Part III: Path Integrals & Perturbation Theory
The path integral formulation provides an elegant alternative to canonical quantization, directly connecting quantum field theory to classical physics and enabling powerful perturbative calculations through Feynman diagrams.
Overview
In Part II, we quantized fields by promoting classical observables to operators satisfying canonical commutation relations. Part III presents the path integral formulation, where we sum over all possible field configurations weighted by eiS.
This approach has several advantages:
- Manifestly Lorentz covariant formulation
- Direct connection to classical field theory through the stationary phase approximation
- Natural framework for perturbation theory and Feynman diagrams
- Easier to generalize to gauge theories and curved spacetime
- Provides powerful computational techniques for correlation functions
Course Structure
1. Interacting Theories & S-Matrix
πMoving beyond free field theory: interaction picture, time evolution, and the scattering matrix. Understanding how to compute observable quantities from quantum field theory.
2. Path Integrals in Quantum Mechanics
β«Feynman's sum-over-paths formulation of quantum mechanics. Understanding the transition amplitude as a path integral and its connection to the classical action.
3. Path Integrals in QFT
β«πΟExtending path integrals to field theory: functional integration over field configurations. The generating functional and vacuum-to-vacuum transitions.
4. Time-Ordered Correlation Functions
β¨Tβ©Computing n-point correlation functions using path integrals. Time-ordering and the connection to physical observables through the LSZ formula.
5. Feynman Diagrams & Wick's Theorem
β·Visual representation of perturbation theory through Feynman diagrams. Wick's theorem for computing Gaussian integrals and systematic diagrammatic expansion.
6. More on Perturbation Theory
βAdvanced techniques in perturbation theory: connected vs. disconnected diagrams, vacuum bubbles, and the linked cluster theorem. Setting up calculations for Οβ΄ and QED.
π Prerequisites
Before starting Part III, you should be comfortable with:
- Part I: Classical Field Theory - Lagrangian formalism and symmetries
- Part II: Canonical Quantization - Creation/annihilation operators and Fock space
- Propagators - Feynman propagator and Green functions
- Functional Analysis - Basic functional differentiation
π― Key Concepts You'll Master
Fundamental Techniques:
- Path integral quantization
- Functional derivatives
- Generating functionals
- Wick's theorem
- Feynman rules
Physical Applications:
- Scattering amplitudes
- n-point correlation functions
- Perturbative expansions
- Connected diagrams
- Systematic QFT calculations