Quantum Field Theory Course

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

Part III, Chapter 6

More on Perturbation Theory

Connected diagrams, vacuum bubbles, and systematic QFT calculations

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πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part III
Feynman Diagrams
Diagrammatic expansion and Wick's theorem
β†’
QFT Part III
S-Matrix
Dyson series and perturbative expansion
β†’
▢️

Video Lecture

Lecture 12: More on Perturbation Theory and Feynman Diagrams - MIT 8.323

Advanced perturbation theory techniques and the linked cluster theorem (MIT QFT Course)

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

6.1 Connected vs Disconnected Diagrams

A Feynman diagram is connected if you can reach any part from any other part by following lines. Otherwise, it's disconnected.

πŸ’‘Physical Interpretation

Connected diagrams: All particles actually interact with each other. This is the "interesting" physics!

Disconnected diagrams: Independent processes happening simultaneously. Example: Two particles scatter in one region while two others scatter far away.

For scattering amplitudes, we only care about connected diagrams - disconnected ones factor out and cancel in properly normalized S-matrix elements!

The generating functional decomposes as:

$$Z[J] = e^{iW[J]} \quad \Rightarrow \quad W[J] = -i\ln Z[J]$$

W[J] generates only connected diagrams! This is the linked cluster theorem.

6.2 Vacuum Bubbles

Vacuum bubbles are diagrams with no external lines - they're disconnected from the rest of the diagram and represent vacuum-to-vacuum fluctuations.

$$Z[0] = \langle 0|0\rangle = \int \mathcal{D}\phi \, e^{iS[\phi]} = e^{i\Gamma_{\text{vac}}}$$

where Ξ“vac is the sum of all vacuum bubble diagrams. Key fact: Vacuum bubbles cancelin normalized correlation functions!

$$\langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle = \frac{1}{Z[0]}\frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

The Z[0] in the denominator exactly cancels all vacuum bubble contributions! This is crucial - we don't have to compute them.

6.3 Linked Cluster Theorem

The linked cluster theorem states:

Linked Cluster Theorem

For properly normalized S-matrix elements and correlation functions:

  1. Only connected diagrams contribute
  2. Vacuum bubbles cancel in the normalization
  3. Disconnected diagrams factor into products of connected pieces

This dramatically simplifies calculations - we only compute connected, non-vacuum diagrams!

Mathematically:

$$\langle f|S - \mathbb{1}|i\rangle = \text{(sum of connected, amputated diagrams)}$$

Key Concepts (This Page)

  • Connected diagrams: only these contribute to the S-matrix
  • $W[J] = -i\ln Z[J]$ generates connected diagrams (linked cluster theorem)
  • Vacuum bubbles cancel in normalized correlation functions
  • Z[J] factorizes as $Z[J] = e^{iW[J]}$ where W sums connected pieces
  • Disconnected diagrams = products of connected subdiagrams with symmetry factors
Next: Loop Integrals & Dimensional Regularization