In 1662 Fermat amended Hero’s law to be the path of least time, as when light enters water where it travels slower, it refracts to take the fastest path not the shortest path. In Figure 3.15, light takes the path that “bends” as it enters water not the shortest path.
To understand this, imagine the photon is a lifeguard trying to save a drowning swimmer as quickly as possible. The fastest path to the swimmer isn’t the dotted straight line but the solid bent line because lifeguards can run faster than they swim, so it is faster to run further down the beach then swim a shorter distance. The dotted line is the shortest path but the solid line is the fastest and that is the path light takes (Figure 3.15). But again, how does a photon of light know in advance to take this faster path?
In 1752, Maupertuis generalized further that:
“The quantity of action necessary to cause any change in Nature always is the smallest possible”.
Euler, Leibnitz, Lagrange, Hamilton and others then developed the mathematics of this law of least action, that nature always does the least work, sparking a furious theoretical debate on whether we live in “the best of all possible worlds”. Despite Voltaire’s ridicule, how light always finds the fastest path remains a mystery today.
For example, light bouncing off the mirror in Figure 3.16 could take any of the dotted paths shown but by the principles of optics, it always takes the solid line fastest path. But as the photon moves forward in time to trace out a complex path, how does it at each stage pick out the fastest route? As Feynman says:
“Does it ‘smell’ the neighboring paths to find out if they have more action?” (Feynman et al., 1977) p19-9
To say that a photon chooses a path so that the final action is less is to get causality backwards. That a photon, the simplest of all things, with no known internal mechanisms, always takes the fastest route to any destination, for any media combination, any path complexity, any number of alternate paths and inclusive of relativity, is nothing short of miraculous.