Stringy Spacetime
Partice physics utilises group theory to a large extent, the theory of elektroweak
interactions has a
U(1) x SU(2) structure and the the theory of strong interactions has a
SU(3) structure.
The impact of group theory on general relativity is not as obvious but
when delving deeper in the subject one finds that solutions of the Einstein equations is
dependent on 'Lie groups' (that can be said to be axiomatic of noneuclidean geometry)
A theory of everything should accordingly incorporate both the standard model and general
relativity, as candidates for such a theory has been proposed the E8 x E8 and SO(16) x SO(16)
string theories that assumes supersymmetry as the vital ingredient of a TOE.
A still more powerfull symmetry is indicated by workers in knot theory by the transhyperbolic group
(denoted as 'E' with the 'infinity sign'). This group is a continous extension of the 'monster group' that
is the largest of the 26 sporadic finite groups that together with 18 families of finite groups constitutes
the 'prime numbers' of group theory (each family has an infinite number of individuals).
Mathematicians describe this monster group as extremely rich in symmetries so from a Platonic view
of what a TOE should be like it seems very attractive.
When this large symmetry is then broken down into smaller parts an effective GUT would be formed.
An analogue to this could be seen in snowflakes that share a basic hexagonal symmetry but where
initial conditions at the phase transition gives each snowflake an individual shape. In the same way
different parts of the universe could have different setups of quantum fields and particles.
The mechanism of inflation would however separate such regions from each other by very large distances.