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1.2 The module MATH4017
The aim of this module is to provide an introduction to QFT. We shall focus mainly on QFTs that are defined on the Minkowski spacetime, as required for applications to high-energy physics, but similar techniques apply to QFTs that are defined on
other spaces or spacetimes \(M\) as well, although there are sometimes new effects such as e.g. particle creation in curved spacetimes. Since QFT is one of the cornerstones of modern physics, every course on theoretical and/or mathematical physics
should expose students to this topic, and this is precisely what we are going to do. Due to the immense relevance of QFT in various areas of physics, as well as its rich and fruitful interplay with modern mathematics, see Section 1.1, it is likely that you will benefit from this module in your future career, whether this will be in physics, mathematics or something else.
The way how we approach QFT is by focusing first on the simplest examples in order to understand the underlying mechanisms. After a brief introduction to classical field theory, we study the quantization of the free Klein-Gordon field on the
Minkowski spacetime, which is a scalar field that satisfies a linear equation of motion. In this context we shall substantiate the claim from Section 1.1 that quantum fields are operator valued
functions \(\hat {\Phi }(x)\) whose representations on a Hilbert space \(\HH \) give rise to a concept of particles. For the free Klein-Gordon quantum field, these particles do not interact with each other as a consequence of linearity of the equation
of motion. We shall then study more sophisticated examples of quantum scalar fields that display nonlinear features and thereby lead to interactions between particles. Treating these nonlinearities perturbatively, we will derive graphical techniques,
called Feynman diagrams and Feynman rules, that allow us to compute transition amplitudes for particle interactions, such as scattering processes and particle decays.
After understanding the foundations of QFT for the simplest case of scalar fields, we will spend some time to generalize our techniques to other types of fields that appear in nature. Most notably, we will develop the free quantum Dirac field, which
includes the electron as an example, and also study the quantization of the electromagnetic field, leading to photons. These new examples will come with new difficulties that have to be overcome, but this will pay off because there will be interesting
new features.
After the basics are settled, we will dive into more advanced topics in QFT. The first goal is to make sense of the infinities we encounter at higher orders in perturbation theory, which leads us to the development of renormalization techniques. The
second goal is to understand the structure of the standard model of particle physics, in particular its particle content and their interactions. Additional advanced topics will be discussed during the student projects/presentations at the end of the
second semester.