This site contains the notes for the QFT tutorial sessions
(rules for hand-ins are here)

You can reach us at:
Francesco: francesco.torsello (at) Room: A5:1031
Jorge: jorge.laranaaragon (at) Room: A5:1054
Julius: julius.engelsoy (at) Room: A5:1054


Tutorials (done)

Tutorial #1   [Jo]   2018-11-11 Handout   Solutions
Notation, Lorentz invariance, Index gymnastics, Dirac delta function.

Tutorial #2   [Fr]   2018-11-13 Handout   Notes   Solutions  
Natural Units and Dimensional Analysis. SR recap II. Ladder operators and Heisenberg algebra.

Tutorial #3   [Jo]   2018-11-18 Handout   Fields overview   Solutions
Fourier Transform. The Lagrangian of the real scalar (KGF) field.

Tutorial #4   [Fr]   2018-11-25 Notes
Group theory. Invariance in form and value. Rotations in Euclidean 3D space and the deffnition of a group. The Lorentz group. The unitary group. The generators of a group of differentiable transformations.

Tutorial #5   [Ju]   2018-12-02 Handout   Part 1   Part 2
Part I. The quantization procedure. The inverse operator expansion. The evaluation of the commutator [a,a†]. The evaluation of the equal time commutator for the fields.

Tutorial #6   [Fr]   2018-12-09 Handout   Solutions   Overview
Part II. The expansion of the Hamiltonian. An introduction to the infinite contributions to the energy and the normal ordering of operators. (Additional references: Casimir effect.)

Symmetries and Conservation Laws. Noether’s theorem. (Additional references: Chapter 8 from Hagen Kleinert's Book on Particles and Quantum Fields)

Tutorial #7   [Fr]   2018-12-16 Handout I & II   Solutions   Notes
The Dirac field, Part I. Gamma matrices (Dirac equation), Dirac algebra. Part II. Lorentz invariance of the Dirac equation. The Dirac equation for a free particle. Properties of the energy projection operator. Additional problems.

Tutorials (scheduled)

Tutorial #8   [Fr,Ju]   2019-01-08
Dirac field and gamma matrices (contd).
Maxwell field, gauge fixing. Highlights from Problem Set 1.

Tutorial #9   [Ju]   2019-01-13 Handout
The Schrödinger, Heisenberg and Interaction Pictures. Perturbation Expansion of the S-Matrix. Wick’s Theorem. QED processes at 2nd order. (Additional references: The Schrodinger, Heisenberg and Interaction Pictures in QFT, Invariant commutation and propagation functions)

Tutorial #10   [Fr]   2019-01-20
Problems 7.1, 7.3 and 7.5 (Mandl & Shaw), Feynman amplitudes for QED.

Tutorial #11   [Fr]   2019-01-27) Handout
Cross-sections. The relativistic definition of flux. The spin-sums lemma. The e-e+ production in electromagnetic field.

Tutorial #12   [Jo]   2019-02-03) Handout
gamma^5 matrix and Left/Right-chiral fields. Weyl fields (and problem of coupling by mass). Transformation properties of vectors/axial vectors.

Tutorial #13   [Fr]   2019-02-10
Massive vector bosons (the Proca equation). Propagating degrees of freedom for spin-1 fields. EoM for the Yang-Mills Lagrangian. SU(2) charges of weak interactions.

Tutorial #14   [Fr]   2019-02-17 Handout
Higgs mechanism: generation of mass for Z bosons and photons, Higgs decays
Suggested reading: Schwartz ch 29.1-29.2, Mandl and Shaw chapter 18-19

Additional notes on spontaneous symmetry breaking

Tutorial #15   [Jo]   2019-02-24
Path Integrals. Preliminaries. Gaussian integrals. Grassmann numbers

Tutorial #16   [Jo]   2019-03-02
Path Integrals, Part II. Perturbation expansion for the phi^4-theory.

Tutorial #17   [Fr]   2019-03-09
Old exam problems.

Useful external resources

Tutorial Plan

Textbook: Mandl, Franz and Shaw, Graham - Quantum Field Theory (2nd ed., 2010),
ISBN: 9780471496830

Chapters in Mandl and Shaw
2 Lagrangian field theory
3 Klein-Gordon field
4 Dirac field
5 Photons
6 S-matrix expansion
7 Feynman diagrams
8 QED at lowest order
(9)(Radiative corrections)
11 Gauge theories
13 Path integrals
(15)(Asymptotic freedom)
16 Weak interactions
17 Gauge theory of weak interactions
18 Spontaneous symmetry breaking
19 Standard electro-weak theory
Chapters in Schwartz
2 Lorentz transformations
5 Cross-sections
10 Spinors and Dirac equations
Additional topics
Group theory(join with Schwartz, Ch 2)
Heisenberg algebra (ladder operators)(join with KG-field)

Note: Four chapters in parentheses (9, 10, 14 & 15) — if there is time.

Supplementary reading:

  • "Classical Mechanics" by H. Goldstein, C. P. Poole, J. L. Safko (chapter 13)
  • "Quantum Theory and the Standard Model" by Matthew D. Schwartz (Cambridge University Press)
  • "Quantum Field Theory" by Mark Srednicki. (You can try a prepublication draft of this book on the author's webpage here. Note that the author follows different signature convention for the metric than one usually used in Mandl and Shaw.)
  • Another book for path integrals in QFT is "Field Theory: A Modern Primer" by Pierre Ramond, section 3.1, 3.2, 4.1.