Quantum Field Theory Course

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

Part III, Chapter 5

Feynman Diagrams & Wick's Theorem

Visual perturbation theory: from path integrals to pictures

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part III
Correlation Functions
n-point functions and connected diagrams
β†’
QFT Part III
Path Integrals in QFT
Generating functional Z[J]
β†’
▢️

Video Lecture

Lecture 11: Computation of Correlation Functions & Feynman Diagrams - MIT 8.323

Wick contractions and the diagrammatic expansion of perturbation theory (MIT QFT Course)

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

5.1 Wick's Theorem

Wick's theorem is the master formula for computing Gaussian path integrals. It states that time-ordered products can be written as sums of contractions:

$$\boxed{T\{\phi(x_1)\cdots\phi(x_n)\} = \text{(all possible full contractions)}}$$

A contraction (or Wick contraction) of two fields is:

$$\boxed{\wick{\c\phi(x)\c\phi(y)} = D_F(x-y) = \text{Feynman propagator}}$$

πŸ’‘What is a Contraction?

Think of a contraction as "pairing up" two field operators. In Feynman diagrams, each contraction becomes a line (propagator) connecting two points!

Example: For 4 fields, there are 3 ways to pair them:

  • (φ₁φ₂)(φ₃φ₄) β†’ two propagators: x₁-xβ‚‚ and x₃-xβ‚„
  • (φ₁φ₃)(Ο†β‚‚Ο†β‚„) β†’ two propagators: x₁-x₃ and xβ‚‚-xβ‚„
  • (φ₁φ₄)(φ₂φ₃) β†’ two propagators: x₁-xβ‚„ and xβ‚‚-x₃

Example: 4-Point Function

$$\langle 0|T\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\}|0\rangle = $$
$$= D_F(x_1-x_2)D_F(x_3-x_4) + D_F(x_1-x_3)D_F(x_2-x_4) + D_F(x_1-x_4)D_F(x_2-x_3)$$

Each term corresponds to a different pairing = different Feynman diagram!

5.2 Feynman Diagram Basics

A Feynman diagram is a pictorial representation of a term in the perturbative expansion.

Visual Elements

  • β€’ Lines (propagators): Connect two points, represent DF(x-y)
    Internal line = virtual particle, External line = real particle
  • β€’ Vertices: Where lines meet, represent interaction
    For φ⁴: vertex = 4 lines meeting, factor = -iΞ»
  • β€’ External points: Incoming/outgoing particles
    Where we measure asymptotic states
  • β€’ Loops: Internal closed paths
    Represent quantum fluctuations, require momentum integration

5.3 Example: φ⁴ Theory

The interaction Lagrangian is:

$$\mathcal{L}_{\text{int}} = -\frac{\lambda}{4!}\phi^4$$

In the path integral:

$$Z[J] = \int \mathcal{D}\phi \, e^{iS_0[\phi]} \exp\left[-i\frac{\lambda}{4!}\int d^4x \, \phi^4(x)\right] e^{i\int J\phi}$$

Expand the interaction exponential:

$$e^{-i\frac{\lambda}{4!}\int \phi^4} = 1 - i\frac{\lambda}{4!}\int d^4x \, \phi^4(x) + \frac{1}{2!}\left(-i\frac{\lambda}{4!}\right)^2\left[\int d^4x \, \phi^4(x)\right]^2 + \cdots$$

Order-by-Order Expansion

  • Order λ⁰: Free theory, only propagators
    Tree-level diagrams, no vertices
  • Order λ¹: One vertex (4 fields meeting)
    First interaction correction
  • Order λ²: Two vertices (8 fields total)
    Loop diagrams start appearing!
  • Order λⁿ: n vertices
    Higher-order quantum corrections

5.4 Feynman Rules for φ⁴ Theory

Position Space Rules

  1. External point at xi: Factor of 1
  2. Internal propagator: DF(x-y) for line connecting x and y
  3. Vertex at x: Factor of -iλ, integrate ∫d⁴x
  4. Symmetry factor: Divide by S (# of equivalent diagrams)
  5. Multiply all factors together

Momentum Space Rules

  1. External line with momentum p: Factor of 1
  2. Internal propagator: i/(pΒ² - mΒ² + iΞ΅) for momentum p
  3. Vertex: Factor of -iΞ», enforce momentum conservation
  4. Loop with momentum k: Integrate ∫d⁴k/(2Ο€)⁴
  5. Overall momentum conservation: (2Ο€)⁴δ⁴(Ξ£pin - Ξ£pout)
  6. Symmetry factor: Divide by S

5.5 Symmetry Factors

The symmetry factor S counts how many ways a diagram can be drawn identically by permuting vertices and propagators.

Common Symmetry Factors

  • β€’ Simple φ⁴ vertex: S = 4! = 24 (from the 1/4! in β„’int)
  • β€’ Two-point tadpole: Additional factor from loop symmetry
  • β€’ Figure-8 diagram: S = 2 (can flip the two loops)
  • β€’ General rule: S = (# of automorphisms of the diagram)

Pro tip: Symmetry factors are tricky! They arise from careful combinatorics in the Wick expansion. In practice, we often use Feynman rules that absorb these factors.

5.6 Loop Diagrams & Divergences

Loop diagrams have closed paths of internal propagators. They require integrating over loop momentum:

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

Problem: This integral diverges at large k! This is the famous UV divergenceof quantum field theory.

⚠️ Divergences

Loop integrals often diverge and require renormalization:

  • UV divergent: Integral ∞ as k β†’ ∞ (high energy)
  • IR divergent: Integral ∞ as k β†’ 0 (low energy, massless particles)
  • Solution: Regularize + Renormalize (Part VI of this course!)

🎯 Key Takeaways

  • Wick's theorem: Time-ordered products = all full contractions
  • Contraction = propagator = line in Feynman diagram
  • Feynman diagrams = visual representation of perturbation theory
  • Vertices = interactions, Lines = propagators, Loops = quantum corrections
  • φ⁴ theory: Each vertex = 4 lines, factor -iΞ»
  • Momentum space: Assign momenta, enforce conservation, integrate over loops
  • Symmetry factors: Count equivalent diagrams (tricky!)
  • Loops β†’ divergences β†’ need renormalization
  • Next: Organizing perturbation theory systematically!