3 Grand Unified Theories

Not all of the symmetries of ${G_{\mbox{\rm SM}}}$, the gauge group of the Standard Model, are actually seen in ordinary life. This is because some of the symmetries are `spontaneously broken'. This means that while they are symmetries of the laws of physics, they are not symmetries of the vacuum. To see these symmetries we need to do experiments at very high energies, where the asymmetry of the vacuum has less effect. So, the behavior of particles at lower energies is like a shadow of the fundamental laws of physics, cast down from on high: a cryptic clue we must struggle to interpret.

It is reasonable to ask if this process continues. Could the symmetries of the Standard Model be just a subset of all the symmetries in nature? Could they be the low energy shadows of laws still more symmetric?

A grand unified theory, or GUT, constitutes a guess at what these `more symmetric' laws might be. It is a theory with more symmetry than the Standard Model, which reduces to the Standard Model at lower energies. It is also, therefore, an attempt to describe the physics at higher energies.

GUTs are speculative physics. The Standard Model has been tested in countless experiments. There is a lot of evidence that it is an incomplete theory, and some vague clues about what the next theory might be like, but so far there is no empirical evidence that any GUT is correct--and even some empirical evidence that some GUTs, like ${\rm SU}(5)$, are incorrect.

Nonetheless, GUTs are interesting to theoretical physicists, because they allow us to explore some very definite ideas about how to extend the Standard Model. And because they are based almost entirely on the representation theory of compact Lie groups, the underlying physical ideas provide a marvelous playground for this beautiful area of mathematics.

Amazingly, this beauty then becomes a part of the physics. The representation of ${G_{\mbox{\rm SM}}}$ used in the Standard Model seems ad hoc. Why this one? Why all those seemingly arbitrary hypercharges floating around, mucking up some otherwise simple representations? Why do both leptons and quarks come in left- and right-handed varieties, which transform so differently? Why do quarks come in charges which are in units $\frac{1}{3}$ times an electron's charge? Why are there the same number of quarks and leptons? GUTs can shed light on these questions, using only group representation theory.