Chapter 2
Symmetries in physics
Perfectly balanced, as all things should be. — Thanos
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 .
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 is one in understanding it as well.
In recent years, symmetries have also guided the development of machine learning algorithms in becoming more powerful and efficient. A particular focus is placed in this dissertation on such equivariant algorithms, which respect the symmetries and inductive biases of our high energy physics data. This chapter lays the foundation for these ideas, which we discuss in more detail in Chapter 7 and contribute to in Chapter 16.
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.