Quantum Field Theory Course

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

Part II, Chapter 3

Fock Space & Particle States

The Hilbert space of quantum field theory: variable particle number

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part II
Free Scalar Field
Creation/annihilation operators
β†’
▢️

Video Lecture

Lecture 5: Complex Scalar Field & Antiparticles - MIT 8.323

Fock space structure and multi-particle states (MIT QFT Course)

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

3.1 What is Fock Space?

In quantum mechanics, we had a fixed number of particles. In QFT, particle number is variable! Fock space is the direct sum of all n-particle Hilbert spaces:

$$\mathcal{F} = \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots = \bigoplus_{n=0}^\infty \mathcal{H}_n$$

where:

  • β„‹0 = vacuum sector (no particles)
  • β„‹1 = one-particle states
  • β„‹2 = two-particle states
  • β„‹n = n-particle states

πŸ’‘Why Variable Particle Number?

In special relativity, energy can create particle-antiparticle pairs: E = mcΒ². At high energies, particles are constantly being created and destroyed!

Example: A photon with E > 2mecΒ² can create an electron-positron pair. We need a framework where particle number can change. That's Fock space!

3.2 The Vacuum State

The vacuum |0⟩ ∈ β„‹0 is the state with no particles:

$$\hat{a}_k |0\rangle = 0 \quad \forall \mathbf{k}$$

Properties:

  • Unique (up to phase)
  • Lorentz invariant
  • Lowest energy state: Δ€|0⟩ = 0 (after normal ordering)
  • Not "empty"! Filled with quantum fluctuations

⚠️ Vacuum is Not Nothing!

The quantum vacuum has zero-point fluctuations. Virtual particles constantly pop in and out of existence! These have real physical effects like the Casimir force and Lamb shift.

3.3 One-Particle States

Create a particle with momentum k:

$$|\mathbf{k}\rangle = \hat{a}_k^\dagger |0\rangle$$

These states are normalized as:

$$\langle \mathbf{k}|\mathbf{k}'\rangle = (2\pi)^3 \delta^3(\mathbf{k} - \mathbf{k}')$$

The energy and momentum of |k⟩:

\begin{align*} \hat{H}|\mathbf{k}\rangle &= \omega_k |\mathbf{k}\rangle \\ \hat{\mathbf{P}}|\mathbf{k}\rangle &= \mathbf{k} |\mathbf{k}\rangle \end{align*}

3.4 Multi-Particle States

Two-particle state:

$$|\mathbf{k}_1, \mathbf{k}_2\rangle = \hat{a}_{k_1}^\dagger \hat{a}_{k_2}^\dagger |0\rangle$$

For bosons (like scalar particles), the state is symmetric:

$$|\mathbf{k}_1, \mathbf{k}_2\rangle = |\mathbf{k}_2, \mathbf{k}_1\rangle$$

This follows from [Γ’k₁†, Γ’k₂†] = 0 (creation operators commute).

General n-particle state:

$$|\mathbf{k}_1, ..., \mathbf{k}_n\rangle = \hat{a}_{k_1}^\dagger \cdots \hat{a}_{k_n}^\dagger |0\rangle$$

3.5 Occupation Number Representation

For bosons, we can have any number of particles in the same state. Define the number operator:

$$\hat{n}_k = \hat{a}_k^\dagger \hat{a}_k$$

It counts particles in mode k:

$$\hat{n}_k |n_k\rangle = n_k |n_k\rangle$$

A general state can be written as:

$$|n_{k_1}, n_{k_2}, n_{k_3}, ...\rangle = \frac{1}{\sqrt{n_{k_1}! n_{k_2}! n_{k_3}! \cdots}} (\hat{a}_{k_1}^\dagger)^{n_{k_1}} (\hat{a}_{k_2}^\dagger)^{n_{k_2}} \cdots |0\rangle$$

The factorials ensure proper normalization.

3.6 Algebra of Creation/Annihilation

Key identities:

\begin{align*} \hat{a}_k^\dagger |n_k\rangle &= \sqrt{n_k + 1} |n_k + 1\rangle \\ \hat{a}_k |n_k\rangle &= \sqrt{n_k} |n_k - 1\rangle \\ \hat{n}_k &= \hat{a}_k^\dagger \hat{a}_k \\ [\hat{n}_k, \hat{a}_k] &= -\hat{a}_k \\ [\hat{n}_k, \hat{a}_k^\dagger] &= \hat{a}_k^\dagger \end{align*}

3.7 Completeness Relation

The Fock space states form a complete orthonormal basis:

$$\mathbb{1} = |0\rangle\langle 0| + \int \frac{d^3k}{(2\pi)^3} |\mathbf{k}\rangle\langle\mathbf{k}| + \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3} |\mathbf{k}_1,\mathbf{k}_2\rangle\langle\mathbf{k}_1,\mathbf{k}_2| + \cdots$$

This allows us to expand any state in Fock space!

🎯 Key Takeaways

  • Fock space β„± = β„‹0 βŠ• β„‹1 βŠ• β„‹2 βŠ• ... (all n-particle sectors)
  • Vacuum: Γ’k|0⟩ = 0, unique lowest energy state
  • One-particle: |k⟩ = Γ’k†|0⟩ with energy Ο‰k
  • Multi-particle: |k₁,...,kn⟩ = Γ’k₁†...Γ’kn†|0⟩
  • Number operator: nΜ‚k = Γ’k†Òk counts particles in mode k
  • Bosons: Symmetric states, any occupation number allowed
  • Completeness: Fock states span the entire Hilbert space
  • Next: How do particles propagate? β†’ Propagators!