Path Integrals and Perturbation Theory

The path integral expresses quantum amplitudes as a sum over all possible field configurations, providing the foundation for perturbative calculations organized by Feynman diagrams.

Definition

The path integral is a formulation of quantum theory in which the transition amplitude between states is given by a weighted sum over all field configurations, and perturbation theory is the expansion of interacting amplitudes in powers of the coupling constant, represented diagrammatically by Feynman diagrams.

Scope

This topic covers Feynman's path-integral formulation of quantum mechanics and field theory, in which probability amplitudes are computed by summing contributions from every possible history weighted by the action. It treats the systematic expansion of interacting theories in powers of the coupling, the translation of each term into Feynman diagrams with propagators and vertices, and the extraction of scattering cross sections and decay rates from these amplitudes.

Core questions

• How does summing over all possible histories reproduce quantum dynamics?
• How is an interacting field theory expanded as a series in the coupling constant?
• How do Feynman diagrams encode the terms of the perturbative expansion?
• How are measurable cross sections and decay rates extracted from scattering amplitudes?

Key concepts

• Sum over histories
• Action and the phase factor
• Feynman propagators
• Interaction vertices
• Tree-level and loop diagrams
• Cross sections and decay rates

Key theories

Path integral formulation

Quantum amplitudes are obtained by integrating the phase factor exp(iS) over all field configurations, with the classical path recovered in the limit where the action is large compared to Planck's constant.

Diagrammatic perturbation theory

Each order in the coupling expansion corresponds to a set of Feynman diagrams whose lines and vertices are translated by fixed rules into mathematical contributions to the scattering amplitude.

Clinical relevance

Path integrals and perturbation theory furnish the standard machinery for predicting collider observables, underlie lattice gauge theory and Monte Carlo simulation of the strong interaction, and provide methods that carry over to statistical mechanics and condensed-matter physics.

History

Building on a suggestion of Dirac, Feynman formulated the path-integral approach to quantum mechanics in 1948 and developed the diagrammatic rules that bear his name for quantum electrodynamics. Dyson showed the equivalence of Feynman's diagrams with the operator methods of Schwinger and Tomonaga, and the path integral later became the preferred framework for quantizing gauge theories and formulating lattice field theory.

Key figures

• Richard Feynman
• Paul Dirac
• Freeman Dyson

Seminal works

• feynman1948
• feynmanhibbs1965

Frequently asked questions

What does it mean to sum over all paths?

In the path integral, every conceivable history connecting the initial and final states contributes a complex phase to the amplitude. The paths interfere, and the dominant contribution near the classical path emerges when the action is large.

Are Feynman diagrams literal pictures of particle paths?

No. Feynman diagrams are bookkeeping devices for terms in the perturbative expansion. Their lines represent propagators and their vertices represent interactions, not actual trajectories of particles in space.