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 2 – Notes 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
Introducing the Fields
Introducing the fields 1 – Notes 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 2 – Notes and video – i epsilon physics, intuition on propagator, zero point energy, the complex scalar field, quantization, conserved charge and current, particles and antiparticles.
Interactions 1 – Notes 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 2 – Notes 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 4 – Notes 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 1 – Notes 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 2 – Notes 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 3 – Notes 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 4 – Notes 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 5 – Notes and video – mass renormalization, wavefunction renormalization, logic and formal techniques, general techniques, renormalizable and nonrenormalizable theories, example of issues with nonrenormalizable theories
QED 1 – Notes 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.
Introduction to Path Integrals
Path Integrals 2 – Notes 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 3 – Notes 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 4 – Notes 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 5 – Notes and video – Connections between Path Integrals and quantum statistical mechanics, quantum mechanics and quantum field theory, effective field theory, integrating out heavy fields