Fermat's Principle Explained by Feynman

[dEdt]

In his QED lectures, Feynman demonstrates, in a way, how Fermat's principle follows from adding up the amplitudes for all possible paths, and then noting that removing the amplitude for a path near the path-of-least-time from the calculation will have a greater effect on the total amplitude than if some other path were removed. But how does this explain why we see light as coming from a direction corresponding to the path-of-least-time? If the situation were laser light reflecting off a mirror, the eye can't remove sections of the mirror, so how would the eye know that removing the amplitude for a path near the path-of-least-time will have a big effect on the total amplitude?

 

[Jacques_L]

In his QED lectures, Feynman demonstrates, in a way, how Fermat's principle follows from adding up the amplitudes for all possible paths, and then noting that removing the amplitude for a path near the path-of-least-time from the calculation will have a greater effect on the total amplitude than if some other path were removed. But how does this explain why we see light as coming from a direction corresponding to the path-of-least-time? If the situation were laser light reflecting off a mirror, the eye can't remove sections of the mirror, so how would the eye know that removing the amplitude for a path near the path-of-least-time will have a big effect on the total amplitude?

Augustin Fresnel (1788-1827) had already demonstrated that in the years 1815-1821.
And Louis Victor de Broglie has explained in 1924 why it is true also with fermions, like the electron : because the real path(s) is/are in phase accord with its/their close neighbours.
So Broglie gave the physical reason of the union between optics and mechanics, made mathematically by W. R. Hamilton in 1834 : the iso-action surfaces are also iso-phase.

I wonder whether Feynman gave or did not give the width of the Fermat spindle, depending on the wavelength and the distance between emittor and absorber. I bet he did not.

 

[ralqs]

Hmmm, sorry, that was pretty confusing. My understanding of quantum-anything is at a minimum :)

Would you be able to clarify that for someone as ignorant as I?

 

[Jacques_L]

Hmmm, sorry, that was pretty confusing. My understanding of quantum-anything is at a minimum :)

Would you be able to clarify that for someone as ignorant as I?

You do not need to know it is "quantic". You only need to know it is optic. In the example given above by dEdt, there are lots of photons in a bunch. Only the emission of photons in the laser depends on quantic phenomena, not the propagation at all. And in his eye, only the activation of the opsine by a photon is quantic again, with a yield of more or less 25 %. Up to the retina, the phenomenon of propagation remains in the field of optics.

And in physical optics, whose laws were given by Thomas Young and Augustin Fresnel, it was already known that no real optical path can be of null width. Remember diffraction. Remember the maximum theoritical angular discriminating power of a telescope for a given entry pupil, for a given wavelength of light.

And what is the difference between the macrophysics, the only one known by Fresnel in 1821, and microphysics ? In microphysics you are interested on the destiny of ONE photon, which has ONE absorber at the end. So you have both geometrical constraints : by the emitter, and by the absorber. Plus of course the constraints on frequency, phase and polarization ; of course nor Huyghens, nor Snellius, nor Descartes, nor Fermat, nor Young, nor Fresnel could know these last constraints, in their time.