The Double-Slit Experiment Revisited
The double-slit experiment is quantum mechanics' central mystery: a single electron passes through two slits simultaneously and creates an interference pattern on a screen — as if it is a wave — yet when we detect which slit it went through, the interference vanishes and it behaves as a particle. Feynman's path integral explains this: the electron's amplitude is the sum of all paths, including paths through both slits simultaneously. Detecting which slit eliminates the contributions of paths through the other slit — collapsing the interference.
All Paths, Weighted by Action
In Feynman's formulation, each path has an amplitude equal to e^(iS/ħ), where S is the classical action along the path and ħ is the reduced Planck constant. Most paths cancel each other out through destructive interference. The paths that contribute most are those near the classical path — the one that minimizes action (Hamilton's principle). In the classical limit (large objects), quantum uncertainty disappears and only the classical path survives. Classical mechanics emerges from quantum mechanics as an approximation.
The Connection to Quantum Field Theory
Feynman path integrals are the natural language of quantum field theory, where one sums not just over all paths of a particle but over all possible field configurations in spacetime. Every particle interaction is represented as a sum over Feynman diagrams — each diagram corresponds to a path integral contribution. The Standard Model of particle physics is written in this language.
Frequently Asked Questions
What is Feynman's path integral?
Feynman's path integral formulation of quantum mechanics says the probability amplitude for a particle to go from one point to another is the sum of contributions from all possible paths between those points, each weighted by e^(iS/ħ) where S is the classical action. Paths near the classical trajectory dominate; others cancel. It gives the same predictions as Schrödinger's equation but from a different conceptual perspective.
What does 'a particle takes all paths' mean?
In quantum mechanics, before measurement, a particle does not have a definite trajectory. Feynman's formulation captures this by summing amplitudes over all conceivable paths — no matter how wild or physically implausible. Most cancel through interference; the observed trajectory emerges from the paths that constructively interfere. It is not a metaphor but a calculational procedure with exact predictions.
What is quantum action?
In classical mechanics, the action S of a path is the integral of the Lagrangian (kinetic minus potential energy) along the path. Hamilton's principle says particles follow the path that minimizes action. In quantum mechanics, Feynman showed that all paths contribute with amplitude e^(iS/ħ), and the classical path minimizes the phase variation — it is the path that 'survives' the quantum interference.
