# Feynman physics on least time of light

What does light checks all paths mean by Feynman? Especially the statement is labeled by yellow. Why there is only one path that leads radiowaves to D’? And how wave check all paths, that is, why it can stop radiation checking paths by closing the slot down? I have another question, also labeled by yellow, I don’t understand what he means. How we can ensure that all lights go to the point P after passing through a piece of glass? Why the path to the point P is the least time?

• Taking the path that requires the least time is equivalent to minimizing a certain integral. This can be described in terms of the light wave solving an Euler-Lagrange equation. The solution changes, with some regions increasing in their resulting amplitude, if we narrow the slit.
– J.G.
Feb 27, 2023 at 16:49
• Thanks, I thought there was some more basic ways to explain it instead of integral because he just introduced basic knowledge of lights in this chapter. Feb 27, 2023 at 18:55
• Oh, the integrals aren't so bad. Here's a warm-up.
– J.G.
Feb 27, 2023 at 19:14
• Ok, I’ll go through it. Many thanks. Feb 28, 2023 at 1:31

There is something in physics called the action.

$$S = \int{dt L}$$

Where $$L$$ is what is called a lagrangian. The lagrangian is a tool that helps capture the physics of our system. The principle of least action says that the path a particle takes is the one that minimizes (or more properly, extremizes) this action. This boils down to saying that $$\delta S = 0$$. Each possible path light takes will have different values for the action.

When Feynman says light “checks the paths” I think he means this figuratively. He means that light will take the path that corresponds to the smallest value of the action. When you make the opening smaller, you have introduced new conditions in the system, and the resulting path that minimizes the action will be different to reflect these new conditions.

• Thanks for your comment. In this chapter, he just introduced some basic knowledge of lights, I thought there was some more basic way to explain these things, such as geometric ways. Feb 27, 2023 at 18:53

The speed of light is different in different materials. The number you divide by relative to light in a vacuum is called the index of refraction. What Feynman is saying is that all those paths have the same total time. This may seem counterintuitive, because they are different lengths. If you think about it, each path has a different set of speeds, the speed in air, the speed in glass, and then the speed in air again. Some paths spend more time in glass, others spend more time in air. So, even though the paths have different distances, they also have different average speeds. He is saying that the shape of glass that is needed is the one that makes all the times the same.

While people are talking about Euler Lagrange and least action and all that, and that is fine, you should be able to show that along each path the time is the same given the geometry of the lens and the refractive index. I think that is the "more basic way" you were looking for.

• Thanks for your moment. I want to know why lights know where they should go, that is, how they know the destination is point P instead of a point above or below P or anywhere else. How lights know they should choose the fastest path to the Point P. Feb 28, 2023 at 1:31
• The light does not "know" that it should choose the fastest path to P, that is what people here are talking about. What I (and Feynman) are saying about this particular example, though, is that if you take the shortest path property of light as a given, then when a diffracting lens bends all the light from a source at P to a target at P', this is because all the times are equal. Actually finding the paths given a lens shape is harder, and finding a lens shape given a desired set of points even harder. But in the picture, both the straight line and the Q-Q' paths should take the same time. Feb 28, 2023 at 3:11
• Ok, I think I start understanding. Could you please explain a bit the first question of light smelling all paths and so on. Thanks. Feb 28, 2023 at 3:46
• I don't know that I would say light "smells all the paths" personally. Feynman is talking about Fermat's principle and you can read about it many places, including here: en.wikipedia.org/wiki/Fermat's_principle That article links to Huygens's principle as well, which is also worth looking at. The math for such a path is found by Euler Lagrange and is basically a geodesic. As one answer notes, the so called "Lagrangian" extends this principle of light to mechanical systems, making it a very general principle of physics. That is a bit deeper than Feynman is talking about, though. Feb 28, 2023 at 4:15
• Ok, many thanks. Feb 28, 2023 at 5:26

If we have a light source at A, we know that light arriving at B will have gone in a straight line, the principle of least time tells us that. But what if we put a mirror underneath, what path will the light take now?

Light can now go from A to B in two ways, in a straight line and via reflection in the mirror, where the angle of incidence = angle of reflection. The principle of least time explains the straight line path, but what about the reflected path - do we need an additional law of nature? No, it turns out that the two allowed paths that light can take, have one property in common that only those two paths have, and all other hypothetical paths dont - the property that if you make a small change to the allowed path, i.e. consider a slightly distorted neighbouring path, this neighbouring path takes "about" the same time as the oringinal path (to second order to be precise.) This is the common principle the two paths obey, and is called the principle of stationary action. Note that this solves a mystery inherent in the original principle of least time - we know that light arriving at B without going via the mirror will have gone in a straight line - but how does light leaving the light source at A know that it will end up at B and is required to take the least time? It seems it can only know when arriving at B, and then it is to late to make any changes to its path to obey the principle. But with the stationary action principle, light now has a way of knowing already at A what to do: it can sort of "probe" the start of a path, and then check out a couple of neighbouring paths - if the neighbouring paths take very different times, light knows not to go down the original path, and looks for other paths. The principle brings back locality so to speak. In this model, when you make light go through a very narrow slit, then inside the slit light is still trying to probe what path to choose, but if the slit is too narrow there isnt room for neighbouring paths to check against, so the light could now choose a path that for example is bent compared to the direction heading into the slit, and light will spread out more.
In a later chapter, Feynman explains what light actually does - light in fact physically takes all imaginable paths between A and B, paths that go faster and slower than the speed of light, paths that makes loops, paths that get reflected at arbitrary points in the mirror - but all these crazy paths in the end cancel each other in a way, and so we dont see these paths.

• Thank you. That makes a lot sense. So if the slit is too narrow, then lights can’t check nearby path and light would thought the time heading straight equal to heading to some points above and below middle point? Hence some lights go straight, some go up and some go down, which causes lights spread out. Is this right? Mar 1, 2023 at 10:59
• Yes, if the slit is wider, when light checks if e.g. a bent path is OK to take, it will check the neighbouring paths but these will have veryt different times to the bent path, so light doesnt choose that, and only goes straight. When the slit is very narrow, there are no neighbouring paths that tells light not to go in a bent path (beacuse there is no room for those paths), so it spreads out/bends. Mar 1, 2023 at 20:28
• Ok, many thanks Mar 2, 2023 at 1:22