Quantum Field Theory Course

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

Part II, Chapter 4

Propagators & Green's Functions

How particles propagate through spacetime: the heart of Feynman diagrams

🔗Course Connections

📚Prerequisites

QFT Part II
Fock Space
Vacuum state and particle states
→
Quantum Mechanics
Time Evolution
Time-ordering and evolution operators
→
▶️

Video Lecture

Lecture 6: Propagators & Green Functions - MIT 8.323

Feynman propagator, causality, and time-ordering (MIT QFT Course)

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

4.1 Why Do We Need Propagators?

In QFT, we want to calculate scattering amplitudes: the probability that particles in state |i⟩ scatter into state |f⟩. The key quantity is:

$$\langle f | \hat{S} | i \rangle$$

where Ŝ is the S-matrix (scattering matrix). To compute this, we need to know how fields propagate between interaction points. That's where propagators come in!

💡Propagator = Field Correlation

The propagator tells us: "If I create a particle at spacetime point y, what's the amplitude to find it at point x?"

It's the quantum field theory generalization of the classical Green's function, which tells you how a source at y affects the field at x.

4.2 The Feynman Propagator

The Feynman propagator for a scalar field is defined as:

$$D_F(x-y) = \langle 0 | T\{\hat{\phi}(x) \hat{\phi}(y)\} | 0 \rangle$$

where T is the time-ordering operator:

$$T\{\hat{\phi}(x) \hat{\phi}(y)\} = \begin{cases} \hat{\phi}(x) \hat{\phi}(y) & \text{if } t_x > t_y \\ \hat{\phi}(y) \hat{\phi}(x) & \text{if } t_y > t_x \end{cases}$$

Time-ordering ensures that later operators are always to the left!

⚠️ Why Time-Ordering?

Without time-ordering, we'd get the wrong causality! The Feynman propagator automatically handles both particles going forward in time AND antiparticles going backward in time (Feynman-Stueckelberg interpretation).

4.3 Explicit Form in Position Space

For tx > ty (x is later than y):

\begin{align*} D_F(x-y) &= \langle 0 | \hat{\phi}(x) \hat{\phi}(y) | 0 \rangle \\ &= \int \frac{d^3k}{(2\pi)^3} \frac{1}{2\omega_k} e^{-ik \cdot (x-y)} \end{align*}

where k0 = ωk = √(k² + m²) (on-shell energy).

For tx < ty (y is later than x):

\begin{align*} D_F(x-y) &= \langle 0 | \hat{\phi}(y) \hat{\phi}(x) | 0 \rangle \\ &= \int \frac{d^3k}{(2\pi)^3} \frac{1}{2\omega_k} e^{ik \cdot (x-y)} \end{align*}

Combining both cases with the time-ordering:

$$D_F(x-y) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{2\omega_k} \left[ \theta(t_x - t_y) e^{-ik \cdot (x-y)} + \theta(t_y - t_x) e^{ik \cdot (x-y)} \right]$$

where θ(t) is the Heaviside step function.

4.4 Momentum Space Propagator

The real power of the Feynman propagator comes in momentum space. Fourier transforming:

$$\tilde{D}_F(p) = \int d^4x \, e^{ip \cdot x} D_F(x)$$

After careful contour integration (see problem set), we get the famous result:

$$\boxed{\tilde{D}_F(p) = \frac{i}{p^2 - m^2 + i\epsilon}}$$

This is the fundamental formula for scalar propagators! The iε term (where ε → 0+) is crucial for correct causal behavior.

💡Understanding the iε Prescription

The iε term shifts the poles slightly off the real axis:

  • p² = m² - iε has poles at p⁰ = ±ωp ∓ iε/2ωp
  • The +ω pole is slightly below the real axis
  • The -ω pole is slightly above the real axis

When doing the p⁰ integral for t > 0, we close the contour in the lower half-plane → picks up +ω pole → particle propagates forward in time.

For t < 0, close in upper half-plane → picks up -ω pole → antiparticle backward in time!

4.5 Green's Function Interpretation

The Feynman propagator is the Green's function for the Klein-Gordon operator:

$$(\Box + m^2) D_F(x-y) = -i\delta^4(x-y)$$

where □ = ∂μ∂μ is the d'Alembertian operator.

This means: "The propagator tells you how the field responds to a point source at y."

In momentum space, this becomes:

$$(-p^2 + m^2) \tilde{D}_F(p) = -i \quad \Rightarrow \quad \tilde{D}_F(p) = \frac{i}{p^2 - m^2 + i\epsilon}$$

4.6 Other Types of Propagators

Besides the Feynman propagator, there are other useful propagators:

Retarded Propagator

$$D_R(x-y) = \theta(t_x - t_y) \langle 0 | [\hat{\phi}(x), \hat{\phi}(y)] | 0 \rangle$$

Only propagates forward in time (tx > ty). Used for classical field responses.

