Path integral formulation: Difference between revisions

Revision as of 05:21, 13 September 2005

Revision as of 12:36, 9 October 2005

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=== Recovering the action principle ===

Feynman was initially attempting to make sense of a brief remark by [[Paul Dirac]] about the quantum equivalent of the [[action (physics) | action principle]] in classical mechanics. In the limit of action that is large compared to [[Planck's constant]] <math>\hbar</math>, the path integral is dominated by solutions which are [[stationary point]]s of the action, since, there, the amplitudes of similar histories will tend to constructively [[interference|interfere]] with one another. Conversely, for paths that are far from being stationary points of the action, the complex phase of the amplitude calculated according to postulate 3 will vary rapidly for similar paths, and amplitudes will tend to cancel. Therefore the important parts of the integral&mdash;the significant possibilities&mdash;in the limit of large action simply consist of solutions of the [[Euler-Lagrange equation]], and classical mechanics is correctly recovered.

Action principles can seem puzzling to the student of physics because of their seemingly [[teleology|teleological]] quality: instead of predicting the future from initial conditions, one starts with a combination of initial conditions and final conditions and then finds the path in between, as if the system somehow knows where it's going to go. The path integral is one way of understanding why this works. The system doesn't have to know in advance where it's going; the path integral simply calculates the ''probability amplitude'' for a given process, and the stationary points of the action mark neighborhoods of the space of histories for which quantum-mechanical interference will yield large probabilities.