Of All Paths
It is fascinating to watch ants scuttling around searching for food, be it breadcrumbs, spilled sugar granules or a dead insect. In experiments it has been found that foraging ants take the path of shortest time to get to their food. To achieve this, when they encounter a boundary transitioning from a smooth surface to a rough one that slow their walking, they take a bent path with an angular shift. Why? Because it reduces the time spent on the rough surface, minimizing the total time taken to get to the meal. But how do they discover this optimal route?
Ants are intelligent indeed, but surely not enough to solve sophisticated optimization problems. While this article is not about ant biology or intelligence, we will soon see that Nature astonishes us with recurring themes across diverse realms, revealing a deeper underlying unity. Let’s leave the living world for a moment and come back to it again later.
Infinite paths
Like the ants, light also does something similar when it passes from one medium to another. It bends when it encounters a boundary between two different media, such as air and glass, or air and water. This happens because light gets slowed down when it enters a denser medium. The bending of light therefore effectively reduces the time spent in the medium where its speed gets reduced. The total time taken to get across therefore is minimized.
This property of light bending called refraction is what enables lenses in our eyes and phone cameras to focus images. It is also responsible for the beautiful phenomenon of a rainbow when sunlight passes through raindrops in the sky. In this case, light of different colors or wavelengths bend by different amounts because they travel at different speeds in water. The bending of light that minimizes the time taken to reach a point in one medium (say air) to a point in another medium (say glass) is called Fermat’s principle of least time. Out of the infinitely possible paths, why does light favor certain ones? Fermat’s principle hints at a special case of a grand unifying principle in all of physics. It’s called the principle of least action which governs the workings of the microscopic to the macroscopic.
A number for every path
Throw a ball in the air, it climbs to a maximum height before falling back to the ground. The path taken by the ball is a parabola which is a shape that looks like the letter U inverted. Why does the ball follow this specific shape and not some other random one? There is a quantity that can be calculated for every possible path the ball can take. It is calculated by taking the difference between the kinetic and potential energy of the ball at each position and summing over the entire duration of the path. As you might have learnt in school, kinetic energy is the energy of motion and potential energy is the energy associated with the height of an object under Earth’s gravity. The summed-up value or number thus obtained is called the action. When this sum is done for every possible path the ball can take, it turns out that the parabolic path is the one that produces the smallest value.
We can roughly understand this if we carefully observe the motion of the ball. When the ball leaves our hand with a certain speed, it quickly climbs to a maximum height. Its speed is reduced thereby reducing its kinetic energy while the potential energy increases since the ball is now at a greater height. This makes the action smaller since a bigger number (the potential energy) is subtracted from the kinetic energy. Similarly, during the downward motion, as the ball starts slow, more time is spent higher keeping the potential energy greater. When the correct mathematical calculations are done, the parabolic path comes out as the one that minimizes the action.
Note that in our example the ball was under the force of gravity that gave us the parabolic path that minimizes action. What about objects that are not under any forces like gravity? For such free objects the path that minimizes the action is a straight line. The general principle is that any physical system that starts from an initial state and ends in a final state evolves in such a way that minimizes the action. This is called the principle of least action. In our example, the initial and final states are the starting and the ending positions of the ball respectively. Out of the infinite conceivable paths the ball could have taken between the two states all have a higher value for the action compared to the natural parabolic path.
Path of Light
The action principle extends to physical fields as well. A field can be thought of as something that is spread out in space. For example, temperature in a room can be thought of as a field as it can be assigned a value at every point. Another familiar example is Earth’s magnetic field which can be detected by a magnetic compass which aligns itself to it. Although we cannot talk about paths for fields that is spread out in space, the principle of least action still holds.
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Unlike definite paths of objects, it is the field configuration whose evolution minimizes the action. Here configuration means the values of the field at every point in space at a given time. How the field values evolve over time is determined by the principle of least action. For example, the natural evolution of the electromagnetic field is governed by the action principle which in turn determines how light propagates through space or in any medium. Thus, light bending or refraction like in an air/glass boundary (like in a lens) is a consequence of the principle of least action. Similarly, reflection of light in a mirror follows the path of least time leading to minimum action. This is the path in which when light approaches a mirror at an angle, it reflects off the mirror at the same angle.
