###### 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

#### Other books by this author

### The Leanpub 60-day 100% Happiness Guarantee

Within **60 days of purchase** you can get a **100% refund** on any Leanpub purchase, in **two clicks**.

See full terms

### 80% Royalties. Earn $16 on a $20 book.

#### We pay **80% royalties**. That's not a typo: **you earn $16 on a $20 sale**. If we sell **5000** non-refunded copies of your book or course for **$20**, you'll earn **$80,000**.

*(Yes, some authors have already earned much more than that on Leanpub.)*

In fact, authors have earnedover $12 millionwriting, publishing and selling on Leanpub.

**Learn more about writing on Leanpub**

### Free Updates. DRM Free.

If you buy a Leanpub book, you get free updates for as long as the author updates the book! Many authors use Leanpub to publish their books in-progress, while they are writing them. All readers get free updates, regardless of when they bought the book or how much they paid (including free).

Most Leanpub books are available in PDF (for computers) and EPUB (for phones, tablets and Kindle). The formats that a book includes are shown at the top right corner of this page.

Finally, Leanpub books don't have any DRM copy-protection nonsense, so you can easily read them on any supported device.

Learn more about Leanpub's ebook formats and where to read them