**The Standard Model** is the name particle physicists give to their unified theory of electromagnetism, and the weak and strong nuclear forces. The Standard Model is an example of a gauge theory (unrelated to the gauge integral). Gauge theories are parametrized by Lie groups. Particles in gauge theories possess internal state that does not correspond to a classical observable; this internal state is described by an element in the Lie group. The group for the Standard Model is U(1) x SU(2) x SU(3).

This is a subject that I’ve always meant to learn more about, but I’ve never had the chance. Gauge theories make the subject of Lie algebra representations more vivid. For example, the part of the Standard Model that describes hadrons (particles such as protons and neutrons) is SU(3). SU(3) was found by fitting the existing hadron data to an 8-dimensional representation of the Lie algebra su(3). The 8-dimensional representation is not the smallest possible representation of su(3); there is (pretty obviously) a 3-dimensional representation. Taking that representation seriously led to the discovery of quarks.

Here are a few survey articles about the Standard Model from ArXiv: