Canonical Quantization

Quantum field theory (QFT) extends quantum mechanics to systems with infinitely many degrees of freedom — quantum fields. Canonical quantization promotes classical fields to operators satisfying commutation or anticommutation relations, and particles emerge as field quanta.

Key Concepts

Fields φ(x,t) as quantum operators
Klein-Gordon equation: (□ + m²)φ = 0 (scalar field)
Canonical commutation: [φ(x), π(y)] = iδ³(x-y)
Mode expansion: φ = ∫ d³k (a_k e^{ikx} + a†_k e^{-ikx})
Particle number operator: N = ∫ d³k a†_k a_k

Key Equations

Klein-Gordon equation

(\partial_\mu\partial^\mu + m^2c^2/\hbar^2)\phi = 0

Canonical commutation

[\hat{\phi}(\mathbf{x}), \hat{\pi}(\mathbf{y})] = i\hbar\delta^3(\mathbf{x}-\mathbf{y})

Relativistic dispersion

E^2 = p^2c^2 + m^2c^4

Zero-point energy (per mode)

E_0 = \frac{\hbar\omega_k}{2}

Worked Example

Example Problem

Problem

A Klein-Gordon field has m = 0 (massless). Find the dispersion relation.

Solution

Setting m=0: ω² = c²k². So ω = c|k|, the same as a photon. Massless scalar particles travel at the speed of light.

Practice

Exercises

7 problems

1 of 7

A real scalar field has mass m (in natural units c=ℏ=1). The dispersion relation is ω²=k²+m². For k=3m, find ω in units of m.

Your answer

2 of 7

In natural units (ℏ=c=1), the electron mass is mₑ = 0.511 MeV. Find the Compton wavelength λ_C = ℏ/(mₑc) in fm.

Your answer

3 of 7

The commutator [a_k, a†_q] = δ³(k-q). For a state |n_k⟩ with n particles in mode k, find the eigenvalue of N_k = a†_k a_k.

Unlock Exercise 3