The passage of light

Light travels in straight lines. We are taught this from a very young age – light will not wiggle its way through things, but travel in straight lines reflecting and refracting. Hopefully you like me find this curious; light is an electromagnetic wave for which the fundamental unit (we believe) is the photon which is of course a particle. There are plenty of other examples where particles are not bound to travel in straight lines so what is the inside scoop? Is it the wavelike behavior?

There is an overarching principle that light takes the shortest path possible from A to B. From the below diagram, a simple right angled triangle, you can appreciate there are in infinite number of paths I could draw to go from A to B. I could go from A to C to B, or I could totally ignore the lines – go from A to Amsterdam, enjoy a lovely coffee by the canal and return to B after a little airport shopping. My path is still A to B. Of course what we can say is that there is only one shortest path, which is the path from A to B labelled c – the hypotenuse of the triangle, calculable with some before Christ mathematics.

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To say that light takes the shortest path possible isn’t wrong – but it is the kind of thing often said by grown ups when they don’t fully understand something. Perhaps if you placed me at A and asked me to get to B as quickly as possible I would run in a straight across c to B, because I am a sentient being and it makes sense. But what if I were to be robbed of my senses? What if you were to place me in a densely overgrown forest with no clear sight of B, C or anything but shrubbery and thousands of different paths – then I would need to simply try out the different paths until I found B. So assuming light is not sentient, how does it know the shortest path from A to B? Does it send out scoutons (scout photons) to work out the path and report back to HQ before the troops ride on?

As much as I wish I could spend the rest of this post talking about the curious adventures of the scouton, the real truth lies by considering the wavelike quantum nature of the photons – i.e. light as an electromagnetic wave. If we are to accept that light travels at a constant speed, then it makes sense that it takes the least time for light to travel directly from A to B – this is a given by virtue of the fact it is the shortest length. If light were to take other paths we would expect it to take more time to reach B. The following diagram has been taken from the popular science book QED: The strange theory of light and matter by Richard Feynman.

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The first part of the diagram shows us an illustration of some paths light may take to get from one place to another, as discussed. It is not exhaustive, there are an infinite number of possibilities. The graph at the bottom left shows the various times light takes to travel those paths, which is in a U-shape with paths C,D and E the shortest. This is intuitive from looking at the diagram.

We begin by explaining the passage of light in simple terms, in a fashion similar to Feynman before we go on to explain the complexity that has allowed the simple explanation to be offered. The arrows beneath the time graph are imagined to be the whirling hand of a stopwatch, which stops when the photon lands at the source. If you want to imagine this stopwatch hand in real time you better let it whirl 30,000 times for every inch light travels! The direction of the arrow can be considered similar to the direction of travel when the light reaches the source. Now the arrows can be added together to form a path, not just here but anywhere in physics . When we add these arrows by butting the head of one by the tail of the other, it isn’t so important where each one takes us rather the overall direction we have traveled. As such when we finish adding arrows we just draw one big one from start to finish.  What we see is that some arrows cancel each other out, for example B and F are almost exact opposites of each other so their sum is nothing exciting at all. Similarly A and B are quite close to cancelling, so if you net these together you haven’t really gone anywhere at all. The arrows in the middle however do not cancel – these arrows are the more horizontal ones.

Now the above illustration is of course a small snapshot; we are considering many different possible paths so we will end up with many more arrows than this although we do not need any more to understand. The core idea here is that the when we say light travels in a straight line what we are really saying is the light would probably fail to make it from the source to the detector if we were to remove the horizontal arrows. In fact the length of the arrows is representative of the probability of occurrence, which as you can see is exactly the same! So we are not saying that the other paths do not occur, they are just as a likely to occur however they cancel with other paths. The ones we are left with, the ones that actually mean something are the straight line paths which are the ones which give us that age old adage – light travels in straight lines.

