Quantum Field Theory

PHYS4040 (Quantum Field Theory - scalar fields)

First part of the course, focus on quantising scalar fields

Background: Classical field theory

You may want to quickly remind yourself of some aspects of classical field theory before we go on to develop the quantum version. Note: We’ll cover everything that’s strictly required in class, so these notes will not be expected/asumed knowledge. But, it might be interesting or helpful for you to go through them, or refer back to them as we go

  • Classical Field Theory
    • They cover: Relativity, classical field theory, scalar and vector fields, interactions between fields and particles
    • Then: Electrodynamics, covariant Maxwell’s equations, EM waves and radiation, motivation for general relativity

Whiteboards from the 2024 lectures

The lectures follow the course notes quite closely, and cover much the same material as David Tong’s lecture notes (linked below).

  • Klein Gordon feild
    • Introduction to QFT: from classical to quantum to quantum fields
    • Canonical quantisation: harmonic oscillator to Klein Gordon
    • Hamiltonian, energy, momentum, particles, Hilbert and Fock spaces
    • Dealing with our first infinity: normal ordering, vacuum energy (dark energy and the famous $10^{120}$ discrepancy)
  • Mathematics interleude
    • Motivation: interacting classical fields
    • Distributions, Dirac delta “function”, Green’s functions, complex integration
  • Propogators
    • Reminder: Propagators in QM
    • Correlation functions and propagators in QFT
    • Klein-Gordon Feynman propagator
    • First look at interacting quantum fields (semi-classical)
  • Noether’s Theorem
    • Symmetries, conservation laws and Noether’s theorem
    • Lorentz and coordinate transformations
      • Stress energy tensor: Hamiltonian, momentum, Angular momentum
    • Internal symmetries: charge, spin
    • Transformations, representations, and generators
      • Conserved charges as generators of transformations

Retarded/Advanced potentials (in classical theory)

I think you didn’t get up to seeing advanced/retarded potentials in your classical electrodynamics course. It’s funny that we often see these first in quantum theory despite the techniques being developed first for classical field theory. The classical case may also be more intuitive to understand, so might provide some extra insight. Even the “crazy” contour integrations we do are not special to quantum field theory: the exact same techniques were first used for classical fields. The below classical notes are beyond the direct scope of the course, but you might find interesting or useful:

Texts

PHYS6004 (Advanced Quantum Field Theory - quantum electrodynamics)

Extremely brief overview to QED. First, covers a recap of Dirac theory. Then states quantisation of Dirac and photon fields (without derivation). Feynman rules are motivated and stated (not derived) – we will see the derivation in the later parts of the course. Finally, a semi-relativistic treatment of the Lamb shift is given (following the groundbreaking paper of Bethe), which shows an example of infinities in the theory and how to deal with them (renormalisation)

RQM/QED Lecture Notes

  • Relativistic Quantum Mechanics
    • Contains: quick overview of Dirac theory
    • QED overview, summary of Feynman rules
    • Semi-relativistic treatment of Lamb shift, renormalisation

Textbooks

  • Peskin & Schroeder, An Introduction To Quantum Field Theory
    • Main book for the course, available as pdf from library
  • Sakurai, Advanced Quantum Mechanics
    • A little dated… (ok, very dated – written before the Standard Model was finished!). But still quite nice. Available from library.