Definitions

homotopic paths

Sum-over-paths

Sum-over-paths, also known as Feynman sum-over-paths, is an approach to visualizing the movement of particles that is mathematically described by the equations of quantum mechanics. This model competes with the concept of probability waves, though it is numerically identical. Richard Feynman first articulated this idea and contended that fast moving subatomic particles travel from point A to point B not by a single path but by all possible paths.

By taking the sum of all possible paths, he reached the same conclusions he would have reached by associating probability waves with each travelling particle. That is, he found the exact probabilities for the outcomes predicted by other theories and by experimental results.

The sum-over-paths approach has become an especially useful analogy for visualizing some of the strange phenomena observed in the subatomic world. Feynman's approach is often used to explain in words the seemingly bizarre conclusions derived by mathematics and experimental results, in particular the famous double slit experiment, which was a starting point for the investigation of quantum mechanics (and is often a starting point for students of physics).

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