Quantum Field Theory I

In QFT I, the goal is to transition from the basic Quantum Mechanics style of thinking to that of Quantum Field Theory. The course is designed for students of all backgrounds and all research interests. This is a topic that everyone should understand. As a result, some of the more specialized topics that are relevant for particle and nuclear physics are saved for the QFT II course. A summary of the topics covered can be found here.


Basics 1 Notes and video – constructing a field, mass points, continuum limit, Lagrangian for the field, the wave equation, quantum commutation rules, solving with creation operators, quanta.

Basics 2Notes and video – Solving the Hamiltonian, States and quanta, why equal time commutators, connection to normal modes, continuous momentum notation, the Quantum Field, taking matrix elements, intuition

Basics 3Notes and video– inverting the field, conservation of energy, the momentum operator, natural units, four vectors, dimensional analysis

Introducing the Fields

Introducing the fields 1Notes and video – listing the common fields, the real scalar field, more general lagrangian, why phonons are massless, Feynman propagator, representation in terms of solutions, Fourier integral representation.

Introducing the fields 2Notes and video – i epsilon physics, intuition on propagator, zero point energy, the complex scalar field, quantization, conserved charge and current, particles and antiparticles.

Introducing the fields 3Notes and video – the Schroedinger Lagrangian, non-relativistic reduction, quantization, Fermions, Review of the electromagnetic field, quantization, propagator

Introducing fields 4Notes and video – the Dirac Lagrangian, Hamiltonian, useful identities, quantization, the particle number current, the Dirac propagator.


Interactions 1Notes and video – the covariant derivative and interactions in E&M, gauge invariance, identifying the electromagnetic current, matrix elements of the current, transition matrix elements, Rules for matrix elements

Interactions 2Notes and video – crossing, Dirac rules, other interactions, perturbation theory plan, review of the interaction picture and the time development operator, first example scattering amplitude in φ^4

Interactions 3Notes and video – review, toy model for QED, matrix elements with propagator, connections to first Born approximation and to QM perturbation theory, Wick’s theorem.

Interactions 4Notes and video – Feynman rules for φ4, second order example, self energies, disconnected diagrams, loop integrals, dropping disconnected diagrams, Feynman rules for other theories, photons and fermions.

 Calculating in field theory

Calculating 1Notes and video – Review of Feynman rules, plan for “Calculating section”, decay rate and cross section formulas, review of Fermi’s Golden Rule, adapting FGR to field theory decay rates, using FGR for cross sections.

Calculating 2Notes and video – Identical particle effects, generalization to more particles, Lorentz invariant phase space, cross section for φ4, correspondence to non-relativistic results, low energy expansion of a propagator, QED scattering cross section

Calculating 3Notes and video
– review of Feynman rules for Dirac particles, finishing QED cross section – connection to non-relativistic QM, Noether’s theorem, calculating the Noether current, example, spacetime symmetry and the energy momentum tensor, interactions and symmetries.

Calculating 4Notes and video – Ground state energies and masses, symmetry breaking and Goldstone’s theorem, calculating with two different names for the fields, example, names don’t matter – Haag’s theorem, the Higgs mechanism

 Renormalization and loop diagrams

Renormalization 1 Notes and video – the philosophy of renormalization: measuring the electric charge, expressing cross-section in terms of measured value, the bare charge and the physical charge, the counterterm method for QED, renormalizing phi-phi scattering.

Renormalization 2Notes and video – calculating a loop integral: Feynman parameterization, completing the square, Wick rotation, 4 dimension spherical coordinates, the final integral and results.

Renormalization 3Notes and video – review, two procedures for renormalization, imaginary parts, unitarity, useful identities, infinities, why quantum calculations work, philosophy of infinities

Renormalization 4Notes and video – regularization, Pauli-Villars (with example), dimensional renormalization, gamma function review, calculation of loop integral, expansion about d=4.

Renormalization 5Notes and video – mass renormalization, wavefunction renormalization, logic and formal techniques, general techniques, renormalizable and nonrenormalizable theories, example of issues with nonrenormalizable theories

 Quantum Electrodynamics

QED 1Notes and video – QED, charge quantization, gauge invariance in matrix elements, the renormalization program, fermion mass and wavefunction renormalization, the vacuum polarization, calculating the vacuum polarization, photon wavefunction renormalization.

QED 2Notes and video – review, diagrams for charge renormalization, the vertex function, the full charge renormalization, residual predictions, the running coupling, the Lamb shift

QED 3Notes and video – infrared divergences, solution via bremsstrahlung, Lamb shift and off-shell g=2 in Dirac theory, Gordon decomposition, calculation of g-2, practical group theory.

QED 4Notes and video – SU(N) group theory, SU(N) gauge theory, the gauge covariant derivative, the field strength tensor, the Yang-Mills lagrangian, interactions, examples

 Introduction to Path Integrals

Path Integrals 1Notes and video – Gaussian integrals needed for functional methods

Path Integrals 2Notes and video – example for functional methods, PI in quantum mechanics, derivation, wavefunctions and matrix elements, projecting out the ground state, functional differentiation, the generating functional, Example: the harmonic oscillator

Path Integrals 3Notes and video – review of QM work, path integral for fields, functional differentiation again, Greens functions, solving the free field generating functional, the two point function, the four point function

Path Integrals 4Notes and video – Interactions in PI framework, perturbative expansion, working to first order in λ, the two point function and the propagator, the four point function and the scattering amplitude, the generating functional contains all amplitudes, the LSZ reduction formula

Path Integrals 5Notes and video – Connections between Path Integrals and quantum statistical mechanics, quantum mechanics and quantum field theory, effective field theory, integrating out heavy fields

Research and teaching pages for John Donoghue