Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part II Overview

Canonical Quantization

From classical fields to quantum operators: Creating and annihilating particles

πŸ”—Course Connections

πŸ“šPrerequisites

Quantum Mechanics
Canonical Quantization
Understanding [xΜ‚, pΜ‚] = iℏ commutation relations
β†’
Quantum Mechanics
Harmonic Oscillator
Creation/annihilation operators Ò†, Γ’
β†’
QFT Part I
Classical Field Theory
Lagrangian formalism, Euler-Lagrange equations
β†’

🎯 What You'll Learn

Canonical quantization is the bridge between classical and quantum field theory. We promote classical fields Ο†(x,t) to quantum operators Ο†Μ‚(x,t) that satisfy commutation relations, just like in quantum mechanics.

The revolutionary insight: particles are excitations of quantum fields! What we call "particles" are actually quanta - discrete energy bundles of the underlying field.

The Big Picture

In Part I, we studied classical fields described by Lagrangians. Now we quantize these fields to create a quantum theory. The process mirrors what you learned in quantum mechanics:

Quantum Mechanics

  • β€’ Observable: position x(t)
  • β€’ Quantize: x β†’ xΜ‚
  • β€’ Commutator: [xΜ‚, pΜ‚] = iℏ
  • β€’ States: |ψ⟩ in Hilbert space
  • β€’ Particles: Fixed number

Quantum Field Theory

  • β€’ Observable: field Ο†(x,t)
  • β€’ Quantize: Ο† β†’ Ο†Μ‚
  • β€’ Commutator: [Ο†Μ‚(x), Ο€Μ‚(y)] = iδ³(x-y)
  • β€’ States: |n,k⟩ in Fock space
  • β€’ Particles: Variable number!

Chapter Roadmap

1. Quantization Procedure

Canonical commutation relations, promoting fields to operators, equal-time commutators

2. Free Scalar Field

Mode expansion, creation/annihilation operators, vacuum energy, normal ordering

3. Fock Space & Particle States

Multi-particle states, occupation number basis, creation/annihilation algebra

4. Propagators & Green's Functions

Feynman propagator, time-ordering, causality, i prescription

5. Quantizing the Dirac Field

Fermionic anticommutators, Pauli exclusion, spinor modes, antiparticles

6. Spin-Statistics Theorem

Why bosons commute and fermions anticommute, connection to spin

7. Photon Field Quantization

Gauge fixing, polarization vectors, transverse modes, QED vacuum

πŸ”‘ Key Concepts

  • Fields become operators: Ο†(x,t) β†’ Ο†Μ‚(x,t)
  • Canonical commutation relations: [Ο†Μ‚(x), Ο€Μ‚(y)] = iδ³(x-y)
  • Particles are quanta: excitations of the quantum field
  • Fock space: states with definite particle number |n,k⟩
  • Creation operators Ò†k create particles with momentum k
  • Annihilation operators Γ’k destroy particles
  • Vacuum |0⟩: lowest energy state (not empty!)
  • Propagators: ⟨0|T{`Ο†Μ‚(x)Ο†Μ‚(y)`}|0⟩ describe particle propagation