Introduction to the Quantum Field Theory

2026/2027

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered at:

Faculty of Mathematics

Course type:

Optional course (faculty)

When:

3, 4 module

Instructor

Dunin-Barkowski, Petr

Course Syllabus

Abstract

"Today, the standard model of particle physics is a quantum field theory (QFT). Besides its main role in modern physics, QFT has many uses in pure math. For example, quantum knot invariants and Gromov-Witten invariants of symplectic manifolds came from QFT. ""Normal"" quantum mechanics studies systems with a fixed number of particles. However, QFT studies fields (like the electromagnetic field, not the math field of complex numbers). The small changes in these fields act like quantum particles. These particles can appear and disappear (""be born"" and ""die""), and they have an infinite number of degrees of freedom. This course will introduce the basic ideas of QFT from scratch. These include the Fock space, operators on it, and ""path integration."" We will strictly define all the math objects we use. The main example will be the quantum theory of a scalar field. In physics, a classical scalar field is defined via just a single scalar value at each space point. This means its state at any time is just an ordinary real scalar function over space (unlike a vector field, such as the electromagnetic field). In the standard model of particle physics, only the Higgs boson field is a scalar field (but it is a complex scalar). Still, it is very useful to study the scalar field, even one simpler than the Higgs field, on its own. It helps you understand the math and effects of QFT using the simplest possible example. The course will cover ""perturbation theory"" for the scalar field (first-order changes using a small parameter). It will also show ways to calculate the probabilities of different particle events. The math we will use includes operators on Hilbert spaces and generalized functions (distributions). All the math you need for this will be explained. You do not need to have taken classical mechanics, classical field theory, or quantum mechanics before. However, knowing them will definitely help. All the necessary details from these courses will be explained."