Chapter 2
Symmetries in physics

Symmetry is a powerful and beautiful way to understand nature. Intuitively, a symmetry is a transformation that leaves an object unchanged. For example, a plain square has a four-fold rotational symmetry: it looks identical rotated once, twice, thrice, or four times by $90^{\circ }$.

Similarly, in physics, a symmetry is a transformation that leaves the laws of physics unchanged. Electromagnetism, for example, is invariant to translations in space or time: electric charges and currents should behave the same in San Diego 5 years ago as in Geneva today. Understanding such symmetries, and accounting for them in our mathematical formulation, has been a guiding principle in the development of the SM over the 20th century, and will be a guiding principle in our understanding of it as well.

In this chapter, we first introduce the framework for describing symmetries, group theory, in Section 2.1. We then describe Lie algebras for continuous symmetries, and derive representations for the algebra corresponding to 3D rotations, in Section 2.2. We conclude in Section 2.3 with a discussion of the Lorentz and Poincaré groups, comprising the fundamental symmetries of spacetime, whose irreducible representations are what we call particles.