Advanced Propagator

$$D_A(x-y) = -\theta(t_y - t_x) \langle 0 | [\hat{\phi}(x), \hat{\phi}(y)] | 0 \rangle$$

Only propagates backward in time (tx < ty). Less commonly used in practice.

Wightman Functions

\begin{align*} D^+(x-y) &= \langle 0 | \hat{\phi}(x) \hat{\phi}(y) | 0 \rangle \\ D^-(x-y) &= \langle 0 | \hat{\phi}(y) \hat{\phi}(x) | 0 \rangle \end{align*}

No time-ordering. Used in axiomatic QFT and thermal field theory.

The Feynman propagator is the sum:

$$D_F(x-y) = \theta(t_x - t_y) D^+(x-y) + \theta(t_y - t_x) D^-(x-y)$$

4.7 Causality and Spacelike Separation

For spacelike separated points (|x - y|² > (tx - ty)²), we have:

$$[\hat{\phi}(x), \hat{\phi}(y)] = 0 \quad \text{for spacelike } (x-y)$$

This is the microcausality condition: fields at spacelike separation commute, so measurements at these points don't interfere. No faster-than-light signaling!

However, the propagator itself is non-zero for spacelike separations:

$$D_F(x-y) \neq 0 \quad \text{for spacelike } (x-y)$$

This is okay! The propagator is not an observable—it's an internal quantum amplitude. Only commutators of observables must vanish for spacelike separation.

4.8 Connection to Feynman Diagrams

Feynman diagrams are the most powerful tool in perturbative QFT. Invented by Richard Feynman in 1948, they transform complicated integrals into intuitive pictures that encode complete mathematical expressions. The propagator is the heart of every diagram—it tells us how particles travel between interaction points.

💡What Feynman Diagrams Really Are

Feynman diagrams are not literal pictures of particle paths! They're graphical representations of terms in a perturbation expansion. Each diagram corresponds to a specific mathematical expression (amplitude), and the total amplitude is the sum of all diagrams.

Think of them as bookkeeping devices that help us organize and visualize incredibly complex calculations.

Building Blocks of Feynman Diagrams

Every Feynman diagram is built from three basic elements:

───────

Propagator (Internal Line)

Virtual particle traveling between vertices

$\frac{i}{p^2 - m^2 + i\epsilon}$
●

Vertex

Interaction point where particles meet

$-ig$ (coupling constant)
→───

External Line

Real incoming/outgoing particles

On-shell: $p^2 = m^2$

Example: φ⁴ Theory Scattering

In φ⁴ theory, the simplest interaction is 4 scalars meeting at a point. Consider 2→2 scattering:

     p₁    p₃
      \    /
       \  /
        \/
        /\
       /  \
      /    \
     p₂    p₄

Tree-level diagram: 4 external lines meeting at one vertex

$$i\mathcal{M} = -i\lambda$$

At tree level (no loops), the amplitude is just the coupling constant! But at higher orders, we get more interesting diagrams with internal propagators.

Loop Diagrams and Propagators

At one-loop level, we encounter diagrams where the propagator creates a closed loop:

     p₁         p₃
      \   ___   /
       \ (   ) /
        ● ⟲ ●
       / (___) \
      /         \
     p₂         p₄

One-loop correction: internal propagator forms a closed loop

$$\text{Loop} \sim \int \frac{d^4k}{(2\pi)^4} \frac{i}{k^2 - m^2 + i\epsilon}$$

⚠️ Loop Integrals and Divergences

Loop integrals often diverge! This leads to the profound subject of renormalization—the systematic handling of infinities in QFT. The propagator's pole structure (1/(p² - m²)) is directly responsible for both the physics AND the mathematical challenges of QFT.

💡Virtual Particles

Internal lines represent virtual particles—they don't satisfy the on-shell condition E² = p² + m². They exist only during the interaction and can "borrow" energy from the vacuum temporarily (Heisenberg uncertainty: ΔE·Δt ≈ ℏ).

The propagator 1/(p² - m²) blows up when p² = m² (on-shell), reflecting the long lifetime of real particles!

Complete Feynman Rules for Scalar QFT

Feynman Rules (φ⁴ Theory)

───────

Internal Propagator

For each internal line with momentum p:

$\frac{i}{p^2 - m^2 + i\epsilon}$
●

Vertex (4-point)

For each vertex where 4 lines meet:

$-i\lambda$
→───

External Line

For each external particle:

$1$ (scalars have no polarization)
∮

Loop Integration

For each undetermined loop momentum k:

$\int \frac{d^4k}{(2\pi)^4}$
$\Sigma p$

Momentum Conservation

At each vertex, total momentum is conserved:

$(2\pi)^4 \delta^4\left(\sum p_{\text{in}} - \sum p_{\text{out}}\right)$
$\frac{1}{S}$

Symmetry Factor

Divide by the symmetry factor $S$ of the diagram

(accounts for equivalent configurations)

From Diagrams to QED

In Quantum Electrodynamics (QED), we have three types of lines:

→──────→

Electron (Fermion)

$\frac{i(\not{p} + m)}{p^2 - m^2 + i\epsilon}$
∿∿∿∿∿∿

Photon (Gauge Boson)

$\frac{-ig_{\mu\nu}}{k^2 + i\epsilon}$
●

QED Vertex

$-ie\gamma^\mu$

The electron propagator includes the Dirac structure $(\not{p} + m)$, which handles spin. The photon propagator is massless $(m = 0)$, leading to long-range electromagnetic interactions!

