Ben Allanach

Symmetries, Particles and Fields Ben Allanach

Symmetries, Particles and Fields is a textbook for an online course of the same name, freely available on youtube. It grew out of a Master's level course at the University of Cambridge to prepare students for a Ph. D. in theoretical particle physics.


Lie groups and Lie algebras are important in the construction of quantum field theories that describe interactions between known particles. One particle states are described in terms of irreducible representations of the Poincare group, a Lie group. Quantum fields may be acted on by operators of the Poincare group. Gauge theories, which describe many of the interactions in the Standard Model of particle physics, also rely on Lie groups.


The book assumes knowledge of quantum mechanics, linear algebras, and vector spaces at the undergraduate level. Knowledge of quantum field theory is not required, although the book was designed with the assumption that some basic quantum field theory is studied simultaneously (in particular, the construction of Lagrangian densities in terms of fields); then, a few applications will make more sense.


After some basic properties and preliminaries, we introduce matrix Lie groups, which rely on continuous parameters. Differentially, these act as a Lie algebra. The exponential map connects the Lie algebra to the Lie group.


We then introduce representations in terms of square matrices, describing how to construct various new representations in terms of combinations of others.


The group of rotations in three-dimensional space SO(3) is examined, along with SU(2) and the connection to angular momentum states in quantum theory. Representations of each are covered.


The relativistic symmetries (the Lorentz group and the Poincare group in four dimensions) are studied from the point of view of their group elements and Lie algebras.


Analysis of compact simple Lie algebras and their finite representations comes from mapping them to a geometrical picture involving roots and weights via the Cartan matrix. An overview of the results of the Cartan classification of simple Lie algebras is included.


An application in terms of representations of a global SU(3) F flavour symmetry explains some features of the spectrum of hadronic particles. Further properties of the spectrum lead one to introduce an additional local SU(3)c colour symmetry leading to a particular gauge theory called quantum chromodynamics.


We cover abelian and non-abelian gauge theories before returning to irreducible induced representations of the Poincare group, which are used to describe one-particle states.