About the Feynman Path Integral and Principle of Least Time

[EsbMtrx]

I don't entirely get why we usually say that only the shortest path contributes in the path integral. If you calculate the volume of nth fresnel zones which is the locus where the path length is between n-1 and n wavelengths from the shortest path in 3 dimensions, they are the same I believe. So I figured each path length is contributed by the same number of paths and the shortest one wouldn't dominate. My question pertains to Fermat's principle.

ps: this is my first thread here so I apologize in advance if I did anything wrong

 

[stevendaryl]

It is not correct that only the shortest path contributes in the path integral. All paths contribute. However, you can use the method of stationary phase to approximate the path integral, and then you find that the largest contribution is from the paths that satisfy the classical equations of motion.