# Photons always take the quickest route

• Green dwarf
In summary: The second question is about the behaviour of photons in general in a classical situation and that's just not something you can answer.

#### Green dwarf

My understanding is that the path of a photon between any two points A and B can be worked out by finding the route which will get it there in the least time. This seems to be true both when a photon passes near a heavy mass and deviates because of gravity (as predicted by general relativity) and when it travels through different media and is refracted (as predicted by classical optics). I don’t know why this would be the case, but, intuitively, it seems like it should be something fundamental about the nature of space-time. Is it? Or is it just a coincidence? Or am I just wrong?

Green dwarf said:
My understanding is that the path of a photon between any two points A and B can be worked out by finding the route which will get it there in the least time.
Note that you can have multiple possible photon paths between two points. Not all of them are the quickest possible route.

Be careful not to think of your photon as a little bullet. It can be considered as existing Everywhere between the emitter and the receiver. You cannot think classically here.

Thanks A.T. Could you give me an example? I maybe should have said that the photon travels directly from A to B without being reflected anywhere.

Photons traveling from a point at infinity to the focal point on the far side of a lens can take many routes through the lens, but I think they would all take equal time. The closer to the edge of the lens they pass, the further they have to go, but the less time they spend moving slowly in the glass. No photon would travel between the two points via any route that would take longer than this minimum time.

My understanding of a geodesic along which a photon would travel is that it is the shortest distance between A and B (i.e. the one that takes the photon the least time).

I'm not at all sure that what I'm saying is correct. If I'm wrong, could you show me where in such a way that I can see my error?

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Thank you sophiecentaur. If the photon does exist everywhere between the emitter and the receiver, am I still right in saying that it does take a particular route in the sense that if you placed an obstacle anywhere on that route, it would stop the photon, while an obstacle placed anywhere else wouldn't?

Green dwarf said:
Thank you sophiecentaur. If the photon does exist everywhere between the emitter and the receiver, am I still right in saying that it does take a particular route in the sense that if you placed an obstacle anywhere on that route, it would stop the photon, while an obstacle placed anywhere else wouldn't?
All you can say is that the obstacle will alter the probability of the photon reaching a destination that's in the direction of a classical 'ray' of light. The wave nature of the light will completely spoil your simple model of a photon's path being blocked or not. Once you get multiple objects put in the way (e.g. a grid), you can end up with many photons arriving at points 'behind' the bars of your grid. Light (all EM waves) follows the rules of diffraction, whether you choose to regard it as photons or waves.

1oldman2
Sophiecentaur, could we define a curve from A to B along which the placement of obstacles would have the greatest impact on the probability that the photon will make it to point B? Could we then call this the 'route' that the photon takes?

Green dwarf said:
Sophiecentaur, could we define a curve from A to B along which the placement of obstacles would have the greatest impact on the probability that the photon will make it to point B? Could we then call this the 'route' that the photon takes?
In no way. If you put some sort of detector in various places across the path, it may or may not interact with the light. Its very presence has altered the experiment and when (and if) it detects a photon, it will have collapsed the wave function . Your "route" is a multiple one and cannot be described by a single line.
Of course, in the case of a large hole in a plate, most of the photons can be considered as going 'straight through' it in a classical way but that 'bulet model' falls down at the edges of any hole (or blocking object) and the wave will be more spread out.
You really can't make up your own rules for QM and the behaviour of EM waves. You either have to take it all on board or stay in ignorance of what happens in non classical situations.
Your first question, above is all about the 'rules' of diffraction and that involves the interaction across the whole of any wave front (from -∞ to +∞.

Green dwarf said:
it seems like it should be something fundamental about the nature of space-time. Is it? Or is it just a coincidence? Or am I just wrong?
No coincidence. It turns out to be the result of something fundamental about quantum mechanics. Feynman explains it in lecture 2 of this series...

If it's not making sense, and you need some more background go back and watch #1 first.

Green dwarf said:
My understanding is that the path of a photon between any two points A and B can be worked out by finding the route which will get it there in the least time. This seems to be true both when a photon passes near a heavy mass and deviates because of gravity (as predicted by general relativity) and when it travels through different media and is refracted (as predicted by classical optics). I don’t know why this would be the case, but, intuitively, it seems like it should be something fundamental about the nature of space-time. Is it? Or is it just a coincidence? Or am I just wrong?