We could build a really complex path using lots of funny arrows inching us closer and closer which is nothing like a straight line at all! This path would be expensive though, all those probability arrows mean that the overall path is going to be highly unlikely to occur. If we want to start worrying about things like that then for goodness sake buy a lottery ticket.  Of course it is highly possible that one of the very improbable routes from A to B may be traversed by a photon at some point; indeed to should if you allow the clock to run for long enough but this is almost certain. But what’s a rouge photon among friends? A photon here or there isn’t enough to stimulate sight, but it is enough to change our belief that light travelling in straight lines is simply a general probabilistic expression of quantum electrodynamics rather than an absolute fact inherent in the properties of light.

So if you are still unclear, light is most likely to travel in a straight line path – which means over reasonable distances we can consider the straight line path. If you want to get really quantum reduce the distance to less than a stopwatch turn and watch things get weird. Another day.

Disclaimer: This is an optional end to the post. For the interested reader we elucidate the mathematical interpretation of the above explanation. This is done briefly for completeness and is unlikely to be of interest without a mathematical background. 

I couldn’t leave you without putting a little more flesh on the bones – spinning arrows and adding together seemingly random paths may seem unsatisfying to the curious or mathematical mind. What Feynman is getting at in his book is path integral formulation. In classical mechanics we can consider a unique path for a particle such as a cannonball flying from A to B or a person jumping from the ledge of a tall building trying to understand quantum mechanics. In quantum mechanics however, we have to use mathematics that allows us to compute the sum over all possible trajectories. This is powerful mathematics which evaded human comprehension for some time – understandably too for it is a fairly out there idea to consider all possible paths rather than just one based on initial condition. It was Feynman who first worked out this precise description – and for this he should be remembered.

It was found that the quantum action could often be considered as a number of discrete classical actions. This redefines the way we can look at such problems and greatly enhances our toolkit for solving them. The probability assigned to a particular event is given by the modulus of the probability amplitude, which is a complex number. The probability amplitude itself is given by adding together all the paths in the configuration space we consider. We can then compute the contribution of the path by considering the time integral of the Lagrangian over a particular path (the process involves considering an exponential function, but this will suffice here). For any given process, we add up all of the paths which then gives us a wonderful elaborate anything can happen picture. We find that the amplitudes assign equal weight (modulus) but different phases (arguments) to the paths which is what allows the path which differ considerably from classical paths to cancel in a very similar way to interference. This is analogous to what we considered in the diagram above.

 

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20 responses to “The passage of light

  1. Very good explanation Joseph. Its indeed quite difficult to explain the most basic principle of quantum theory (apart from the wave-particle duality), the Principle of Least Action so intuitively without being tempted to include the math. Many writers (like me) tend to slip to mathematics while doing so.

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    • Thank you very much! It is always a fine balance between including enough maths for the curious, but not putting off those who are not interested. I do try to leave some to allow the interested reader to go on and find out more. Thank you for reading!

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  2. Tricky stuff which seems to me to explain what we already knew by intuition. Since light is only a very narrow band of electromagnetic waves are we to assume they all can have a particle nature? Is it not true that light is bent by strong gravitational fields such as that of the sun? Does that mean all electromagnetic radiation would be bent on the same way? In the bending of light by gravity are we to conclude that photons have mass?

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    • Indeed it is interesting that the seemingly simple actually has a rater complex explanation which makes it simple again! I think the orientation of the sentence is wrong – light is comprised of photons, which are particles and behave (as do all quantum particles) with wave-like nature. I know it’s confusing because it is an electromagnetic wave, but the photoelectric effect showed light to most certainly be comprise of particles. Light continues to travel in straight lines when it goes around a massive object such as the sun – it’s just the geometry of the surface has change – straight lines on curved surfaces don’t look the same anymore! But yes it does mean other em radiation would bend in the same way in theory, there is nothing special about light except the human ability to detect it. Photons are however massless particles.

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  3. Very timely Joseph! Light is what defines fluorite. Because of the perfect symmetrical arrangement of its structure, the color that we see is light bouncing on the small inclusions trapped in the crystal. Lot of food for thought beside the shortest way from A to B when you consider refraction, reflexion, angles, intensity of light – and our perception of it. I’ll keep one of your “scouton” nearby for my next design!

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    • Hello! Very interesting indeed, the tricks we can play with light. Natural crystals truly are wonderful in the way in which they can form natural diffraction gratings. Light must be up there in the physical phenomena which is most taken for granted. Thank you very much for reading!

