways to remove it and
perhaps many more to be discovered. Sometimes the background on which the string
can propagate has no obvious geometrical description, just an algebraic one. The
space in which the strings moves is called the Target Space. The interactions of
strings are described by them parting or joining, this is described (in a Euclidean
formulation) by a compact two dimensional surface with non-trivial topology. These
manifolds are called Riemman manifolds and are distinct by their topology. The
freely propagating string is described by a world sheet 2 sphere, the first interactions
occur when the world sheet is a torus. The theory is defined perturbatively by
summing with a certain weight over all such two dimensional Riemann surfaces.
The non-perturbative structure is non-trivial but less understood at this stage. But
setting that aside, one would like to detect extra dimensions experimentally. As the
idea of having extra dimensions was raised originally in field theory, upper bounds
on the length of such dimensions were already available, more recently under the
security net of string theory these bounds have been revised (they have been relaxed
by more than 10 orders of magnitude and brought up to the sub micron regime),
in fact I am not aware of any bound on a fundamental quantity in physics that
has been altered to such an extent by a theoretical idea. This fact about strings
illustrates the tension inherent in attempts to search for experimental verification
of extremely basic but very weakly coupled phenomena. Sitting in our chairs we
sometimes forget how frail gravity is. It is all the planet earth that gravitationally
pulls us towards its center, yet all this planetary effort is easily counterbalanced
by the electromagnetic forces applied by the few tiles beneath our chair. Although
string theory sets constraints on the possible number of dimensions, that is not to
505
always possible to associate this number with the number of dimensions. In fact
the numbers appearing are 15 and 26 and they reflect from one point of view a
conformal anomaly in the theory describing first quantized string theory. In first
quantized field theory of particles can be expressed as a sum over one dimensional
field theories, i.e. quantum mechanics, of a particle moving in space time and
interacting along its space time trajectory. The action of the quantum mechanical
system is geometrical and describes in its simplest form the length of the particle’s
trajectory. The result should not depend on the parameterization of trajectory and
this should be general coordinate invariant. In a more fancy manner the action
describes a particle coupled to gravity evolving in one space time dimension. The
generalization to a description of the motion of a one dimensional object, a string,
follows. As the string is a one dimensional object, its motion in space time spans
a two dimensional manifold. The theory describing its motion is that of a particle,
coupled to gravity, moving in two space-time dimensions. The two dimensions are
called the world sheet and the theory describing them using first quantization is
called the World Sheet theory. The usual counting of quantum degrees of freedom
leads to the conclusion that gravity has minus one degrees of freedom in both one
and two dimensions. In other words the system is over constrained and will require
a higher degree of symmetry to be consistent. In addition the two dimensional
gravitational system has an anomaly which restricts the algebraic structure of the
conformal system describes by the string. A motion of a (super) string in (10) 26
flat dimensions removes the anomaly. There are many other ways to remove it and
perhaps many more to be discovered. Sometimes the background on which the string
can propagate has no obvious geometrical description, just an algebraic one. The
space in which the strings moves is called the Target Space. The interactions of
strings are described by them parting or joining, this is described (in a Euclidean
formulation) by a compact two dimensional surface with non-trivial topology. These
manifolds are called Riemman manifolds and are distinct by their topology. The
freely propagating string is described by a world sheet 2 sphere, the first interactions
occur when the world sheet is a torus. The theory is defined perturbatively by
summing with a certain weight over all such two dimensional Riemann surfaces.
The non-perturbative structure is non-trivial but less understood at this stage. But
setting that aside, one would like to detect extra dimensions experimentally. As the
idea of having extra dimensions was raised originally in field theory, upper bounds
on the length of such dimensions were already available, more recently under the
security net of string theory these bounds have been revised (they have been relaxed
by more than 10 orders of magnitude and brought up to the sub micron regime),
in fact I am not aware of any bound on a fundamental quantity in physics that
has been altered to such an extent by a theoretical idea. This fact about strings
illustrates the tension inherent in attempts to search for experimental verification
of extremely basic but very weakly coupled phenomena. Sitting in our chairs we
sometimes forget how frail gravity is. It is all the planet earth that gravitationally
pulls us towards its center, yet all this planetary effort is easily counterbalanced
by the electromagnetic forces applied by the few tiles beneath our chair. Although
string theory sets constraints on the possible number of dimensions, that is not to