“Lie groups, symmetry breaking, and particle physics”

This talk, given by a physicist to an audience of mathematicians, will be split into two parts. We will start by introducing Lagrangian mechanics and the action principle, which is central to many branches of physics research. We will then move forward into applying the action principle within the framework of the so-called Standard Model of particle physics (SM), which has been hugely successful in describing and predicting microscopic phenomena. Broadly speaking, there are two types of particles in nature: fermions (half-integer spin) and bosons (integer spin). The SM is most easily understood as a theory of fermions charged under the Lie groups SU(3) x SU(2) x U(1), which are regularly referred to as gauge groups. For the second part of the talk, after giving a concrete definition for what it means to be charged under a gauge group, we will show how the notion of local gauge invariance leads to the emergence of gauge bosons. We will then discuss symmetries between a subset of SM particles, and show how the process of symmetry breaking gives rise to composite particle states at low energies. Finally, after briefly summarizing the so-called dark matter puzzle, we will connect the symmetry breaking ideas to our recently proposed dark matter model.