PHY2403F:
Quantum Field Theory I

Office hours

Office hours are by appointment. Please send me e-mail to set up a time.

Prerequisites

You'll need a solid background in quantum mechanics, special relativity, classical mechanics and electromagnetism. Specific topics you need to be comfortable with are as follows. Special relativity: Lorentz transformations, 4-vectors, indices. Quantum mechanics: the Schrodinger, Heisenberg and interaction pictures; scattering theory and the Born approximation; spin angular momentum and SU(2). Classical mechanics: Lagrangians, Hamiltonians, coordinates and canonical momenta. Math: contour integrals. Please see me if any of these terms are unfamiliar. If you haven't seen one or two of these, you should be able to catch up, but if you aren't familiar with most of them then you may need to choose another course. Either way, please discuss any unfamiliarities with me first.

Syllabus

Canonical quantization of field theory

failure of single-particle quantum mechanics; review of Lagrangian formulation of classical mechanics, quantum mechanics, classical field theory, quantum field theory; canonical quantization, review of simple harmonic oscillator, creation and annihilation operators, free Klein-Gordon theory, field expansion;

Symmetries and conservation laws

Noether's theorem in field theory: example of energy-momentum tensor; Goldstone's Theorem; internal symmetries: example U(1) and antiparticles; C,P,T.; introduction to group theory: Lie groups, concept of spin of a field, representations of Lorentz group;

Scattering theory, Feynman Diagrams

Dyson's formula, Wick's theorem; diagrammatic perturbation theory for scalar field theory; examples;

Cross-sections, phase space, decay rates

examples from scalar QFT;

Spin 1/2 fields, Dirac Lagrangian

Weyl Lagrangian; Dirac Lagrangian, Dirac matrices, chirality; spin-statistics theorem; perturbation theory for spinor-scalar coupling;

Vector fields and QED

spin 1 fields; quantization of massive vector fields; massless limit and gauge invariance; QED and examples; massive vector bosons and weak interactions: effective four-fermi theory, SLAC experiment and neutral currents; concept of renormalizability, non-renormalizability of massive vector bosons.

Texts

Graduate courses giving you an entry to research never have one hand-holding textbook from which you can learn everything you need. Useful basic introductory texts I have put on reserve in the Physics and Main Libraries are Quantum Field Theory, by Ryder (especially the first four chapters), and Quantum Field Theory by Mandl and Shaw. In addition, Professor Luke, who taught a similar course two years ago, has a set of TeXed notes based on part of Sidney Coleman's Harvard introductory QFT course and he has kindly agreed to make it available in postscript form as another introductory text for you. The "recommended" textbook for this course, An Introduction to Quantum Field Theory by M. Peskin and D. Schroeder, is significantly more comprehensive than the above three sources. (Copies are on sale at the UofT Bookstore.) We will use only about the first five chapters in this course; this text will also be used in the Spring course Quantum Field Theory II, PHY2404S. Students interested in a future in particle/string theory research will find Peskin&Schroeder a very useful reference, and may also want to take a peek at Warren Siegel's massive free online textbook Fields .

Notes

I have a severe case of computer use injuries, and so I cannot typeset my class notes. I will scan in my own handwritten notes, or sometimes use excerpts from Luke's notes if I follow them closely. From the combination, you will get a complete orderly set of class notes here. [Note, however, that attendance and participation in-class will be 100% necessary if you want to obtain a passing grade for this course.]

• Background on special relativity indices, Fourier transforms, and the Dirac delta .
• 13Sep: Natural units. Why (&when) single-particle QM - nonrelativistic or relativistic - isn't good enough.
• 18Sep: Why we need QFT. Lightning review of simple harmonic oscillator and classical particle mechanics. Classical field theory. The classical relativistic scalar (spin-zero a.k.a. Klein-Gordon) field and its action.
• 25Sep: Quantum Field Theory. Fock space and relativistic normalization. The quantum Klein-Gordon field. Zero-point energy and normal-ordering. Meaning of the field operator. Causality and locality.
• 02Oct: Symmetries and Conservation Laws. Noether's Theorem, Goldstone's Theorem.
• 09Oct: Symmetries and Conservation Laws II. Internal Symmetries, U(1) and Antiparticles; Groups, Representation Theory of Poincare Group.
• 16Oct: Interacting QFT I. Dyson's Formula and Wick's Theorem.
• 23Oct: Scattering Theory and Feynman Diagrams.
• 30Oct: Cross-Sections and Phase Space.
• 06Nov: Spinors: Weyl and Dirac Reps and their Lagrangians, Gamma Matrices, Chirality, Spin-Statistics Theorem.
• 13Nov: Spinors: perturbation theory. Causal observables. Wick's Theorem and Dyson's Formula. Dirac propagator. External legs. Scalar-spinor coupling. Spin sums and unpolarized cross-sections.
• 20Nov: Vectors: Classical Field Theory. Proca Equation. Vector-Scalar Interactions, the Noether Procedure. Vector-Spinor Interactions. Massless vector (gauge) fields. Quantization for massive field. Feynman rules. Decoupling of longitudinal mode for massless vector field, I.
• 27Nov: Vectors: Decoupling of longitudinal mode for massless vector field, II. Ward Identity. Example of QED scattering amplitude.
• 04Dec: Review / Worked Problem(s) / !

Note: for Feynman diagram external leg spinors and polarisation vectors, and gamma-matrix stuff, I am using Mike Luke's conventions (not Peskin&Schroeder's!).

Assessment

The final grade will be 60% on the homeworks (five at 12% each), and 40% on the final exam. There will be no midterm. Each homework assignment will be given out two weeks before its due date. Homeworks will be due at 11am (the beginning of class) on:

• October 02 [Solutions]
• October 16 [Solutions]
• October 30 [Solutions]
• November 13 [Solutions]
• November 27 [Solutions]

The final exam will be held in MP408 on Thursday 13th December from 12-3PM. The format of the exam will be three hours in-class, and you may bring in one sheet of paper with normal-size (un-reduced) text/equations on both sides. Here is the 2001 exam and solutions:

• 2001 Final Exam [ Solutions ]

For those who wanted them, old homework questions from 2000 are here:

• Homework 1
• Homework 2
• Homework 3
• Homework 4
• Homework 5

For other important deadlines and dates, see the Faculty of Arts and Sciences Calendar .