Notes on symmetries in particle physics

TL;DR

This paper provides an introductory overview of symmetries in quantum field theory, focusing on group theory, representations, and their application to the Standard Model, including the Higgs mechanism.

Contribution

It offers a comprehensive pedagogical account of symmetry groups, their representations, and their role in formulating the Standard Model of particle physics.

Findings

Detailed exposition of SU(N), Lorentz, and Poincare group representations

Explanation of gauge and global symmetries in quantum field theory

Description of the Higgs mechanism and mass generation in the Standard Model

Abstract

These are introductory notes on symmetries in quantum field theory and how they apply to particle physics. The notes cover the fundamentals of group theory, their representations, Lie groups, and Lie algebras, along with an elaborate discussion of the representations of SU(N), Lorentz, and Poincare groups and their respective algebras. We spend a lot of time on the realisation of these symmetry groups in quantum field theory, as both global and gauge symmetries, as well as their spontaneous breaking and the Higgs mechanism. In the end, we culminate all the lessons from the course to enumerate the symmetries and field content of the Standard Model of particle physics and write down the Standard Model Lagrangian. Special consideration is given to how the weak-force gauge bosons and the matter fields obtain their mass via the Higgs mechanism.