Is ACTION The Most Fundamental Property in Physics?(Script) (Patreon)

It’s about time we discussed an obscure concept in physics that may be more fundamental than energy and entropy and perhaps time itself. That’s right - the time has come for Action.

In order to wrench the laws of physics from the reluctant clutches of nature, we need to watch for clues - patterns in its behavior that can reveal deeper truths.  Here’s an example: around 40 AD, Heron of Alexandria noticed that when light moves between two points, out of all the possible paths between them, it follows the shortest one. He proposed what you might call a Principle of Least Distance to define the path of light. Of course we now know that light doesn’t always travel in a straight line - like when it’s refracted by glass or traveling through a gravitational field. It took a millennium and a half following Heron for Pierre de Fermat to propose a solution for the case of refraction: what if light didn’t travel the path of least distance, but rather of least time?. If light changes speed between media, then the fastest trajectory is no longer a straight line. With this principle of least time, Fermat was able to explain all of the field of optics.

It seems as if we’d stumbled on a powerful guiding pattern in nature - minimizing principles. What if there was a similar property that could be minimized to determine the trajectories of matter as well as light? There were some efforts - for example Euler considered properties involving momentum and kinetic energy - but to no avail. In the meantime Isaac Newton came up with his laws of motion which gave us the instant and miraculous power to explain the motion of all particles in the universe. To do this, all you needed was to know the exact vector forces on each body at each time, which lets you calculate its exact vector positions, velocities, and accelerations at each time. That’s doable for, say, a ball flying through the air, but it gets frankly pretty hellish for complicated systems. So maybe we were still in the market for a simple guiding principle after all.

It was Joseph-Louis Lagrange who found that principle. Consider again the ball flying through the air. We can describe that motion without the painful vectors if we use energy. The ball starts out moving fast - it has a lot of kinetic energy, which it trades for potential energy as it rises, and then back to kinetic as it falls. If you need a refresher on this energy stuff, check out our previous video when you’re done here. You can figure out a lot using just energy - like the maximum height the ball will reach. But because energy doesn’t include directional information - vectors - it seems you shouldn’t be able to determine the exact path. Except you can.

Lagrange realized that a particular combination of kinetic and potential energy had the same minimizing property as does time for the path of light. He found that the ball will always follow the path that minimizes the time-averaged difference between these energies. In other words, add up the difference between kinetic and potential energy at all time steps during the flight and you get a number. Then try the same for any other path you can imagine. The result will come out higher for every path other than the path the ball actually travels. Lagrange called this new quantity the Action. Formally, it’s the integral over time of the kinetic minus potential energies. And that difference in energies itself got the name Lagrangian, which itself turns out to be an enormously important and useful quantity.

Lagrange proposed that everything in Nature happens in such a way that Action is always minimized. And so was discovered the Principle of Least Action. A century later William Hamilton made the important correction that Action is always minimized OR maximized. Here’s an imaginary graph of how the action can change between paths - you get paths where it’s at its lowest and at its highest. Moving objects tend to choose paths where the variation of the action between nearby paths doesn’t change much - in other words, where the slope of this curve is zero, and that’s true at both the minima and maxima. Most of the time it’s the minimum, which is why most people still call this the Principle of Least Action, but others call it the Stationary Action Principle.

If the principle of Least Action is true, then it should be possible to figure out the path that any object will take between two points as long as you can find a way to determine which has the minimum action. That sounds hard, but thankfully Lagrange and Euler already did most of the work for us by finding the Euler-Lagrange equations, allowing both Physics Students and Nature to minimize their action.

The Principle of Least Action in combination with the Euler-Lagrange equations leads to Lagrangian mechanics. When you study Newton’s mechanics and then move on to Lagrangian mechanics, it seems almost like a miracle. The hellishly complex equations needed to solve problems using pure Newton seem to just melt away into much simpler forms. This simplicity comes from the fact that with Lagrangian mechanics we can dispense with forces and vectors in general, and only consider the energies.

The Principle of Least Action is so powerful that it really seems like it must be telling us something deep about the universe. But what? It’s hard to interpret what this Action property really signifies. Well, if it really is so fundamental then it should apply well beyond classical mechanics and into our modern theories - in particular general relativity and quantum mechanics. And perhaps in these we’ll find clues as to what the Action is really about.

Let's start with Einstein’s General Theory of Relativity, which, as we’ve discussed before, describes the force of gravity in terms of the bending of the fabric of space and time. One of Einstein’s motivations in developing general relativity was the fact that Newtonian mechanics failed to correctly predict the orbit of Mercury. If you calculate that orbit using the Einstein equations in their full glory and full gory detail, you get the right answer. But you don’t even need to go that far. If the Principle of Least Action is really a fundamental law, then there should be a relativistic version of the Action that we can minimize to find Mercury’s orbit. After all, planets trade between potential and kinetic energy as they move around their orbits, just like the ball.

It turns out that we only need to make a couple of relativistic corrections to do this. The most important is writing the Lagrangian as a function of something called proper time. In relativity, clocks run at different rates depending on things like relative speed and position in a gravitational field. Proper time is the time that an object will perceive in its own reference frame. If you plug the relativistic Lagrangian into the Euler-Lagrange equation, you get equations of motion correctly describing the orbit of Mercury around the Sun.

