# Tutorials (done)

 Tutorial #1   [mk]   2018-11-05 Handout   Solutions Notation, Lorentz invariance, Index gymnastics, Dirac delta function. Tutorial #2   [ft]   2018-11-12 Handout / Notes 1, Notes 2 Natural Units and Dimensional Analysis. SR recap II. Ladder operators and Heisenberg algebra. Tutorial #3   [mk]   2018-11-19 Handout   Fields overview   Solutions Fourier Transform. The Lagrangian of the real scalar (KGF) field. Tutorial #4   [ft]   2018-11-26 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   [mk]   2018-11-28 Handout   Part 1 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   [mk]   2018-12-10 Handout 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   [ft]   2018-12-17 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. Tutorial #8   [ft]/[mk]   moved to 2018-12-17 Dirac field and gamma matrices (contd). Maxwell field, gauge fixing. Highlights from Problem Set 1. Tutorial #9   [mk]   moved to 2018-12-19) Handout   Solutions 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   [ft]   2019-01-21 Solutions, Feynman diagrams Problems 7.1, 7.3 and 7.5 (Mandl & Shaw), Feynman amplitudes for QED. Tutorial #11   [ft]   2019-01-28) Handout Solutions Cross-sections. The relativistic definition of flux. The spin-sums lemma. The e-e+ production in electromagnetic field. Tutorial #12   [mk]   2019-02-04) Handout Solutions gamma^5 matrix and Left/Right-chiral fields. Weyl fields (and problem of coupling by mass). Transformation properties of vectors/axial vectors. Tutorial #13   [ft]   2019-02-11 Solutions 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   [ft]   2019-02-18 Handout Solutions 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   [mk]   2019-02-25 Path Integrals. Preliminaries. Gaussian integrals. Grassmann numbers Tutorial #16   [mk]   2019-03-04 Path Integrals, Part II. Perturbation expansion for the phi^4-theory. Tutorial #17   [ft]   2019-03-11 Old exam problems.