Types of quantum field thories
The standard model of particle physics asserts that the fundamental quantum physical fields and particles are modeled as sections of and connections on a vector bundle that is associated to a -principal bundle, where the Lie group – called the gauge group – is the product of (special) unitary groups (or rather a quotient of this by ) and where the representation of used to form the associated vector bundle looks fairly ad hoc on first sight.
A grand unified theory (“GUT” for short) in this context is an attempt to realize the standard model as sitting inside a conceptually simpler model, in particular one for which the gauge group is a bigger but simpler group , preferably a simple group in the technical, which contains as a subgroup. Such a grand unified theory would be phenomenologically viable if a process of spontaneous symmetry breaking at some high energy scale – the “GUT scale” – would reduce the model back to the standard model of particle physics without adding spurious extra effects that would not be in agreement with existing observations in experiment.
The terminology “grand unified” here refers to the fact that such a single simple group “\hat G” would unify the fundamental forces of electromagnetism, the weak nuclear force and the strong nuclear force in a way that generalizes the way in which the electroweak field already unifies the weak force and electromagnetism, and electromagnetism already unifies, as the word says, electricity and magnetism.
Since no GUT has been fully validated yet, GUTs are models “beyond the standard model”. Often quantum physics “beyond the standard model” is expected to also say something sensible about quantum gravity and hence unify not just the three gauge forces bot also the fourth known fundamental force, which is gravity. Models that aim to do all of this would then “unify” the entire content of the standard model of particle physics plus the standard model of cosmology, hence “everything that is known about fundamental physics” to date. Therefore such theories are then sometimes called a theory of everything?.
(Here it is important to remember the context, both “grand unified” and “of everything” refers to aspects of presently available models of fundamental physics, and not to deeper philosophical questions of ontology.)
An original article with an eye towards supergravity unification is
A survey of arguments for the hypothesis of grand unification with an eye towards heterotic string unification is given in
A good introduction to GUTs for mathematicians is
Blog discussion of this is at
John Baez, The algebra of grand unified theories I
John Baez, The algebra of grand unified theories II
John Baez, The algebra of grand unified theories III