Feynman’s Path Integral Formulation Explained

The beauty and simplicity of summing over all possible paths

In the early 20th century, mathematician Norbert Wiener introduced the Wiener integral as a way to study Brownian motion and diffusion processes. Although he was interested in probability and statistical physics, his work unwittingly formed the basis for an entirely new way of thinking about quantum mechanics: summing over all possible paths a particle could take. Some decades later, physicist Paul Dirac, probably around 1933, decided to take that one step ahead. He sought to explain just how the concept of the Lagrangian should be interpreted-for classical mechanics-applied to quantum physics. There, he said that the amplitudes should be related in an exponential fashion with the classic action-a bold statement, showing that quantum dynamics could be summarised as a weighted sum over distinct histories. I’ll simplify this later in the story.

[Left] Norbert Wiener sitting in front of a chalkboard. MIT Museum, courtesy of AIP Emilio Segrè Visual Archives. [Right] Paul Dirac’s portrait photographed by Coale, Phil, AP Photo. Public Domain Images.

The development of this principle into a systematic framework is really due to Richard Feynman. In particular, whereas in classical physics a single, well-defined path is considered, the treatment of quantum systems involves summing over all possible paths the particle might take from the initial to the final point, with each path contributing according to a general phase factor. In this way…