Description

Quantum field theory is conventionally founded upon Feynman diagrams. Physical processes (eg probabilities of different outcomes of scattering experiments in the LHC) are computed by adding together lots of Feynman diagrams each one of which represents an integral.

Graph theory on the other hand is the study of graphs which in this context is a collection of "vertices" or "nodes" and a collection of edges that connect pairs of vertices.

Feynman diagrams are essentially (generalised) graphs and there are many connections between Feynman diagrams and graph theory (see eg here for an intriguing connection between Feynman diagrams and the four colour theorem from 1973). In this project we will look generally at graph theory and in particular its applications to quantum field theory. We will probably start by learning in parallel some basic graph theory, whilst also building up an understanding of Feynman diagrams and their relation to QFT. The project then has the potential to develop in various different directions according to taste. A more pure inclined student may want to look at the Hopf algebra approach to renormalisation of Kreimer. Alternatively a more physics minded student could look at and compare various current programmes for computing Feynman diagrams and applications in the standard model of particle physics. In particular, for the ambitious, computer-programmer-oriented student, I have one specific application in mind related to my own research: after assessing available graph theory software you can either utilise available programmes or develop your own computer code to help in understanding a specific application of graph theory at the forefronts of current research in quantum field theory.

Prerequisites

Resources

One will first need a basic familiarity with quantum field theory, in particular Feynman diagrams. Introductory texts on quantum field theory such as  Quantum Field Theory in a Nutshell by A.Zee, Princeton University Press, Quantum Field Theory by Itzykson, C. and Zuber, J. B. New York: McGraw-Hill, 1980 or Quantum Field Theory of Point Particles by Brian Hatfield will be useful.

You will also need to introduce yourself to graph theory. Wikipedia is a fine place to start.