You are probably thinking of Fermat's Principle, but it applies to rays of light. It does not apply to photons (which have neither paths nor positions, and for which there is no precise definition of "get there in the least time"). Thus, your premise is mistaken.

It is possible to derive Fermat's Principle for a ray of light from quantum electrodynamics, which is the theory that (among other things) predicts and describes photons. The Feynman stuff linked by MrSpeedyBob will give you a sense of how that might work. However, those shouldn't be understood as Fermat's Principle telling us something fundamental about the nature of spacetime. It's the other way around - we've learned enough to explain why geometrical optics works as well as it does.

A.T. said:
Just pick two points on a photon orbit:
https://en.wikipedia.org/wiki/Photon_sphere
You seem to be introducing black holes very early for someone who is trying to understand photons. ?
Also, the effect of gravity on EM waves is a bit different from the Electromagnetic interactions between systems of electrical charges and Em waves.

sophiecentaur said:
You seem to be introducing black holes very early for someone who is trying to understand photons.
The OP explicitly asks about light bending in curved space-time and General Relativity.

A.T. said:
The OP explicitly asks about light bending in curved space-time and General Relativity.
Yes but he doesn't appear to have got far enough with the basics of photons to cope with that. Whilst he still appears to be using little bullets, I think that should be sorted out first.

Green dwarf said:
Thanks A.T. Could you give me an example? I maybe should have said that the photon travels directly from A to B without being reflected anywhere.

Hi - think about gravitational lensing around a galaxy as an example.
- Pick a point A in a distance galaxy behind that "lensing galaxy"
- Point B can be your eye

There will be oodles (that's a tech term ) of paths that take photons on different paths (of different lengths) between points A and B. One of those paths by definition will be the shortest between points A and B, but photons will travel all of them depending on their initial trajectory.

sophiecentaur said:
Yes but he doesn't appear to have got far enough with the basics of photons to cope with that. Whilst he still appears to be using little bullets, I think that should be sorted out first.
And I think this is missing the point of his question.

Thank you to everyone for the responses. I looked up Fermat's Principle (that is sort of what I had heard of with regard to optics, though I didn't know its name) and am part way through watching the Feynman lectures.

I am just a high-school maths teacher with an interest in astronomy (including relativity and quantum mechanics, but with little real understanding of either).

I will post again when I have finished the lectures and had some time to think about what people have said.

A.T. said:
And I think this is missing the point of his question.
Maybe. But I think the idea of describing the "path" of a photon needs clearing up long before the relativistic effects.
If you look at that "Photon Sphere" article you will find a very large caveat so it may not be the best thing for the OP to be getting a lot of information from.

## 1. What is meant by "photons always take the quickest route"?

This statement refers to the principle of least action, which states that photons (and all particles) will always follow the path of least resistance or shortest time in their motion from one point to another. This is a fundamental principle of physics that explains the behavior of light and other particles.

## 2. How does this principle apply to the behavior of light?

Light travels at a constant speed in a vacuum, but when it encounters a medium such as air or water, it can slow down or change direction. According to the principle of least action, light will always take the path that minimizes the time it takes to travel from one point to another. This explains why light appears to bend when passing through different materials.

## 3. Does this mean light always travels in a straight line?

No, the principle of least action does not necessarily mean that light will always travel in a straight line. Light can also bend or refract when passing through materials with varying densities. However, the path it takes will always be the one that minimizes the time it takes to travel.

## 4. Are there any exceptions to this principle?

In general, the principle of least action holds true for all particles, including photons. However, in certain situations, such as when light is passing through a complex medium with multiple interfaces, the path it takes may not be the most direct one due to interference effects. Additionally, in the realm of quantum mechanics, the behavior of particles can sometimes deviate from classical principles like the principle of least action.

## 5. How does this principle impact our understanding of the universe?

The principle of least action is a fundamental principle of physics that helps us explain the behavior of particles, including light, in the universe. It has been used to develop theories and models that accurately describe and predict the behavior of particles in various scenarios, from the motion of planets to the behavior of light in optical systems. This principle is crucial in our understanding of the fundamental laws that govern the universe.