📺 Video Lectures: Feynman Diagrams

Understanding Feynman diagrams is essential for QFT. These videos provide excellent visual explanations of how propagators connect to particle interactions.

Quantum Field Theory Visualized (ScienceClic)
▶️

Video Lecture

QFT Visualized - ScienceClic

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

Excellent 15-minute overview covering spin, charge, the Standard Model, and how particle interactions are represented through Feynman diagrams.

The Secrets of Feynman Diagrams (PBS Space Time)
▶️

Video Lecture

Secrets of Feynman Diagrams - PBS Space Time

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

Deep dive into what Feynman diagrams really represent: not just pictures, but complete mathematical expressions encoding quantum amplitudes.

Feynman's Infinite Quantum Paths (PBS Space Time)
▶️

Video Lecture

Feynman Path Integrals - PBS Space Time

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

The path integral formulation that underlies Feynman diagrams. Essential for understanding why we sum over all possible histories.

All Particle Physics Explained with Feynman Diagrams (Arvin Ash)
▶️

Video Lecture

Particle Physics with Feynman Diagrams - Arvin Ash

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

Comprehensive 18-minute overview of all fundamental forces and particles using Feynman diagram notation. Covers electromagnetism, weak force, strong force, and Higgs.

Feynman QED Lecture: Photons & Electrons (Original Feynman)
▶️

Video Lecture

Feynman QED Lecture - Original

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

The master himself explaining QED. Feynman's 1979 Auckland lecture on how photons and electrons interact—the foundation of all Feynman diagrams.

📜 Historical Note: The Birth of Feynman Diagrams

In 1948, at the famous Pocono Conference, Feynman presented his revolutionary approach to QED. Initially, his diagrams confused even the great physicists present—Niels Bohr and Paul Dirac were skeptical. But Freeman Dyson soon showed that Feynman's pictorial method was equivalent to the more traditional approaches of Schwinger and Tomonaga.

Today, Feynman diagrams are the lingua franca of particle physics. Every calculation in the Standard Model, from electron scattering to Higgs production, uses these elegant pictures that Feynman invented.

Practice Problems

Practice Problems

Progress: 0/2 (0%)
1
⭐⭐ Medium

Show that in momentum space, the Feynman propagator for a Klein-Gordon field is:

D̃F(p) = i/(p² - m² + iε)

2
⭐⭐⭐ Hard

Explain physically why the Feynman propagator can be interpreted as a particle propagating forward in time OR an antiparticle propagating backward in time.

Problem Set Statistics

Total Problems
2
Attempted
0
Completion
0%

⚠️Common Mistakes to Avoid

❌

Mistake:

Forgetting the time-ordering in the Feynman propagator
🤔

Why it's wrong:

Time-ordering is essential for correct causal behavior and proper particle interpretation.
✅

Correct approach:

The Feynman propagator is ⟨0|T{φ̂(x)φ̂(y)}|0⟩, not just ⟨0|φ̂(x)φ̂(y)|0⟩. Time-ordering is crucial for causality!
❌

Mistake:

Confusing +iε and -iε prescriptions
🤔

Why it's wrong:

The sign of iε determines which poles contribute and affects causal structure.
✅

Correct approach:

The Feynman propagator has i/(p² - m² + iε), NOT -iε. The +iε prescription ensures correct causal boundary conditions.
❌

Mistake:

Thinking the propagator is just for free particles
🤔

Why it's wrong:

Propagators are the building blocks of all scattering calculations in QFT.
✅

Correct approach:

The free propagator is the building block for all interactions in perturbation theory via Feynman diagrams!

🎯 Key Takeaways

  • Feynman propagator: DF(x-y) = ⟨0|T{φ̂(x)φ̂(y)}|0⟩ (time-ordered correlation)
  • Momentum space: D̃F(p) = i/(p² - m² + iε) — THE fundamental formula!
  • iε prescription ensures correct causal boundary conditions
  • Time-ordering handles particles forward + antiparticles backward in time
  • Green's function: (□ + m²)DF = -iδ⁴(x-y)
  • Microcausality: [φ̂(x), φ̂(y)] = 0 for spacelike separation
  • Feynman diagrams: Internal lines = propagators
  • Next: Quantize fermions with anticommutators!