The Lagrangian – possibly the most fundamental mathematical principle in physics

This is something I wrote on Quora, which was ‘upvoted’ by a physics tutor (Mike Milner) and someone with an MSc (Dimitrios Kalemis), which gives it some credence.

I’ve written about all of this before in previous posts, but probably not as succinctly, which hopefully makes it easier to follow.


How does an electron know beforehand that it's a single slit or double slit so it decides whether to create an interference pattern or not?

Obviously it doesn’t. It’s like asking how does a ball thrown in the air know what path to follow? These 2 questions have more in common than you might think.

There is a fundamental principle in physics called the principle of least action, and Richard Feynman used it to describe the trajectory of a ball in a gravitational field and also as the basis for his path integral method of quantum mechanics (QM).

The principle of least action is that the difference between the potential energy and the kinetic energy of a particle will always be a minimum and, mathematically, this is called a Lagrangian. In his book, Six Not-So-Easy Pieces, Feynman demonstrates how this applies to a body in a gravitational field when it follows the path dictated by a geodesic, which, in Einstein’s theory of relativity, is the path of maximum relativistic time. It turns out that this is the shortest path and also the path of least action, as determined by the Lagrangian.

Feynman gives the following analogy. Imagine a lifesaver needing to run along a beach and then swim out to rescue a bather in distress in the surf. The lifesaver could run along the beach (at a diagonal) until he (or she) is perpendicular to the swimmer in the waves and swim out. Or the lifesaver could run straight into the surf and swim diagonally to the swimmer. But the optimum path is something in between these 2 and that’s the path of least action or least time. It’s also the path of light when it refracts through glass or any other medium.

It was Paul Dirac who originally wrote a Lagrangian for QM and Feynman used his result to derive Schrodinger’s equation. Feynman’s approach to the 2 slit problem or any other QM problem was to combine all the possible paths the electron (or a photon) could take. By ‘combine’ this means adding all the phases of the wave function, most of which cancel each other out. Then, using Born’s rule, he derived the probabilities of where the electron would hit the screen on the other side of the slit(s).

In his book, QED, he provides a graphic demonstration using this method to derive the path of a photon hitting a mirror. He says ‘the light goes where the time is least.’

In response to your specific question, the electron’s path is only determined retrospectively after it hits the screen on the other side of the slit(s). Freeman Dyson (who collaborated with Feynman) argues that QM cannot describe the past but only the future. So prior to the electron hitting the screen, QM describes the probabilities of where it will go, which is mathematically dependent on it being able to go everywhere at once. If there are 2 slits then this means it can go through both and if there is only one slit then it can only go through one. So the observation made retrospectively confirms this.

Addendum: Sabine Hossenfelder gives a much more erudite exposition in this video. And I agree with her - it's the closest we have to a 'theory of everything'.