Path integral formulation of wave-optics

**gptejms** · Apr7-06, 04:09 AM

A few years back,I stumbled upon a nice idea which I am reporting below:-

Just as classical mechanics is the $LaTeX Code: h \\rightarrow 0$ limit of quantum mechanics(rather action >> $LaTeX Code: \\hbar$ ,from path integral formulation),so should it be possible to argue from a path integral approach, that ray optics is the $LaTeX Code: \\lambda\\rightarrow 0$ limit of wave optics.What would such a path integral be?Here we go:-

The optical action may be written as
$LaTeX Code: S=\\int \\frac{ds}{v(x,y,z)}=\\int dt,$
where the symbols are self-explanatory.So the prob. amplitude for light/sound(or photon/..) to go from point A to point B may be written as
$LaTeX Code: \\begin{equation} P=\\sum_{all paths}\\exp{\\iota S/T_0}, \\end{equation} $
where

is the time period.

As long as $LaTeX Code: S=\\int dt$ is not very large compared to

,all conceivable ray paths between any two fixed points are possible,so that there is an uncertainty in the ray path taken by light/sound in going from one point to the other.

Specially interesting is the case of a photon.For a photon
$LaTeX Code: \\begin{equation} \\frac{1}{T_0}=\\nu=\\frac{E}{h}. \\end{equation} $
Hence for a photon,prob. amplitude to go from one point to the other is given as
$LaTeX Code: \\begin{equation} P=\\sum_{all paths}\\exp{\\iota ES/h}, \\end{equation} $
i.e.
$LaTeX Code: \\begin{equation} P=\\sum_{all paths}\\exp{\\frac{\\iota E\\int dt}{h}} \\end{equation} $

P.S. I am not able to see the latex graphics that I've generated in my browser--hope others are able to view it!

~~eljose~~ · Apr10-06, 05:59 AM

i think it should be correct..but the action would come from the Eikonal equation:

$LaTeX Code: (\\nabla{S})^{2}=n^{2}$ where n is the refraction index as a function of x,y,z,t...then the path integral for optics would be:

$LaTeX Code: \\int{D[r,t]e^{iS(r,t)/\\hbar}$

**gptejms** · Apr12-06, 04:15 AM

The beauty about the form that I suggest is that for a photon using E=h\nu,we arrive naturally at the particle action(starting with the optical lagrangian/action).
My quick impression is that your S(phase I believe) is nothing but my S(modulo T_0)/T_0*2\pi---don't know where you get the \hbar from.

**gptejms** · Apr12-06, 06:15 AM

Eikonal equation should follow easily from my optical action S . $LaTeX Code: \\delta S=0$ i.e. Fermat's principle should easily lead to the eikonal equation.
The nice thing would be to show that the wave equation follows from this sum over paths approach.