Gravity’s path
We know that planets like Earth orbit the Sun in elliptical paths. Newton described this motion with forces acting between the Sun and the planets. Later Einstein refined this understanding with his theory of Gravity in which there were no forces involved. In this picture, the Sun curves the space around it and planets follow natural paths in this curved space leading to elliptical orbits. How does the action principle enter here?
Einstein replaced Newton’s force with a field like the electromagnetic field for light. But now space (and time) itself is the field. Einstein’s ‘field equations’ then describes how the geometry of space is affected by the presence of massive objects like the Sun. Remarkably, out of all possibilities, nature chooses a specific geometry that is again governed by the action principle. This means the specific way in which space curves around the Sun is the one that satisfies the action principle. Planets then follow the natural elliptical orbits in this curved space.
Leaving the planets aside, when we zoom in to the subatomic world, we again see the same recurring theme but with a strange twist.
Every possible path
At the fundamental level, subatomic particles like electrons do not have a definite path as they are governed by Quantum Mechanics. For example, let’s say an electron travels between two points in space. What happens to the electron between these two points is not well defined. A definite path connecting these two points does not exist (like the parabolic path of a ball). Does this mean the action principle is not applicable in the subatomic world? On the contrary, it is much more fundamental than in the macroscopic world.
According to one of the successful interpretations of Quantum Mechanics by Richard Feynman, an electron traveling from one point to another does not take a single path. All possible paths contribute to the final outcome. The electron behaves as if it explores all possible paths simultaneously to end up in that final location! This could be a straight, curved, zigzag, or even a path that goes back and forth between the two points. That is strange indeed. But what happens then to the action principle?
Like the value for action for every path of a ball we saw earlier, there is an associated value for every possible path for the electron. The main difference is that these values for the different paths are added together to determine the final outcome. Remarkably, of the infinite possible paths of the electron between two points, a special path emerges as predicted by the action principle!
As the contributions from different paths are added together, those paths very close to the special path reinforce each other producing a large contribution, while most of the other deviant paths cancel each other. This is much like how ripples in a pond overlap resulting in either reinforcement or cancellation leading to either a large or small wave or ripple. The final outcome that the electron is found in a specific location therefore is mainly influenced by that special path predicted by the action principle. We again find that the action principle survives even in the microscopic world in a kind of bizarre way.
The lifesaving path
The American physicist Richard Feynman is well known for his ability to explain complex concepts in simple terms. Feynman, in his famous lectures on physics, starts his explanation of Fermat’s principle of least time with the ‘Damsel in distress’ story. Maybe he felt that it was the most efficient way to grab the attention of a young lecture audience. Thus, the story goes: a pretty girl falls from her capsized boat into the water and screams for help. A lifeguard watching this from the shore had to save her from drowning. Let the location of the lifeguard and girl be A and B respectively. Let’s say running on the sand is faster than swimming in the water. So which path should the lifeguard take to reach the girl in the shortest time and save her?
One possibility is to go straight from point A to B. But note that this might involve swimming a longer distance in water. Maybe the lifeguard must take a longer path in the sand to reduce the time spent swimming in water. It turns out that the path that minimizes the time taken is the one that approaches the sand/water boundary at a specific angle. Surprise! This describes Snell’s law, which mathematically relates the refractive indices of two mediums to the angles at which light approaches and bends at their boundary. Refractive index of a medium determines how much light is slowed in it, just like in the analogy how the lifeguard’s speed is reduced swimming in water.
Coming back to the experiment on foraging ants in the beginning of the article. Do ants really optimize paths to their food? One possible explanation is that some ants may have wandered taking a path of least time to the food leaving a chemical or pheromone trail. Compared to other paths, this path gets reinforced by other ants following the same chemical trail. Therefore, it appears as if ants have discovered Fermat’s principle of least time! Strangely, like the foraging ants, an electron seems to sense all the potential paths to find the one that minimizes the action. Even more mind-bending is that it explores every single one of those paths at the same time!
It is fascinating that the same principle governs phenomena across vastly different scales. It seems as though the Universe is revealing a deeper unity beneath the apparent diversity of nature, like an intricate tapestry woven with recurring patterns. Recognizing these patterns leaves us with a profound sense of wonder and awe.
