“Exploring All Possible Paths” Leads to Fermat’s Principle of Least Time

April 1, 2018

1. In 1657 the French lawyer and mathematician Pierre De Fermat (behind Fermat’s Last Theorem) worked out that when light travels from one place to another, it always takes the path of least time. The path of a ray of light going from air to water is shown below.

  • There’s a formula called Snell’s law (shown in the figure) that correctly predicts the exact angle by which the light bends, depending on the materials it’s traveling through and the angle at which it hits the surface.
  • This observation of light taking the “time of least time” to reach a detector in the water by changing its path (called “refraction”) was explained by Fermat by taking into the account that light travels slower in water than in air.
  • But the question of WHY it does that (and how a photon would know there is an interface) has not been answered up to now. As we saw in the post, “Feynman’s Method of “A Particle Exploring All Possible Paths“, wave theory of light cannot explain it.
  • As we also saw in that post, Feynman came up with a technique called “a photon exploring all possible paths”, but admitted that he did not know WHY it worked. In future posts, we will show that it is due to the nonlocality of Nature and the instantaneous establishment of quantum fields for “all possible paths” for the photon.

2. In fact, this is exactly the same procedure followed by an lifeguard (instinctively) in reaching a drowning swimmer in water. The situation is shown in the figure below.

3. When we look at the above figure, at the first glance one may wonder whether a straight line (path A) is the fastest path. This is indeed the shortest one, but it isn’t the quickest. This is because one can run faster along the beach, and cover more distance on land than in water.

  • However, if one runs on path B, make the distance in water to be minimum, that is not the quickest either. This route is too long, and it slows you down.
  • The quickest path is C, a very specific path that lies somewhere between A and B, where the lifeguard jumps in at a distance x before the shortest path in water.
  • Of course, a lifeguard would not even think about all this. He/she would just instinctively choose a path that turns out to be close to this optimum path C.

4. I have not come across anyone doing an experiment on lifeguards, and seeing how close they get to the “optimum path”. But I came across a paper by a math professor who did an experiment on his dog fetching a ball thrown into Lake Michigan.

5. After collecting 35 data points (the x and y values in the figure above, in meters), Professor Pennings plotted them. Along with these data points, he also plotted the optimal trajectory predicted by the Snell’s law, shown by the straight line below (figure from the above paper).

  • Therefore, just like a photon “would know” how to take the “path of least time”, a dog would too!

6. Even more interestingly, even ants seem to be able to find the “optimum path” that takes least time to get to their food.

  • A group of researchers  used a glass surface and a rough green felt surface — analogous to air and water or sand and water in above cases– to separate a colony of ants and some food they placed some distance in to the rough green felt surface.
  • They found that the ant trails were far closer to the quickest path than to the direct path. Like light and lifeguards, these ants seemed to minimize time and not distance. The following figure showing the trail of the ants is from their paper: “Fermat’s Principle of Least Time Predicts Refraction of Ant Trails at Substrate Borders“.

Conclusion

The key  philosophical problem that existed for Newton to Fermat to Feynman with their “particle representation of light” was to explain “how a photon would know” in advance how to determine the path of least time; see, the book by Ivar Ekeland in the References.

  • But that problem goes away when we realize that a photon (or any particle) takes into account “all possible paths” instantaneously due to the nonlocality of Nature. That is the basis of our new interpretation of quantum mechanics. We will discuss this in detail in upcoming posts.
  • Even more interestingly, the observations of humans, dogs, and ants also taking the “path of least time” just instinctively, illustrate that this is how Nature works. Even living beings are guided by this “nonlocality of Nature”. This is a good example to show that there is so much that we DO NOT KNOW about how Nature works.
  • This is closely related to how kamma vipaka are AUTOMATICALLY executed by Nature. That will become more clear as we proceed.

Any questions on these QM posts can be discussed at the discussion forum: “Quantum Mechanics – A New Interpretation“.

References

I. Ekeland, “The Best of All Possible Worlds: Mathematics and Destiny” (University of Chicago Press, 2006).

R. P. Feynman, “QED: The Strange Theory of Light and Matter” (Princeton University Press, 1985).

J. Oettler et al., “Fermat’s Principle of Least Time Predicts Refraction of Ant Trails at Substrate Borders“, PLOS ONE, vol. 8, issue 3, e59739 (2013).

T. J. Pennings, “Do Dogs Know Calculus?”, The College Mathematics Journal, vol. 34, No. 3, pp. 178-182 (2003); link to pdf in #4 above.

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