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It is possible to add a
fifth dimension to space-time and thereby incorporate
Maxwell's equations of electromagnetism into
Einstein's equations of general relativity, but this fifth
dimension is not extended like the other four. Instead it
must be curled up into a very small tight closed loop. |
String theory introduces the possibility of many more
dimensions and the hope is that these extra dimensions can be
curled up, or 'compactified', in some appropriate way to account
for all the forces of nature. |
String theory has been generalized to membrane theory, lump
theory .. in term of the number of dimensions in which an
elementary object can be extended. These theories have
also been generalized into
superspace, a space with added
Grassmann dimensions in which interaction fields can be
rotated into matter fields and are thus on an equal footing. |
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There are now many new ideas, any of which might be involved in
a final theory of everything. For example, a theory of
quantized rigid self-dual
super p-branes in a
supergravity background is just one of many scenarios
that could conceivably lead to new physics. |
Calabi-Yau Compactification |
Many theoretical physicists currently believe that the unique
theory of everything manifests at low energies as an effective
theory of membranes (2-branes) and 5-branes in an 11 dimensional
background. However the theory is actually still
undiscovered, although it has already been named
M-theory. |
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Consider a simple natural generalization of
knots in four dimensions to
quaternionic knots
(quots) in eight dimensions. One can come
tantalizingly close to finding an infinite hierarchy of such
quotted objects. |
One can construct a beautifully elegant expression for a theory
of doubly self-dual
lumps in
eight dimensions. Unfortunately, as with string theory
and M-theory, no testable predictions have been forthcoming. |
A theory of everything in which all forms
of matter and forces are unified into a single description of
the world is also a unification of ideas whereby
all ideas and concepts are reducible to a beautiful deep
mathematical statement with few, if any, free parameters. |
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