This is a table of contents for my notes on field theory (some classical, but mostly quantum). The posts are arranged in an order that might be found in a textbook.

These notes will be of most benefit if read in parallel with Coleman's book *Quantum Field Theory: Lectures of Sidney Coleman* and Lancaster & Blundell's book *Quantum Field Theory for the Gifted Amateur*.

- Fermat's principle of least time and Snell's law
- Functionals and functional derivatives
- Functional derivatives - more examples
- Lagrangians and the principle of least action
- Time-dependent Lagrangian
- Poisson brackets, commutators and Jacobi identity
- Commutator of two hermitian operators
- Lagrangians for elastic media
- Functional derivative - a 4-dimensional example
- Euler-Lagrange equations for particle and field theories
- Position and velocity as independent variables
- Origin of the principle of least action
- Hamilton's equations and Poisson brackets
- Poisson brackets in classical field theory
- Hamilton's equations for relativistic fields - conjugate momentum
- Free particles and interaction terms in Lagrangians
- Lagrangian example - infinite polynomial in the scalar field
- Lagrangian to equation of motion - Three examples

- Discrete Fourier transforms
- Creation and annihilation operators in a perturbed harmonic oscillator
- Harmonic oscillator ground state from annihilation operator
- Creation and annihilation operators in the harmonic oscillator - a few theorems
- Creation and annihilation operators for the 3-d harmonic oscillator
- Coupled oscillators in terms of creation and annihilation operators - phonons
- Occupation number representation; delta function as a series
- Eigenfunctions of position and momentum; unit operators

- Lagrangian for a relativistic free particle
- Euler-Lagrange equations for a free particle in an electromagnetic field
- Momentum and energy for a free particle in an electromagnetic field
- Electromagnetic Lorentz invariant
- Maxwell's equations using the electromagnetic field tensor
- Maxwell's equations from the electromagnetic field tensor
- Maximum space-time path for free particle
- Field operators for the infinite square well
- Second quantizing a single-particle operator
- Second quantizing operators - examples
- Generalized commutator
- Second quantizing the tight-binding hamiltonian
- Hubbard model of a solid
- Hamilton's equations for relativistic fields
- Translation and rotation invariance
- Relativistic invariant integration measure
- Lorentz transformations of relativistic states
- Probability of particle outside light cone
- Quantum field theory representation of non-relativistic quantum mechanics

- Time evolution operator
- Unitary transformations and the Heisenberg picture
- Time evolution of creation and annihilaton operators
- Time evolution of spin in a magnetic field
- Generators of the translation operator
- Generators of the Lorentz group for 4-vectors
- General infinitesimal Lorentz transformation
- Poincaré group transformations

- Fermion wave functions; the Slater determinant
- Harmonic oscillator to Fock space
- Lorentz transformation and translation of creation and annihilation operators

- Motivation for a quantum field theory
- Lorentz transformation and translation of explicit quantum field
- Final form for the scalar quantum field
- Free scalar field as the fundamental object
- Klein-Gordon equation
- Klein-Gordon equation from Lagrangian
- Klein-Gordon equation - plane wave solutions
- Klein-Gordon equation - orthonormality of solutions
- Klein-Gordon equation - probability density and current
- Klein-Gordon equation for fields - derivation from the Lagrangian
- Klein-Gordon equation - continuous solutions
- Klein-Gordon equation from the Heisenberg picture
- Klein-Gordon equation - commutators
- Free scalar Hamiltonian as an integral of the Hamiltonian density
- Vacuum energy in the free Klein-Gordon field
- Creation and destruction operators for a free scalar field
- Creation and annihilation operators - normalization
- Probability density in a Klein-Gordon field
- Momentum of a free scalar Klein-Gordon field
- Real Klein-Gordon fields
- Klein-Gordon equation with anticommutators
- Feynman propagator for scalar fields
- Feynman propagator in terms of commutators
- Feynman propagator - from commutator to integral
- Feynman propagator as a contour integral
- Feynman propagator as a single real integral
- Normal ordering

- Dirac equation
- Dirac equation - matrix properties
- Dirac equation - the gamma matrices
- Anticommutators of gamma matrices
- Dirac equation as four coupled differential equations
- Dirac equation - 4 solution vectors
- Dirac equation - inner products of spinors
- Dirac equation - orthogonality of solutions
- Dirac equation - adjoint solutions
- Adjoint Dirac equation
- Adjoint Dirac equation - explicit solutions
- Dirac equation - conserved probability current
- Dirac equation - positive probabilities and negative energies
- Eigenspinors of the Pauli spin matrices
- Dirac equation - spin of a particle at rest
- Dirac equation - spin of a moving particle
- Dirac equation - spinors near the speed of light
- Dirac equation - non-relativistic limit
- Helicity operator in the Dirac equation
- Dirac equation in relativistic quantum mechanics - summary
- Dirac equation in quantum field theory - Lagrangian density
- Hamiltonian density for the Dirac equation
- Hamiltonian for the Dirac equation
- Anticommutators, creation and annihilation operators in the Dirac equation
- Momentum of particles in a Dirac field
- Outer product of spinors
- Feynman propagator for fermions

- Classical particle mechanics
- Classical field theory
- Noether's theorem in classical mechanics
- Translation symmetry and conservation of momentum
- Time translation symmetry and conservation of energy
- Rotation symmetry and conservation of angular momentum
- Noether's theorem in classical field theory
- Currents from spacetime translations and the energy-momentum tensor
- Noether's theorem and Lorentz transformations

- Maxwell's equations in 4d spacetime form
- Lorenz gauge in 4d spacetime form
- Solutions to the 4d wave equation
- Polarization vectors

- Maxwell's equations with sources in 4d spacetime
- Electromagnetism Lagrangian with sources
- Electromagnetic interaction form of the Dirac equation

- Schrödinger picture
- Heisenberg picture
- Interaction picture
- Interaction picture equations of motion
- Expectation values in the interaction picture
- Scattering and the S-matrix
- The S operator as a time-ordered exponential
- Dyson's formula and time ordering
- First order S operator terms in QED
- Electron-electron scattering in second order S matrix

- Wick's theorem
- Scalar fields - summary
- Wick's theorem for two fermion fields
- Wick diagrams
- Connected and disconnected Wick diagrams
- Exact solution of a one-field model
- Counterterms in interaction hamiltonians
- Ground state energy in a one-field model with static source
- Mass renormalization

- Wick diagrams to Feynman diagrams
- Evaluating Feynman diagrams
- Feynman rules
- Feynman rules for QED diagrams without loops
- Compton scattering with Feynman rules
- Feynman rules for loops
- Feynman diagrams and the Born approximation
- Nucleon-antinucleon scattering