This shows us that the Principle of Least Action holds up in GR - but it reveals a lot more. First, it means that Lagrangian mechanics based on this principle can be made to be “Lorentz invariant” - it gives equations of motion that are valid even in the shifting frames of Einstein’s relativity. And we also start to learn something about the nature of the Action. Once we apply the relativistic corrections, the Action simplifies in to just the integral of proper time. Let's take a moment to think about that. Before Action was a somewhat vague notion of change in energy, but working with a more precise model of the universe we see Action is much easier to understand. Action is now  just how much time is perceived by an observer in their own frame of reference. All objects moving through space time move through paths that minimize the time measured on that path.

Remember that Fermat figured out that light always follows the path that minimizes its travel time - the principle of least time. Well it turns out that this is just a special case of a  Principle of Least Proper Time. The reason for this is that in the Classical limit proper time becomes kinetic minus potential energy for objects with mass, but for objects without mass, like light, proper time and time are the same.

OK, it seems like we’re getting somewhere. Let’s see if quantum mechanics will get us any deeper, or perhaps just make everything more confusing.

Let’s start with the famous double slit experiment, as we so often do when talking quantum mechanics. In it, a stream of quantum particles are launched at a barrier with two slits cut in it. When the stream reaches a detector screen on the other side the individual particles make individual spots, but over time the spots map out this series of bands - an interference pattern.  The quantum interpretation of this is that the particle travels between its source and the screen not as a particle with a well-defined trajectory, but as a quantum wavefunction that represents all possible paths it could take. The wavefunction at the screen represents the range of possible final locations for those paths, with some being more probable than others. The particle then appears to randomly select a position from that distribution.

But there seems to be a conflict here. The principle of least action says that a particle will always land where the action of the trajectory is at a minimum or maximum. That should correspond only to the central points of those bands. However we see particles landing at every location in between - albeit with more likelihood the closer you get to these action stationary points. So perhaps Action isn’t as fundamental as we thought it was. Or perhaps this is telling us more about what the Action really is.

Paul Dirac very nearly figured this out. He realized that a quantum analog of the action existed that was related to the integrated time evolution of the wavefunction. And he realized that this quantity should result in destructive interference in most places - like the dark bands in the double slit experiment. It would lead to constructive interference only where this quantum action varied slowly - near it’s stationary points, just like with classical action.

But the real action hero of this story is Richard Feynman. Remember that the action tells us about a particle’s history along some hypothetical path. Feynman realized that the path of a quantum object could be determined by adding together all possible paths that particle could take weighted by this quantum action. In this case the quantum action doesn’t come from adding up kinetic minus potential energy nor the proper time. Instead, it effectively calculates the phase shift that a particle picks up along each path towards a destination. Then it adds the phases of all conceivable paths to that destination. We find that the most likely destinations are those where the phases line up - constructively interfere. This is Feynman’s path integral formulation of quantum mechanics, and it exactly reproduces the predictions of previous versions of quantum mechanics - for example, Schrodinger’s wave mechanics that talks about the evolution of a single wavefunction. This is analogous to how Lagrangian mechanics reproduced Newton’s mechanics. And it’s no coincidence, because the path integral is the quantum analog of the Principle of Least Action.

So what does this tell us about the Action? Well, the path integral tells us that a particular final destination is likely if the phases of the many paths that could lead to that destination line up. Away from those special points, very tiny changes in the path lead to rapid changes in the phase, causing those paths to cancel out. As Dirac started to guess, particles tend to end up near the stationary points of the quantum action. In the path integral, this happens for paths that take the shortest path … but not the shortest path through space, the shortest path through something called configuration space.

And this “configuration space” is what ties all of this together. Configuration space refers to the space of all possible trajectories given our object’s constraints. In the case of the ball, configuration space refers to all the places the ball could reach with its current Potential and Kinetic Energies. When we take this to Relativity we see the configuration space becomes configuration spacetime, where the shortest path minimizes proper time. In quantum mechanics configuration space could mean phase space - the space of possible positions and momenta - or it could be more general state space, representing all the quantum states that a system could evolve through.

The application of the quantum action principle to the evolution through quantum states underpins modern quantum theory. Back to Paul Dirac for a moment. His eponymous equation was derived to describe the quantum evolution of the electron. Well it turns out that this equation is just the Lagrangian for a spin-½ quantum field. Well, there’s a Lagrangian for each quantum field which describes how that field and its particles tend to evolve. Combining these gives us the Standard Model Lagrangian, which allows us to track the evolution of all quantum fields through configuration space, and so predict the behavior of all known particles.

Remember, this all started with Heron of Alexandria studying light two thousand years ago. He found a very simple and beautiful pattern in nature, this  has been inspiring us ever since. Through Fermat and Lagrange and Dirac and Feynman … Each seeking a path through the configuration space of ideas, guided by mysterious principles, not least of which is the Action - pointing the surest way to the fundamental nature of space time.