Why do the path of light is such that the time travel is minimum?

For most people who study differential calculus, or the first two years of college physics had to solve certain problems that requires them to minimize certain quantities. One such example is to the minimization of the path traveled for light between two points in space. For people who study the calculus of variabtion, and advanced physics, you can obtain the function y(t), given two points in configuration space, and the right lagrange for the problem( associated with the physical system), and obtain the action by integrating the lagrange, and find the stationary points for the action. I suspect it is a mystery to most people why the path of light is such that it minimize the time. It so happens that in quantum electrodynamic(or QED), avoids the problem. According to:
Amazon.com: QED: The Strange Theory of Light and Matter (Princeton Science Library) (9780691125756): Richard P. Feynman, A. Zee: Books.


Given source S( Where light emits), and the distination D( where light reach). The basic question is how does light go from S to D?

These are the steps for light to go from S to D according to QED:

1) Break the event for light from S to D into a finite( or perhaps infinite) many independent events( fancy word is "history"). An event is independent if the occurence of one event does not effect the occurence of others events.

2). Each event is associated with a vector.

3) Each vector has a magnitude, and direction.

3.1) the magnitude is determined by the medium in which light travels.

3.2) The direction is determined as a function of time of traveled in each independent event( or history).

4) The vector from S to D is determined by "vector multiplication" between all the assicated vectors in all the independent events.

4.1) vector multiplication are operations involving adding angles, and scaling the magnitude.


5) The square of the vector between S to D is the probability of a proton going from S to D.




The basic idea is that for light to go from S to D, we can imagine infinite many path( call each path an independent event, some call it "history"). Each path is assicated with a vector. Each vector is determined by the total traveling time, and he medium of travel. By some "rule" of adding these paths( or vectors), we can come to a "final" vector between S to D.
The magnitude of this final vector square is the probability of a proton going from S to D.