Will you understand what I’m going to tell you?
…No, you’re not going to be able to understand it.
… I don’t understand it. Nobody does.
~Richard Feynman on the Path Integral

The “paradox” is only a conflict between reality
and your feeling of what reality “ought to be.”
~Richard Feynman, in his lectures on physics

Quantum mechanics is a very strange beast. Things tunnel and ooze. You can’t know both position and momentum at the same time. These strange properties come from the amazing realization that particles are waves. Not only that, but the amplitude of the wave tells us the likelihood of measuring a particle at a given position! This staggering revelation helps us understand fundamental things, like the very structure of an atom.

In the past, I’ve written extensively about quantum mechanics. The standard equation is called the Schrodinger Equation, after the discoverer, Erwin Schrodinger. However, there are actually a number of ways to think about quantum mechanics. Richard Feynman, who won the 1965 Nobel prize in physics, constructed another way of thinking about quantum particles, called the path integral. Here I try to explain the path integral as I understand it.

Before reading, you might want to read my previous, more introductory articles on quantum mechanics. My trilogy on the fascinating history and motivation of quantum mechanics can be found here:

• http://www.thephysicsmill.com/2012/12/09/the-charming-doubleness-particle-wave-duality/
• http://www.thephysicsmill.com/2012/12/24/unreal-truths-the-bohr-model-of-the-atom/
• http://www.thephysicsmill.com/2012/12/30/the-dice-are-loaded-probability-waves/

I also wrote some ancillary articles on the consequences of quantum mechanics.

• One on the Heisenberg Uncertainty Principle: http://www.thephysicsmill.com/2013/01/13/resolution-fourier-analysis-and-the-heisenberg-uncertainty-principle/
• One on the Pauli Exclusion Principle: http://www.thephysicsmill.com/2013/01/27/binary-unity-the-pauli-exclusion-principle/
• One on quantum tunneling: http://www.thephysicsmill.com/2013/02/24/the-fundamental-oneness-of-nature-quantum-tunneling/

I also wrote an article on band theory and used it in a later article to explain how transistors work. You can find them here:

• http://www.thephysicsmill.com/2013/02/03/im-with-the-valence-band-band-structure-and-the-science-of-conduction/
• http://www.thephysicsmill.com/2013/03/15/field-effect-transistor/

The Principle of Least Action

Imagine you’re a lifeguard at a beach. As you coolly watch the vacationers, you spot an emergency. Pierre Louis Maupertuis is drowning!

But there’s a problem. How can you get to Maupertuis most quickly? It’s quite hard to run on the sand, after all. There’s a cement path that goes towards the left, and that’s faster… but it’s not quite in the correct direction! What do you do?

Because you’re clever, you realize that you should spend some time on the path, and some time on the sand. You run down the cement path for a little ways, and run across the sand the rest of the way. You choose the correct distance on the cement path such that, running at top speed, you minimize the amount of time it takes to go rescue Maupertuis.

When do rescue Maupertuis, he tells you a staggering fact. He believes that when a light ray passes between air and water, or air and glass, it makes a similar decision. The light starts at a given position, call it point , and wants to get somewhere else, call it point . The light knows that it travels slower throw the water than it does through the air, but that the air might not be the most direct route, so it chooses to spend part of its time in the air and part of its time in the water.

You scoff at Maupertuis because it means that the light somehow knows what the quickest path will be. Since this seems to attribute both intelligence and prescience to photons, it’s extremely unintuitive.

But he tells you that he can prove that the idea works… he can reproduce the law of refraction. Besides, because we are part of the universe, we can never really observe it and understand it from the outside. Our models and descriptions reflect our human nature and are limited by it. Why should nature feel obligated to behave in the way you expect?

What Maupertuis just described to you is called the calculus of variations, and it has a wide variety of applications in physics. Calculus of variations is the base technique of Lagrangian mechanics, which (along with its lesser-known cousin, Hamiltonian mechanics) offers an alternative to Newton’s method of solving physics problems. The idea is that every object moves to expend the least energy. (For experts: to make the action extremal.) This is called the principle of least action.

A Quantum Principle of Least Action

In Erwin Schrodinger’s picture of quantum mechanics, quantum particle/waves obey an equation called the Schrodinger wave equation:

   

Don’t worry about what those symbols actually mean. What I want to emphasize is that, although the story it tells is very different, the Schrodinger equation is the quantum analog of the old Newton equation:

   

I wrote Newton’s law to look more similar to Schrodinger’s equation… but what you’re probably used to is

   

(Technically, the Schrodinger equation is the quantum analog of Hamilton’s equations, which can be used to derive Newton’s force law… and are actually very related to the principle of least action. But that’s a story for another day.)

So if Schrodinger’s equation is the analog of Newton’s ideas. Is there an analog to Maupertuis’ principle of least action? Richard Feynman thought about this question rather a lot… and he came up with a solution.

