QR3.6.2 The Physical Law of Least Action

That light always finds the best path to any destination has puzzled thinkers for centuries. As Hero of Alexandria noted, light always takes the shortest path, so how does it find that path? It might seem obvious that it is a straight line but how, at each step, does a photon know what straight is?

Figure 3.15. Light refracts

In 1662 Fermat amended Hero’s law to be the path of least time as when light enters a medium like water where it travels slower, it refracts to take the fastest not the shortest path (Figure 3.15). Imagine the photon as a life guard trying to save a drowning swimmer as quickly as possible. Is the dotted straight line shown the quickest path to the swimmer? If the lifeguard runs faster than he or she swims, it is faster to run further down the beach then swim a shorter distance as shown by the solid line in Figure 3.15. The dotted line is the shortest path but the solid line is the fastest and that is the path light takes. Again, how does a photon of light know in advance to take this faster path?

In 1752, Maupertuis generalized even further that:

The quantity of action necessary to cause any change in Nature always is the smallest possible”.

Figure 3.16. Principle of least action in optics

This law of least action, that nature always does the least work, was developed mathematically by Euler, Leibnitz, Lagrange, Hamilton and others, sparking a furious philosophical 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, e.g. light bouncing off the mirror in Figure 3.16 could take any of the dotted paths shown but the principles of optics are that it always takes the solid line fastest path. 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.