###### Quantum Field Theory I

### Quantum Field Theory I

###### Notes and problems from UofT PHY2403 2018

# About the Book

This is a set of course notes from the Fall 2018, University of Toronto Quantum Field Theory (PHY2403), taught by Prof. Erich Poppitz.

#### Table of Contents

- Preface
- Contents
- List of Figures
- 1 Fields, units, and scales.
- 1.1 What is a field?
- 1.2 Scales.
- 1.2.1 Bohr radius.
- 1.2.2 Compton wavelength.
- 1.2.3 Relations.
- 1.3 Natural units.
- 1.4 Gravity.
- 1.5 Cross section.
- 1.6 Problems.
- 2 Lorentz transformations.
- 2.1 Lorentz transformations.
- 2.2 Determinant of Lorentz transformations.
- 2.3 Problems.
- 3 Classical field theory.
- 3.1 Field theory.
- 3.2 Actions.
- 3.3 Principles determining the form of the action.
- 3.4 Principles (cont.)
- 3.4.1 d = 2.
- 3.4.2 d = 3.
- 3.4.3 d = 4.
- 3.4.4 d = 5.
- 3.5 Least action principle.
- 3.6 Problems.
- 4 Canonical quantization, Klein-Gordon equation, SHOs, momentum space representation, raising and lowering operators.
- 4.1 Canonical quantization.
- 4.2 Canonical quantization (cont.)
- 4.3 Momentum space representation.
- 4.4 Quantization of Field Theory.
- 4.5 Free Hamiltonian.
- 4.6 QM SHO review.
- 4.7 Discussion.
- 4.8 Problems.
- 5 Symmetries.
- 5.1 Switching gears: Symmetries.
- 5.2 Symmetries.
- 5.3 Spacetime translation.
- 5.4 1st Noether theorem.
- 5.5 Unitary operators.
- 5.6 Continuous symmetries.
- 5.7 Classical scalar theory.
- 5.8 Last time.
- 5.9 Examples of symmetries.
- 5.10 Scale invariance.
- 5.11 Lorentz invariance.
- 5.12 Problems.
- 6 Lorentz boosts, generators, Lorentz invariance, microcausality.
- 6.1 Lorentz transform symmetries.
- 6.2 Transformation of momentum states.
- 6.3 Relativistic normalization.
- 6.4 Spacelike surfaces.
- 6.5 Condition on microcausality.
- 7 External sources.
- 7.1 Harmonic oscillator.
- 7.2 Field theory (where we are going).
- 7.3 Green's functions for the forced Klein-Gordon equation.
- 7.4 Pole shifting.
- 7.5 Matrix element representation of the Wightman function.
- 7.6 Retarded Green's function.
- 7.7 Review: ``particle creation problem''.
- 7.8 Digression: coherent states.
- 7.9 Problems.
- 8 Perturbation theory.
- 8.1 Feynman's Green's function.
- 8.2 Interacting field theory: perturbation theory in QFT.
- 8.3 Perturbation theory, interaction representation and Dyson formula.
- 8.4 Next time.
- 8.5 Review.
- 8.6 Perturbation.
- 8.7 Review.
- 8.8 Unpacking it.
- 8.9 Calculating perturbation.
- 8.10 Wick contractions.
- 8.11 Simplest Feynman diagrams.
- 8.12 Phi fourth interaction.
- 8.13 Tree level diagrams.
- 8.14 Problems.
- 9 Scattering and decay.
- 9.1 Additional resources.
- 9.2 Definitions and motivation.
- 9.3 Calculating interactions.
- 9.4 Example diagrams.
- 9.5 The recipe.
- 9.6 Back to our scalar theory.
- 9.7 Review: S-matrix.
- 9.8 Scattering in a scalar theory.
- 9.9 Decay rates.
- 9.10 Cross section.
- 9.11 More on cross section.
- 9.12 d(LIPS)_2.
- 9.13 Problems.
- 10 Fermions, and spinors.
- 10.1 Fermions: R3 rotations.
- 10.2 Lorentz group.
- 10.3 Weyl spinors.
- 10.4 Lorentz symmetry.
- 10.5 Dirac matrices.
- 10.6 Dirac Lagrangian.
- 10.7 Review.
- 10.8 Dirac equation.
- 10.9 Helicity.
- 10.10 Next time.
- 10.11 Review.
- 10.12 Normalization.
- 10.13 Other solution.
- 10.14 Lagrangian.
- 10.15 General solution and Hamiltonian.
- 10.16 Review.
- 10.17 Hamiltonian action on single particle states.
- 10.18 Spacetime translation symmetries.
- 10.19 Rotation symmetries: angular momentum operator.
- 10.20 U(1)_V symmetry: charge!
- 10.21 U(1)_A symmetry: what was the charge for this one called?
- 10.22 CPT symmetries.
- 10.23 Review.
- 10.24 Photon.
- 10.25 Propagator.
- 10.26 Feynman rules.
- 10.27 Example: muon pair production
- 10.28 Measurement of intermediate quark scattering processes.
- 10.29 Problems.
- A Useful formulas and review.
- A.1 Review of old material.
- A.2 Useful results from new material.
- B Momentum of scalar field.
- B.1 Expansion of the field momentum.
- B.2 Conservation of the field momentum.
- C Reflection using Pauli matrices.
- D Explicit expansion of the Dirac u,v spinors.
- D.1 Compact representation of
- E Mathematica notebooks
- Bibliography

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