Part II, Chapter 1

Canonical Quantization Procedure

Promoting classical fields to quantum operators

🔗Course Connections

📚Prerequisites

Quantum Mechanics

Canonical Quantization

Foundation of operator formalism

→

QFT Part I

Lagrangian Field Theory

Classical field equations of motion

→

▶️

Video Lecture

Lecture 4: Canonical Quantization - MIT 8.323

Canonical quantization of free scalar field theory (MIT QFT Course)

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

1.1 Review: Quantization in QM

In quantum mechanics, we learned to quantize a classical system by replacing observables with operators and Poisson brackets with commutators:

QM Quantization Recipe:

Classical variables: x(t), p(t)
Promote to operators: x → x̂, p → p̂
Poisson bracket → Commutator: {A, B} → (1/iℏ)[Â, B̂]
Get canonical commutation: [x̂, p̂] = iℏ

For fields, we follow an analogous procedure, but now our "position" is the field φ(x,t) itself!

1.2 Canonical Momentum for Fields

Starting with a Lagrangian density ℒ(φ, ∂_μφ), the canonical momentum densityconjugate to φ(x,t) is:

$$\pi(x,t) = \frac{\partial \mathcal{L}}{\partial(\partial_0\phi)} = \frac{\partial \mathcal{L}}{\partial \dot{\phi}}$$

💡Why is π the 'momentum' of the field?

Just as p = mẋ is the momentum conjugate to position x in particle mechanics, π is the momentum density conjugate to the field φ at each point in space.

The field φ(x) is like having infinitely many oscillators, one at each point x. Each has its own "position" φ(x) and "momentum" π(x)!

Example: Klein-Gordon Field

For the Klein-Gordon Lagrangian ℒ = ½(∂_μφ∂^μφ - m²φ²):

\begin{align*} \mathcal{L} &= \frac{1}{2}[(\partial_0\phi)^2 - (\nabla\phi)^2 - m^2\phi^2] \\ \pi &= \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = \partial_0\phi = \dot{\phi} \end{align*}

So the canonical momentum is just the time derivative of the field!

1.3 Equal-Time Canonical Commutation Relations

The quantization procedure promotes φ and π to operators and imposes:

\begin{align*} [\hat{\phi}(\mathbf{x},t), \hat{\pi}(\mathbf{y},t)] &= i\delta^3(\mathbf{x}-\mathbf{y}) \\ [\hat{\phi}(\mathbf{x},t), \hat{\phi}(\mathbf{y},t)] &= 0 \\ [\hat{\pi}(\mathbf{x},t), \hat{\pi}(\mathbf{y},t)] &= 0 \end{align*}

These are the equal-time canonical commutation relations (CCR). The Dirac delta δ³(x-y) appears because fields at different spatial points are independent degrees of freedom.

⚠️ Important Note: Equal-Time

These commutation relations hold only for equal times (t = t'). At different times, [φ̂(x,t), π̂(y,t')] ≠ iδ³(x-y) because the fields evolve according to the equations of motion.

1.4 Hamiltonian Formulation

The Hamiltonian density is obtained via Legendre transform:

$$\mathcal{H} = \pi\dot{\phi} - \mathcal{L}$$

The total Hamiltonian is:

$$\hat{H} = \int d^3x \, \mathcal{H}(\hat{\phi}, \hat{\pi})$$

For the Klein-Gordon field:

$$\hat{H} = \int d^3x \left[\frac{1}{2}\hat{\pi}^2 + \frac{1}{2}(\nabla\hat{\phi})^2 + \frac{1}{2}m^2\hat{\phi}^2\right]$$

This looks like the energy density: kinetic energy (π²) + gradient energy + potential energy (mass term).

1.5 Heisenberg Equations of Motion

In the Heisenberg picture, operators evolve in time according to:

$$i\frac{\partial \hat{\mathcal{O}}}{\partial t} = [\hat{\mathcal{O}}, \hat{H}]$$

For the field operator φ̂:

\begin{align*} i\frac{\partial \hat{\phi}}{\partial t} &= [\hat{\phi}, \hat{H}] \\ &= \int d^3y \, [\hat{\phi}(\mathbf{x}), \frac{1}{2}\hat{\pi}(\mathbf{y})^2] \\ &= \int d^3y \, \frac{1}{2}[\hat{\phi}(\mathbf{x}), \hat{\pi}(\mathbf{y})]\hat{\pi}(\mathbf{y}) + \frac{1}{2}\hat{\pi}(\mathbf{y})[\hat{\phi}(\mathbf{x}), \hat{\pi}(\mathbf{y})] \\ &= \int d^3y \, i\delta^3(\mathbf{x}-\mathbf{y})\hat{\pi}(\mathbf{y}) = i\hat{\pi}(\mathbf{x}) \end{align*}

Therefore:

$$\frac{\partial \hat{\phi}}{\partial t} = \hat{\pi}$$

This is just the quantum version of π = φ̇!

1.6 QM vs. QFT Quantization

Quantum Mechanics vs. Quantum Field Theory Quantization

Analogous structures in QM and QFT

Aspect	Quantum Mechanics	Quantum Field Theory
Dynamical Variable	Position x(t)	Field φ(x,t)
Conjugate Momentum	p = mẋ	π(x) = ∂ℒ/∂φ̇
Poisson Bracket	{x, p} = 1	{φ(x), π(y)} = δ³(x-y)
Commutation Relation	[x̂, p̂] = iℏ	[φ̂(x), π̂(y)] = iδ³(x-y)
Hilbert Space	L²(ℝ) for one particle	Fock space (all n-particle states)

Practice Problems

📝 Practice Problems

Progress: 0/2 (0%)

⭐ Easy

For the Klein-Gordon Lagrangian ℒ = ½(∂_μφ∂^μφ - m²φ²), calculate the canonical momentum π(x) conjugate to φ(x).

⭐⭐ Medium

Verify that [φ̂(x), π̂(y)] = iδ³(x-y) implies [φ̂(x), φ̇̂(y)] = iδ³(x-y) for equal times.

📊 Problem Set Statistics

Total Problems

Attempted

Completion

🎯 Key Takeaways

Canonical momentum: π = ∂ℒ/∂φ̇ (analogous to p = ∂L/∂ẋ)
Quantization: Promote φ, π to operators φ̂, π̂
Equal-time CCR: [φ̂(x,t), π̂(y,t)] = iδ³(x-y)
Fields at different points commute: [φ̂(x), φ̂(y)] = 0
Hamiltonian: Ĥ = ∫d³x (π²/2 + (∇φ)²/2 + m²φ²/2)
Heisenberg equation: i∂_tÔ = [Ô, Ĥ] gives time evolution
Next step: Express φ̂ in terms of creation/annihilation operators!

Quantum Field Theory Course