Takeaway

Quantum amplitudes arise by summing contributions from all possible paths, each weighted by e^{i S[path]/ħ}; classical physics emerges from stationary action via interference.

The problem (before → after)

  • Before: Schrödinger’s wave mechanics can obscure the link to classical action and symmetries.
  • After: The path integral unifies dynamics and symmetry through the action, making perturbation theory and field quantization natural.

Mental model first

Imagine many hikers taking every conceivable route between two points. Each route waves a flag whose angle equals S/ħ; routes with similar S reinforce; wildly different S cancel, leaving the classical path dominant.

Just-in-time concepts

  • Action: S[trajectory] = ∫ L dt. Weight is exp(i S / ħ).
  • Stationary phase: Nearby paths with δS ≈ 0 add coherently → classical limit.
  • Correlators: Z[J] = ∫ Dϕ exp(i S[ϕ] + i ∫ J ϕ) generates Green functions.
  • Feynman rules: Expand around free theory; vertices and propagators come from interaction terms.

First-pass solution

Define Z as an integral over configurations. For quadratic actions, integrals are Gaussian → exact propagators. Interactions are treated perturbatively.

Iterative refinement

  1. Wick rotation: t → − i τ makes weights decaying (exp(−S_E)); enables statistical mechanics connections and lattice Monte Carlo.
  2. Gauge fixing and ghosts: For gauge theories, divide by gauge volume to avoid overcounting.
  3. Semiclassics: Instantons and saddle points capture tunneling beyond perturbation theory.

Principles, not prescriptions

  • Symmetries are clearest in the action; quantize by integrating over symmetric configurations.
  • Classical behavior is interference, not “turned-off” quantum effects.
  • Sources and generating functionals organize computations.

Common pitfalls

  • Treating the integral as ordinary: it’s a distributional limit; regularization and care with measures matter.
  • Ignoring boundary conditions, which select the correct propagators.

Connections and contrasts

  • See also: [/blog/renormalization-group] (flows of couplings), [/blog/yang-mills-gauge-theory], [/blog/diffusion-models] (path integrals ↔ stochastic processes via Wick rotation).

Quick checks

  1. Why does the classical path dominate? — Stationary action yields constructive interference.
  2. What does Wick rotation buy you? — Convergent integrals and a link to statistical physics.
  3. Why do we need gauge fixing? — To remove redundant integration over gauge orbits.

Further reading

  • Feynman & Hibbs, “Quantum Mechanics and Path Integrals”
  • Zee, “QFT in a Nutshell,” Part I
  • Review article (source above)