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.
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.
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.