SS18 Advanced Quantum Field Theory



The lecture is aimed at master students with an interest in theoretical physics. It is a crucial preparation for a master thesis in theoretical particle physics. The quantum field theory concepts discussed are however more widely applicable. The focus here will be on methods, rather than on phenomenology (as compared to the 'Theoretical particle physics' course).


  • 12.10: The results of the second exam can be found here.
  • 10.9: The second exam will take place in Hörsaal 2 on Tuesday, 2.10.2018, between 9:00 and 12:00. Please be there at 8:45. The rules remain the same as for the first exam.
  • 18.8: The results of the first written exam can be found here . 
  • 6.8: A couple of missing pages of the script have been added.
  • 6.8: The first exam will take place in Hörsaal 1 on Thursday, 9.8.2018, between 9:45 and 12:45. Please be there at 9:30. You can bring up to 4 pages (i.e. 2 two-sided A4 sheets) of handwritten notes and writing material, but no books or other notes. You will receive scratch paper from us. Please also bring your Student-ID card. If you intend to participate in the exam, please make sure that you are registered for it on TUMonline. Registration has been recently re-opened and will remain open until the day before the exam. If you are currently registered but will not attend, please de-register.
  • 13.7: results of the exercise bonus
  • 30.6: the first exam has been fixed for August 9.
  • 17.5: added bonus requirement for Sheet 5. 
  • 2.5: no lecture today.
  • 26.4: there is no central tutorial today.
  • 23.4: a small typo was corrected in eq. (5) of Exercise Sheet 2. The difference wrt v1 is that g was set to -1, to agree with the conventions adopted in the lecture notes.
  • The first lecture will be on the 11.4.2018. We will have a central tutorial on group theory (Thu 12.4 after the lecture) and hand out the first exercise sheets. 


  • Quantization and renormalization of non-abelian gauge theories
  • Quantum theory of spontaneous symmetry breaking (non-linear symmetry representation, Higgs mechanism, effective potential)
  • General relativity as an effective QFT of gravitons (spin-2 fields)
  • Anomalies (chiral, dilatation)
  • Topology of the vacuum and Instantons
  • Supersymmetry



Wednesday 8:30-10  (Theory seminar room, PH3344)
Thursday 10-11:30 (Theory seminar room, PH3344)

Central Tutorial

Thursday 12-14 (Minkowski room, PH1121


Wednesday 10-12 (Physics container, C.3203)
Wednesday 12-14 (Theory library, PH3343)


Best: QFT1 at TUM, e.g. Martin Benekes lectures, or minimally: Relativity, Particles, Fields (script.pdf)


S. Weinberg - Quantum Field Theory (1 & 2)     
Peskin/Schroeder - An Introduction to Quantum Field Theory
Preskill - Lecture notes (on his Caltech website)
Schwartz - Quantum Field Theory and the Standard Model
Cheng/Li - Gauge theory of elementary particle physics

Sign conventions across QFT books, see here

Lecture Notes

(Please contact Javi or Ennio for the password, it will be also announced during the lecture.)

1 p1-19 (geometry of gauge symmetry, gauge redundancy vs. symmetry, non-abelian gauge symmetry, pure Yang-Mills, Chern-Simons current)

2 p19-32 (pure Yang-Mills, equation of motion, Euclidean YM and BPST, self-dual solutions, winding number)

p32-55 (gravity as a gauge theory, equivalence principle, vierbein, local so(1,3), spinor and gravity, curvature/torsion, Einstein-Hilbert equations)

4 p56-73 (Faddeev-Popov, U(1) warmup and non-abelian gauge theories, BRST invariance for QED and QCD, axial gauges)

p74_86 (BRST and nilpotent Slavnov operator, cohomology, unitarity of restricted S-matrix and BRST, Feynman rules for non-abelian gauge theories, QCD bound states, colored fermion loop)

p87_100 (one loop renormalization of non-abelian gauge theories, determination of counter-terms, beta-function, asymptotic freedom and dimensional transmutation, charge universality)

p101_120 (spin2 fields, longitudinal fields and spin1, removal of dangerous ghosts leads to unique Proca action and gauge redundancy requirement in massless case, longitudinal spin2 ghosts and Pauli-Fierz Lagrangian, local translational symmetry and why no massless spin 3, only non-renormalizable non-linear terms in spin2)

p121_156  p157_ff (spontaneous symmetry breaking, QM vs. QFT, linear and non-linear sigma model, Goldstone theorem, effective action Γ, 1PI, properties of Γ, Higgs mechanism)

anomalies (gauge anomalies, linear divergent diagrams, Fujikawa calculation [E Salvioni], complex rep.'s and Αabc, gravitational anomaly, dilatations and trace anomaly)

10 CCWZ (non-linear sigma model,SO(N)/SO(N-1), SO(4)~SU(2)^2, general G/H and CCWZ, transformation properties under H and G/H, Maurer-Cartan form, SSB of chiral symmetry, effective chiral lagrangian, mass spurions, Gellmann-Okuba, electro-weak interactions, higher-orders and cut-off scale, NDA)

11 WZW (pi0 -> gaga, naive calculation, Sutherland-Veltman paradox, leading order anomaly term via Fujikawa, discrete symmetries of QCD vs. pion parity, magnetic monopole in a sphere, Witten's 5D term, Wess Zumino, winding number N, trickle down gauging, WZW, comparison to Fujikawa N = Nc )

Central Tutorial

Basics of Lie groups  (12.4.2018, Ennio Salvioni)

Gluon scattering and unitarity, Weinberg-Witten Theorem  (19.4.2018, Reuven Balkin)

Transversity of gauge self-energy from BRST  (03.5.2018, Ennio Salvioni)

Exercise Sheets

Exercise coordination: Dr. Javi Serra ( and Dr. Ennio Salvioni (

Sheet 1  (Exercise 2 is due on 18.4.2018 for bonus)

Sheet 2  (Exercises 2 and 3 are due on 25.4.2018 for bonus)  Solution of Ex. 1

Sheet 3  (Exercise 2 is due on 02.5.2018 for bonus)

Sheet 4  (Exercise 2 is due on 09.5.2018 for bonus)

Sheet 5  (Exercise 1 is due on 23.5.2018 for bonus)  Solution of Ex. 2

Sheet 6  (Exercise 1 is due on 30.5.2018 for bonus) 

Sheet 7  (Exercise 2 is due on 13.6.2018 for bonus)

Sheet 8  (Exercise 3 is due on 20.6.2018 for bonus)

Sheet 9  (Exercise 1 is due on 27.6.2018 for bonus)

Sheet 10  (Exercise 1 is due on 04.7.2018 for bonus)

Sheet 11  (Exercise 2 is due on 11.7.2018 for bonus)