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You're describing quantum mechanics, but the phenomena here are classical. There is no mystery with Fermat's law, even if you do not accept wave optics and want
by jules 9y ago
You're describing quantum mechanics, but the phenomena here are classical. There is no mystery with Fermat's law, even if you do not accept wave optics and want a theory of light particles. At each instant in time a light particle can determine how its trajectory will bend based on purely local information. The bending of light is determined by the local variation in the index of refraction. Snell's law is what you get if the index of refraction jumps discontinuously. This is analogous to how in classical mechanics the trajectory of a particle is determined by the local variation in the potential energy function (or more generally, the Lagrangian), and how a particle's trajectory has a kink if the potential energy has a discontinuous jump. The particle bounces back if the jump is too high, which corresponds to reflection of light.
- whatshisface 9y agoWell, here's why I wanted to bring up QM: the principle of least time is really most at home in geometric optics where beams of light are arrows and everything is steady-state. In that view, things are indeed nonlocal: I can alter the arrows drawn for a beam of light going from A to B by doing something over at point C. Wave optics would understand this as interference, but if we insist that we want to keep the arrows then we have to introduce the path integrals. Essentially, geometric optics lets us draw arrows and assert refraction, QM lets us draw arrows and derive refraction, but wave optics doesn't really involve arrows at all.
- jules 9y agoThings aren't nonlocal in geometrical optics though. Something happens at C at the instant in time when the particle is at C. You can start a ray at any point in any direction, and just follow a straight line until you hit something. Then you determine what happens next based on local information at that point, and you continue. Ordinary tennis balls have the same property that the path from A to B can be altered by putting a wall at C. By the way, wave optics is already QM in some sense. Maxwell's equations are the analogue of the Schrödinger equation for light. Geometrical optics is the analogue of classical mechanics.