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      • No Joseph – light is not a trick, nor taken for granted. It is (and has been) the focus of pretty much all human endeavor that is consequential: scientific – engineering – artistic – even religion-wise
        In the absence of light, our world could not exist (think chlorophyll). Being able to explain it is something science has been working on for 5,000 years or more (think astronomy). Fascinating subject indeed – Thanks for bringing it up in your blog!

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      • Oh, by the way – thank you Joseph. Your post on light inspired 02-02- Fluorite #5. I turned off the polyhedra (the surfaces/vertices of the structure) to focus uniquely on atoms & light for this one. Interesting result!
        Keep up the good work, JC

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      • it will be…my son is so into the planets and the universe…one of the things that he wants to witness is the northern lights…he is still 5…he has so many questions about science etc that is sometimes hard to explain to a 5 year old…the other day, he said that maybe somebody can come up how to store energy on the tv, that way when there is no electricity, he can still watch it…

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      • i am trying my best…i am homeschooling him..but the problem is me…i guess he has more info about the world than i do..there will cime a time when i will be the one asking and he will be answering…thats why when i get across ur site, i got really interested…

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  4. I like the way you word things …. Especially the bits of light humor along the way …. “Scoutons” is good ….. I admire someone who can make up words when they are needed.
    I have always thought it would be fun to write a book or teach a class called “Physics without the Math”
    It seems that you could do that well.

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    • Thank you very much! That is great encouragement to have. I am hoping one day to write a book, it would be a most wonderful project to get into – although a very saturated crowded market for sure!

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  5. Let us never forget that gravity bends light. The light from distant stars as it traverses past our sun, Sol, bends ever so slightly. Light is bent and even sucked by black holes. And the claim that we now have proof that dark matters exist because of how light has been bent by it further proves the unreality that light always travels in a straight line.

    Here’s something you might find interesting. The speed of light is not an absolute constant. In a perfect vacuum, yes. But space is not a perfect vacuum, it’s filled millions of particles, including photons, for every cubic centimeter, even in intergalactic space.

    And one other thing that’s interesting, concerning the universe and dark matter. The concept and theory of dark matter was constructed to account for the lack matter in the universe. Apparently, scientists calculations of the known universe found that ninety-five percent of the matter was missing. So, out came the theory of invisible or dark matter, matter does not reflect light. And that’s good, because how could the Flash gain his super power without dark matter once that supercollider that Harrison Wells built blew up?

    Seriously, here’s the the thing, scientists in Great Britain last year announced, after a very thorough study of the universe found that the universe doesn’t have one hundred billion to two hundred billion galaxies but actually one trillion to two trillion galaxies, increasing the amount of matter in universe by almost one hundred percent. So, where’s the need for dark matter theory now?

    A friend of mine who’s a cosmologist told me that the number of galaxies is irrelevant, that we now have empirical proof that the dark matter exists. When I suggested that couldn’t the gravitational lensing that seems to prove dark matter is real actually be caused something else, he laughed at me. “Dark matter exists,” he said. “Get use to it.”

    Apparently, once we have a theory in place, it cannot be deleted. I wonder what future scientists, possibly even alien scientists, will say about our hubris in assuming what we want to believe is actually the truth, rather than we actually can prove.

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    • I have decided to do my next post on the so called bending of light – it is not so much that light does not travel in a straight line but rather the definition of a straight line in non-euclidean geometry. Indeed you are correct in your assertion on the speed of light – and infact the speed is the speed of light in a vacuum; if the speed of light was constant everywhere we would not see things like refraction of light at the surface of water.

      The need for dark matter is still very much present with the galaxies we know about that much I can assure you. Although perhaps that isn’t quite an honest representation of the truth. With all the known matter in the universe and the current laws of physics we have we cannot explain observation. A theoretical extra energy/matter would plug this gap – but is is equally possible the rules are just invalid past a certain domain.

      I must plead my ignorance here – I was not aware we had proven the existence of dark matter beyond reasonable doubt? Do you have a link? I thought it was more an implied existence

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