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I've been reading the book "Geometric Mechancis" by Darryl Holm and the in the first chapter he treats geometric optics. There the author talks about light rays and those light rays looks like trajectories as of particles as we consider in Classical Mechanics. The first thing that the author state is Fermat's principle that seems to define one action and then determine the path that light follows (i.e. the light ray) being the one which extremizes the action.

In all of that discussion, it seemed to me that geometric optics is then all about treating light not as a wave, but rather as a collection of particles. Is that it? In geometric optics we should think of light as a collection of particles?

In that setting, a light ray is just the path followed by one such particle, or is it composed by many particles?

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We usually say that in geometric optics we use the "ray model", which is neither the particle model nor the wave model. Given that particles (subject to no forces...) travel in straight lines and that "light rays" go in straight lines when they are moving through a uniform medium I suppose one could say that the light rays look like particle trajectories. Fermat's principle is a "principle of least time". It says that the path taken by light from point A to point B will be the one which minimizes the transit time. This is analogous to the variational principles that we use in classical mechanics. But I think to say that we are "thinking of light as particles" when we do geometric optics is carrying the analogy past the point where it is useful.

We can, alternatively, formulate geometric optics from the Huygens-Fresnel principle. The underlying ideas of this principle are all to do with waves. So you could say that geometric optics is "agnostic" about whether light is a particle or a wave.

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  • $\begingroup$ I don't know how anybody can think light doesn't consist of waves myself. I've leaned over the rail of a ship watching ocean swell waves. Each is a thing in its own right, a "particle" as it were. But's it's a wave too. $\endgroup$ – John Duffield Aug 10 '15 at 21:10
  • $\begingroup$ To say that "light shows wavelike properties" is true. It interferes, just like the water waves you see over the rail of a boat. But the ocean swell waves are not particles. You can't give a single position for an ocean swell - it is spread out over a large area. But you can persuade light to act like a particle and reveal itself as having a position (such as a flash produced on a phosphor screen). Light is a quantum mechanical object which means you can make it show wavelike properties (with some experiments) and you can make it show particle like properties (with other experiments). $\endgroup$ – gleedadswell Aug 10 '15 at 22:58
  • $\begingroup$ Light has an E=hf wave nature. The fact that I can perform an optical Fourier transform (see Steven Lehar's web page) with a lens doesn't mean it doesn't have a wave nature. $\endgroup$ – John Duffield Aug 11 '15 at 7:18
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    $\begingroup$ All points noted gleedadswell. But I must disagree that E=hf is part of the particle nature of light. Can I draw your attention to arxiv.org/abs/0803.2596 and to weak measurement work by Aephraim Steinberg et al. IMHO when you detect a photon at one slit you perform something akin to an optical Fourier transform on it. So it's converted into something pointlike and goes through that slit only. Then when you detect it at the screen you again perform something akin to an optical Fourier transform on it. No magic or multiverse is required. $\endgroup$ – John Duffield Aug 13 '15 at 7:00
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    $\begingroup$ Noted re the arXiv paper, gleedadswell. See Jeff Lundeen et al for more on weak measurement. Meanwhile I must insist that E=hf is not at odds with the idea of a wave. Nor is E=hλ/c. A photon has a frequency and wavelength because it has a wave nature. As for quantization, do remember that the dimensionality of action h can be expressed as momentum x distance. Take a look at some depictions of the electromagnetic spectrum. What's always the same? $\endgroup$ – John Duffield Aug 14 '15 at 12:37
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The equations of geometrical optics (also called Hamiltonian optics) are completely equivalent to the equations of Hamiltonian mechanics for a single, newtonian, non-relativistic particle . In that sense you are correct; geometrical optics is a single particle theory, not a theory of a collection of particles. The light ray of geometrical optics is the path of a single "classical" light particle. This can also be seen from the fact, that the "rays" of geometrical optics never split (in contradiction to experiment, which shows us, that light rays do split at the border of a media).

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