References: Tom Lancaster and Stephen J. Blundell, Quantum Field Theory for the Gifted Amateur, (Oxford University Press, 2014) – Problem 1.1
One of the guiding principles of quantum field theory is that a particle travelling between two points actually traverses all possible paths between these two points, although with varying probabilities for different paths. Although this idea is expressed mathematically using the calculus of variations, a simpler example of the same idea is that of Fermat’s principle of least time applied to the derivation of Snell’s law of refraction in optics.
The idea is that given that the speed of light in a medium with index of refraction is , if a light beam starts at a point in medium 1 and hits the interface between mediums 1 and 2 at an angle to the normal, and continues through into medium 2 at an angle to the normal eventually arriving at point , then these angles are such that the travel time from to is a minimum. There isn’t any particular reason why this assumption is made (apart from the the fact that it gives the right answer!).
To see how it works, suppose we orient the interface so that it lies in the plane, so that the normal to the interface is the axis. We’ll take the incident beam of light starting at point to lie in the plane, as does the refracted beam which travels from the interface to point . We’ll let be the difference in coordinate of the points and , and let be the difference in coordinate of the point where the beam hits the interface. Thus the difference in coordinate between and is . Similarly, let and be the differences in coordinates between the corresponding points. Finally, let be the distance from to , and the distance from to .
Then by Pythagoras
The total travel time of the light beam is
Since the points and are fixed, as is the location of the interface, the only thing we can vary is coordinate of the point where the light beam hits the interface, that is, . We can therefore minimize with respect to :
where the last line uses the trigonometric definition of the sine from the sides of the triangles.