Course outline: The outline can be downloaded here.
Class timings: 10:30 to 11:45 am, Monday and Wednesday, Room: 202 SSE Complex.
Marking scheme: Homeworks 40%, Midterm 30%, Final 30%
Pre-mid term
To access the video recording, click on the numbered links below.
Introduction to Quantum Field Theory: Prerequisites
1: Overview and Special Relativity (Lecture 1)
- Overview
- 4-Vectors, Minkowski space
- Lorentz transformation, Lorentz boost
- Natural units vs relativistic units
- Space time diagrams, Simultaneity
- Lorentz invariants
2: Special Relativity and Classical Mechanics (Lecture 2)
- Special relativity: 4-velocity
- Mechanics of relativistic particles
- Final momentum from langrangian
- Energy of the system: Hamiltonian
- Minkowski tensor and inner product
3: Classical Field Theory (Lecture 3)
- Simple concept of field
- Example of a single particle
- Non-relativistic particle equation
- Basic principles of field theory
- Klein-Gordon equation
- Solution to Klein-Gordon equation
- Particles inside two interacting scalar fields
Homework 1: Hw1 (pdf) (Due date: 6 February, 2018 10:00 am) Solution: Solution 1 (pdf)
4: Lagrangian Formulation of Electromagnetism (Lecture 4)
- Charged particle inside an electromagnetic field (Relativistic Equation of Motion)
Homework 2: HW2 (pdf) (Due date: 15 February, 2018 04:00 pm) Solution: Solution 2 (pdf)
5: Electromagnetic Field Tensors (Lecture 5)
- Electric field components
- EM field tensor
- Maxwell’s equations (homogeneous and in homogeneous)
- Locality principle
- Lorentz co-variant form of in homogeneous Maxwell’s equations
- Gauge invariance
Homework 3: HW3 (pdf) (Due date: 19 February, 2018 04:00 pm) Solution: Solution 3 (pdf)
Introduction to Quantum Field Theory: Second Quantization
6: Noether’s Theorem, Second Quantization (Lecture 6)
- Derivation, Noether’s current, Noether’s momentum
- Energy momentum tensor
- Uncoupled and Coupled harmonic oscillators
- Second quantization
7: Second Quantization continued: Quantization of Operators (Lecture 7)
- Occupation number
- Continuum limit
- Field operators
- Second quantization of single particle operators
- Examples
Homework 4: HW4 (pdf) (Due date: 26 February, 2018 04:00 pm) Solution: Solution 4 (pdf)
8: Examples of Second Quantization (Lecture 8)
- Second quantized form of Hamiltonian (in position and momentum space)
- Interaction (2 – particles) – Feynman diagram
- Magnetic interactions (String operators, Jordan-Wigner transformations)
- Further examples – Magnons, Tight binding model, Hubbard model.
Homework 5: HW5 (pdf) (Due date: 5 March, 2018 04:00 pm) Solution: Solution 5 (pdf)
Introduction to Quantum Field Theory: Canonical Quantization
9: Scalar Field – Canonical Quantization (Lecture 9)
- Time evolution
- Heisenberg picture
- Canonical momentum
- Hamiltonian density
- Field in terms of creation and annihilation operator, Mode expansion
- Hamiltonian
- Normal ordering
- Positive and negative energies
10: Complex Scalar Field – Canonical Quantization (Lecture 10)
- Canonical quantization of complex scalar field
- Klein-Gordon equation for non-relativistic regime
- Canonical quantization of truncate the complex scalar field
Homework 6: HW6 (pdf) (Due date: 13 March, 2018 10:00 am) Solution: Solution 6 (pdf)
11: Multicomponent Quantum Field – Canonical Quantization (Lecture 11)
- 3-component field (electromagnetic field) canonical quantization
- Fictitious particle – Massive photon canonical quantization
- Polarization
- Internal symmetry
Midterm exam: midterm + solution (pdf) (13 March 2018)
Post-mid term
Introduction to Quantum Field Theory: Gauge Transformation
12: Gauge Transformations (Lecture 12)
- Introduction
- Gauge transformation of classical complex massive fields
- Electromagnetism as a U(1) gauge theory
- Choice of gauge
- Lorenz gauge
- Coulomb gauge
Homework 7: HW7 (pdf) (Due date: 2 April, 2018 10:00 am) Solution: Solution 7 (pdf)
Introduction to Quantum Field Theory: Propagators
- Introduction
- Green’s functions
- Propagators
- Free particle
- Green’s function in the Fourier domain
14: Feynman Propagators (Lecture 14)
- Space-time translations
- Transformation of quantum fields
- Feynman propagator
- Interacting system
- Free propagator
15: Interaction Propagators (Lecture 15)
- Interaction picture
- Scattering matrix (S – matrix)
- Wick’s time ordering
Homework 8: HW8 (pdf) (Due date: 10 April, 2018 10:00 am) Solution: Solution 8 (pdf)
16: Computing the Terms in the Dyson Expansion of Scattering matrix (Part A) (Lecture 16)
- Zeroth order term
- Wick’s theorem
- Feynman diagrams
17: Computing the Terms in the Dyson Expansion of Scattering matrix (Part B) (Lecture 17)
- Phi-4 theory
- Examples (Bosonic fields)
- Yukawa’s theory
18: Dirac Equation – Quantum Field Theory (Lecture 18)
- Dirac equation
- Representing gamma (or Dirac) matrices
- Alternative way of expressing Dirac equation
- Chirality operator
- Dispersion relation for massive Dirac particles
- Non relativistic version of Dirac equation – Pauli equation
Homework 9: HW9 (pdf) (Due date: 19 April, 2018 10:00 am) Solution: Solution 9 (pdf)
Introduction to Quantum Field Theory: Spinor Fields
19: Transformations (Lecture 19)
- Continuous transformations: Spacetime, Rotation
- Representation e.g, Spin 1 particle
- Quantum fields
- Scalar fields and Vector fields
- Lorentz boost
- Discrete transformation: Charge conjugation, Parity operation, Time reversal operation
- Time reversal symmetric (TRS) system
20: Transformation of Spinors (Lecture 20)
- Rotations
- Boosts
- Pure boost
- Spin and particle helicity
- Orthogonality of the spinors
21: Quantization of the Dirac Field (Lecture 21)
- Hamiltonian for the dirac field
- Feynman propagators for fermionic field
Homework 10: HW10 (pdf) (Due date: 2 May, 2018 10:00 am) Solution HW 10
22: Dirac Lagrangian and path integral formalism (Lecture 22)
- Gauge invariance of the Dirac Lagrangian
- Lagrangian for QED
- Minimal coupling scheme
- Path integral formalism
- Transition amplitudes for a harmonic oscillator
Homework 11: HW 11 (pdf) (Due date: 7 May, 2018 4:00 pm) Solution HW 11
23: Bosonic Path Integrals (Lecture 23)
- Bosonic path integrals
- Relationship to bosonic propagator
- source terms, links with statistical mechanics
- generating functionals, normalized generating functionals
- See my notes on derivation of the path integral for a massive photon in the framework of the theory of massive electromagnetism
24: Fermionic Path Integrals (Lecture 24)
- Fermionic path integrals
- Grassmann numbers and their properties
- Grassmann fields
- Coherent states for fermions
- See Ch:44, Quantum Field Theory by Mark Srednicki for a formal derivation of fermionic path integrals