Quantum Field Theory Course

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

Part III, Chapter 4

Time-Ordered Correlation Functions

Computing n-point functions: the building blocks of scattering amplitudes

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part III
Path Integrals in QFT
Generating functional Z[J]
β†’
QFT Part III
S-Matrix
LSZ formula connecting Green functions to S-matrix
β†’
▢️

Video Lecture

Lecture 10: Time-Ordered Correlation Functions - MIT 8.323

Computing n-point functions using path integrals and their physical interpretation (MIT QFT Course)

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

4.1 What Are Correlation Functions?

The n-point correlation function (or Green function) is:

$$\boxed{G^{(n)}(x_1,\ldots,x_n) = \langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle}$$

where T is the time-ordering operator:

$$T\{\phi(x_1)\phi(x_2)\} = \begin{cases} \phi(x_1)\phi(x_2) & t_1 > t_2 \\ \phi(x_2)\phi(x_1) & t_2 > t_1 \end{cases}$$

πŸ’‘Why Time-Ordering?

In the interaction picture, fields at different times don't commute: [Ο†(x), Ο†(y)] β‰  0.

Time-ordering ensures we evolve from the past to the future consistently. It's crucial for connecting to the S-matrix via the LSZ formula!

Physical interpretation: Later time fields are measured after earlier ones - this is causality!

4.2 Path Integral Formula

From the generating functional Z[J], we get:

$$G^{(n)}(x_1,\ldots,x_n) = \frac{1}{Z[0]}\frac{1}{i^n}\frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

Equivalently, in path integral form:

$$\boxed{G^{(n)}(x_1,\ldots,x_n) = \frac{1}{Z[0]}\int \mathcal{D}\phi \, \phi(x_1)\cdots\phi(x_n) \, e^{iS[\phi]}}$$

This formula is the foundation of perturbative QFT! We expand in powers of the coupling and evaluate Gaussian integrals.

4.3 Free Field Theory

For free fields (S = Sβ‚€), all correlation functions reduce to products of 2-point functions!

2-Point Function

$$\boxed{G^{(2)}(x,y) = \langle 0|T\{\phi(x)\phi(y)\}|0\rangle = D_F(x-y)}$$

This is the Feynman propagator we met before!

4-Point Function

For Gaussian integrals (free fields), we have Wick's theorem:

$$G^{(4)}(x_1,x_2,x_3,x_4) = D_F(x_1-x_2)D_F(x_3-x_4) + D_F(x_1-x_3)D_F(x_2-x_4) + D_F(x_1-x_4)D_F(x_2-x_3)$$

Three terms = three ways to pair up four points! This generalizes to all even n.

Odd Point Functions

For free fields with Ο† β†’ -Ο† symmetry:

$$G^{(2n+1)} = 0 \quad \text{(all odd point functions vanish)}$$

4.4 Connected vs Disconnected

Consider G(4) above. The three terms represent:

  • Disconnected diagrams: Two independent 2-point functions (first two terms if points separated)
  • Connected diagrams: All points linked by propagators

The connected Green function Gc(n) has only connected contributions.

Define the generating functional for connected diagrams:

$$\boxed{W[J] = -i\ln Z[J]}$$

Then:

$$G_c^{(n)}(x_1,\ldots,x_n) = \frac{1}{i^n}\frac{\delta^n W[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

The logarithm picks out only connected diagrams - very useful!

4.5 Momentum Space

Fourier transform to momentum space:

$$\tilde{G}^{(n)}(p_1,\ldots,p_n) = \int \prod_{i=1}^n d^4x_i \, e^{ip_i \cdot x_i} G^{(n)}(x_1,\ldots,x_n)$$

Translation invariance gives:

$$\tilde{G}^{(n)}(p_1,\ldots,p_n) = (2\pi)^4 \delta^4\left(\sum_{i=1}^n p_i\right) \tilde{G}^{(n)}_{\text{connected}}(p_1,\ldots,p_{n-1})$$

The delta function enforces momentum conservation! This is why Feynman diagrams have momentum conservation at each vertex.

4.6 Physical Interpretation

LSZ Connection to S-Matrix

The LSZ formula relates correlation functions to scattering amplitudes:

$$\langle f|S|i\rangle \propto \text{(amputated, on-shell)} \, G^{(n)}$$

"Amputated" = remove external propagators. "On-shell" = pΒ² = mΒ² for external lines.

Practical Computation Strategy

  1. Compute G(n) using path integrals (Feynman diagrams)
  2. Fourier transform to momentum space
  3. Remove external propagators (amputation)
  4. Set external momenta on-shell: pΒ² = mΒ²
  5. Result = scattering amplitude Mfi
  6. Square to get cross section: dΟƒ ∝ |Mfi|Β²

🎯 Key Takeaways

  • n-point function: G(n) = ⟨0|T{Ο†(x₁)...Ο†(xn)}|0⟩
  • Time-ordering T essential for causality and LSZ formula
  • Path integral: G(n) = functional derivatives of Z[J]
  • Free theory: Wick's theorem reduces to products of propagators
  • Connected diagrams: W[J] = -i ln Z[J] generates only connected pieces
  • Momentum space: Delta function enforces momentum conservation
  • LSZ formula: G(n) β†’ S-matrix (observable scattering)
  • Next: Wick's theorem and Feynman diagrams!