Quantum Field Theory

MSc Physik / MSc Physics

Summer Semester 2023


Begin of course

Wednesday, 12. April 2023

Time & Place

  • Wednesday, 08:00 - 09:30, Seminar Room 5.331, Pfaffenwaldring 57
  • Friday, 09:45 - 11:15, Seminar Room 5.331, Pfaffenwaldring 57

General Information

  • The lecture and the tutorials will be given in English.
  • There are two types of exercises: "Written" exercises are handed in in class and graded/corrected by the tutor. "Oral" exercises are discussed in the tutorials and presented by students on the blackboard.
  • Admission to the final exam requires 80% of the written scores, 66% of the oral scores, and presenting a problem on the blackboard twice.
  • Please register for the exercise groups online. To do so, you need the Lecture Key given in the first lecture.
  • If you assign a password at the registration, you can request your current scores here.

Recordings of the lectures can be accessed via ILIAS.

This course can be used for the following modules:
Course Module Description ECTS Lectures
MSc. Physik 107280 Schwerpunkt 12 1-31 (26+5 lectures)
MSc. Physik 68030 Ergänzung 9 1-26 (26 lectures)
MSc. Physics 68030 Semicompulsory 9 1-26 (26 lectures)

The 5 additional lectures are part of the block course "Quantum Field Theory - Advanced Topics" (LV-Nr. 045575002) which takes place during the first week after the lecture period. These lectures are mandatory if you want to use the lecture as "Wahlpflichtmodul Schwerpunkt". In this case, the contents of these lectures will be part of the oral exam.


There will be an oral examination at the end of the course. Details will be given in the lecture.


This course follows the exposition of Peskin & Schroeder.


The goal is to gain a thorough understanding of relativistic quantum field theory, the concepts of Feynman diagrams, renormalisation for quantum electrodynamics, and to extend this knowledge to non-abelian gauge theories. In particular:

  • Relativistic quantum mechanics and Dirac equation
  • Path integral formalism
  • Quantisation - Free fields
  • Interacting fields and Feynman diagrams
  • Elementary processes and first corrections
  • Renormalisation
  • Non-abelian gauge fields


The concept of second quantization is necessary to understand the quantization of fields. If you did not learn about second quantization in your advanced quantum mechanics course, I suggest that you catch up by self-study (any textbook on advanced quantum mechanics covers this topic). The same goes for the basic concepts of special relativity.

In particular, you should be familiar with the following concepts:

  • Creation and annihilation operators
  • Bosonic and fermionic (anti)commutation relations
  • Constructing the bosonic/fermionic Fock space from the vacuum state (number states)
  • Basics of special relativity: Lorentz group, Minkowski metric, Lorentz scalars and four-vectors ...

Knowledge of relativistic quantum mechanics it not required; however, it is certainly helpful if you have seen the Klein-Gordon and Dirac equation before. I will briefly rederive the Dirac equation from a "field theory" point of view.


These notes follow mostly the exposition of Peskin & Schroeder. They are not an extension of the material covered in the lectures but the script that I use to prepare them. Please have a look at Peskin & Schroeder and the given references for more comprehensive coverage; the corresponding pages are noted in the headers (→ P&S • pp. xx-yy).

Download PDF


No. Date Notes Topics (planned, may be subject to changes)
1 12.04.23 PDF - Lagrangian and Hamiltonian formalism
- Transformations
2 14.04.23 PDF - Symmetries
- Noether's theorem
- Energy-momentum tensor
3 19.04.23 PDF - Quantization of the Klein-Gordon field
- The Klein-Gordon field in spacetime
4 21.04.23 PDF - Causality of the Klein-Gordon field
- Feynman propagator of the Klein-Gordon field
5 26.04.23 PDF - The Dirac equation
6 28.04.23 PDF - Free-particle solutions of the Dirac equation
- Dirac field bilinears
- Quantization of the Dirac field
7 03.05.23 PDF - Conserved charges
- Spin and statistics
- The Dirac propagator
- Causality
8 05.05.23 PDF - Discrete symmetries of the Dirac theory
- Interacting QFTs
9 10.05.23 PDF - Perturbation expansion of correlation functions
- Wick's theorem
10 12.05.23 PDF - Feynman diagrams
- Feynman rules
- Disconnected Feynman diagrams
- Vacuum energy
11 17.05.23 PDF - Scattering cross sections
- S- and T-matrix
12 19.05.23 PDF - S-matrix elements from Feynman diagrams
13 24.05.23 PDF - Feynman rules for scattering amplitudes
- Wick's theorem for fermions
- The photon propagator
- Feynman rules for quantum electrodynamics
14 26.05.23 PDF - Electron-electron scattering
- Electron-positron scattering
- The muon-antimuon production cross section
15     - Overview of radiative corrections
- Soft bremsstrahlung
- Formal structure of the electron vertex function
16     - The Landé g-factor
- Evaluation of the vertex integral
17     - Evaluation of the vertex integral (continued)
18     - Infrared divergence of the vertex function
- Källén–Lehmann spectral representation
19     - Field-strength renormalization
- Physical mass vs. bare mass
20     - The LSZ reduction formula
21     - Electric charge renormalization
- Dimensional regularization
22     - Vacuum polarization
- Lamb shift
- Running of the fine-structure constant
- Landau pole and Dyson's argument
23     - Systematics of UV-divergences
- Mass dimension and renormalizability
- Note on quantum gravity
24     - Bare perturbation theory
- Renormalized perturbation theory for Phi-4-theory
25     - The path integral in quantum mechanics
- Derivation of the Schrödinger equation
- The path integral for fields
26     - Correlation functions from path integrals
- Faddeev-Popov gauge-fixing procedure
- Photon propagator
27*     - Structure of the QED U(1) gauge symmetry
- Generalization to non-abelian gauge groups
28*     - Yang-Mills Lagrangian
- Higgs mechanism for U(1) gauge theory
- Goldstone theorem
29*     - Structure of the Standard Model
30*     - Glashow-Weinberg-Salam Theory
- Higgs mechanism in the Standard Model
31*     - Quantum Chromodynamics
- Summary

Lectures marked with * are part of the block course "Quantum Field Theory - Advanced Topics" (LV-Nr. 045575002) which takes place during the first week after the lecture period. These lectures are mandatory if you want to use the lecture as "Wahlpflichtmodul Schwerpunkt". In this case, the contents of these lectures will be part of the oral exam.

Problem Sets

Warning: We are well aware of the capabilities and limitations of state of the art transformer models. For example, here is how ChatGPT (GPT-4) solves Problem 1.2. It makes mistakes, but the results are undeniably impressive. In case you feel tempted: Contemplate why you decided to study physics in the first place, and why you signed up for this particular course. And, last but not least, remember the final exam.

No. Published Due Download Comments
1 13.04.23 21.04.23 Problem Set 1  
2 20.04.23 28.04.23 Problem Set 2  
3 27.04.23 05.05.23 Problem Set 3  
4 04.05.23 12.05.23 Problem Set 4  
5 11.05.23 19.05.23 Problem Set 5  
6 18.05.23 26.05.23 Problem Set 6  
7 25.05.23 09.06.23 Problem Set 7  


Tutor Room Day Time
Johannes Mögerle 5.331 Friday 11:30 - 13:00
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