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 t_x > t_y (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 = ω_k = √(k² + m²) (on-shell energy).

For t_x < t_y (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 = ±ω_p ∓ iε/2ω_p
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 (t_x > t_y). 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 (t_x < t_y). 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 > (t_x - t_y)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

In Feynman diagrams, the propagator represents an internal line:

───────────

Internal scalar line = Propagator i/(p² - m² + iε)

Every internal line in a Feynman diagram contributes a factor of the propagator in momentum space! This is the foundation of perturbative QFT calculations.

💡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!

Practice Problems

📝 Practice Problems

Progress: 0/2 (0%)

⭐⭐ Medium

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

D̃_F(p) = i/(p² - m² + iε)

⭐⭐⭐ 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

Attempted

Completion

⚠️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: D_F(x-y) = ⟨0|T{φ̂(x)φ̂(y)}|0⟩ (time-ordered correlation)
Momentum space: D̃_F(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)D_F = -iδ⁴(x-y)
Microcausality: [φ̂(x), φ̂(y)] = 0 for spacelike separation
Feynman diagrams: Internal lines = propagators
Next: Quantize fermions with anticommutators!

Quantum Field Theory Course

Propagators & Green's Functions

🔗Course Connections

📚Prerequisites

Video Lecture

4.1 Why Do We Need Propagators?

💡Propagator = Field Correlation

4.2 The Feynman Propagator

⚠️ Why Time-Ordering?

4.3 Explicit Form in Position Space

4.4 Momentum Space Propagator

💡Understanding the iε Prescription

4.5 Green's Function Interpretation

4.6 Other Types of Propagators

Retarded Propagator

Advanced Propagator

Wightman Functions

4.7 Causality and Spacelike Separation

4.8 Connection to Feynman Diagrams

💡Virtual Particles

Practice Problems

📝 Practice Problems

📊 Problem Set Statistics

⚠️Common Mistakes to Avoid

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

🎯 Key Takeaways