Playground 12

Path Integral as Sum Over Histories

Start with many particle paths between fixed endpoints, weight each by e^{iS}, watch interference select the stationary path, then generalize the same idea to a sum over field configurations.

Animation Controls

Current build step

1. Initial and final states are fixed

History spread0.65

tight familymany alternativeswide spread

Phase oscillation1.10

slow phasemoderaterapid cancellation

Classical pull0.70

fully quantumstationary phaseclassical-looking

Weight

e^{iS[\phi]}

Visible histories

Cancellation

12%

Path-Integral Build

Sum over histories

Each curve is one candidate history between the same endpoints.

5 stages

Amplitude recipe

The transition amplitude is built by adding contributions from every allowed history, each weighted by its phase.

\mathcal{A}(A \to B) = \int \mathcal{D}\phi\, e^{iS[\phi]}

Classical limit

When the phase swings rapidly, nearby non-stationary paths cancel each other. The dominant surviving contribution sits near the classical trajectory.

surviving weight0.80

From particles to fields

In field theory, the same idea survives but the “paths” are now entire field configurations spread across spacetime rather than a single particle trajectory.

Quantum mechanics

\sum_{\text{paths}} e^{iS[x(t)]}

→

Quantum field theory

\int \mathcal{D}\phi\, e^{iS[\phi]}