Path integral formulation

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These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t0 to point B at some other time t1.

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.


  • The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. … So when an electron arrives at a point in space—say a target screen—many different histories must be integrated together to create this one event.
    Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits— applying it to gravitation and even cosmology—so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant.
    • Paul Davies (1994), Introduction to Six Easy Pieces by Richard P. Feynman
  • You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game.

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