Imagine a particle is at some initial position in the -plane, and we want to know what path it will take to some final position. By the classical least action principle, the particle will take a path between the two positions that costs the least energy. But, if the particle is a quantum particle, it’s not really localized at a point. Instead, the particle is a wave… and it doesn’t take one path from the initial position to the final position, it takes all possible paths. 

But what does this mean computationally? Well, quantum particles are represented by probability waves, and quantum mechanics is an inherently probabilistic theory. We need some way of connecting all these paths to some notion of “wavy-ness.”

Wave Properties

An important property of waves that we want to preserve is the “superposition principle.” This is an essential piece of the quantum mechanics picture, so we want to preserve this idea. I’ve previously discussed wave interference
and the superposition principle, so if you remember my previous discussions, feel free to skip to the next section.

Imagine waves as wiggles on a very stretchy string. If I try and push up on the string (make a wiggle that goes up) and you try and push down on the string (make a wiggle that goes down) at the same time, neither of us ends up moving the string as much as we intended. This is called destructive interference. Similarly, if I push up on the string at the same time that you push up on the string, we’ll probably stretch it quite a lot. This is called constructive interference. The process of overlaying one wave over another is called superposition.

Waves and Circles

A hint about how to preserve “wavy-ness” comes from looking at how sine waves appear from a circle. If we take a dot, and it travels around the circle. The vertical motion of the dot traces out the shape of a sine wave. Notice that the wave repeats itself every time the dot traces a full circle. The number of times the dot traces the circle each second is called the “frequency” of the wave. If the wave is a sound wave, then this is the frequency of sound—or the pitch. If the wave is a light wave, this is the color of the light.

We can use this. Let’s pretend there’s a little arrow at the center of the circle, and that it spins to point at the place the dot is. We’ll try and use this arrow to represent the wave. Now we take the frequency wave as a definition, and use it to tell us how fast to spin the arrow. The higher the frequency, the faster we spin the arrow.

But how do we encapsulate the superposition principle? The answer comes from vector addition, which I’ve talked about a little bit before. Imagine we have two arrows, a blue arrow and a green arrow, as shown below (on the left). We can make a third arrow, a red one, by taking the the tail of the green arrow and putting it at the tip of the blue arrow. We then use the red arrow to connect the tail of the blue arrow to the tip of the green one.

This allows us to encapsulate the idea of waves adding up on top of each other. If the blue and green vectors represent two waves, then the red vector represents the wave created by interference.

(Astute readers may remember that I gave a similar picture for calculating acceleration vectors, only with the roles of the red and green arrows reversed. That’s no accident. Acceleration calculations use vector subtraction, which is the opposite of vector addition.)

Now, notice that we can totally cancel out two vectors, if they point in opposite directions, as shown below. The resulting sum vector has zero length, and is called the zero vector. In the language of waves, this is the extreme case of destructive interference.

The Feynman Path Integral

Now we have all the ingredients to take the principle of least action and make it quantum. Because quantum mechanics is a probabilistic theory, our principle of least action should make a probabilistic statement. Suppose we have a quantum particle with some wavelength or frequency (either will do—they’re inverses of each other) which we just measured at some initial point. We want to know the probability of finding it at some final point.

To do this, we take the take our little arrows and make them rotate at a speed based on how difficult it is for a particle to travel through a given point—the harder the travel, the slower the arrow spins. (For experts, they spin with frequency equal to the action.) In other words, if the particle starts in a thick fluid, the arrows would spin slower than if the particle started in the vacuum of space.

Now, we discussed before that the particle takes all paths between the initial point and the final point. So we make our arrows follow each path and rotate them as we go along, as shown below.

At the end of the story, when we’ve followed all the paths—and usually there are an uncountable number of them—we take the arrows at the final point and add them up by vector addition. The final vector corresponds to the height (amplitude) of the wavefunction that represents the particle. And this, my friends, gives us the probability that the particle transitions from the initial position to the final position! That’s a Feynman path integral!

Applications And Implications

Just like the principle of least action is completely equivalent to Newton’s force law, the Feynman path integral is completely equivalent to the Schrodinger picture of quantum mechanics. Although it’s much harder to compute by, the path integral is often more intuitive, and it let’s physicists think about particle interactions very quickly and easily without actually having to compute the integral. Feynman path diagrams, which I haven’t explained come from this formalism.

The path integral is also important for quantum gravity, where it is unclear how to merge quantum mechanics and general relativity. The Feynman path integral, which accepts knowledge of the past and the future, seems to mesh well with general relativity where time and space are one. As I’ll explain in a later article, my own research used the path integral in an attempt to understand how quantum gravity works.

Further Reading

Richard Feynman actually wrote a book for the layperson on this very subject. It’s called “Q.E.D. The Strange Theory of Light And Matter.” It’s a wonderful book and you should all read it. Feynman has a gift for explaining things simply. I hope that this article has whet your appetite for the words by the master.

The excellent blog LessWrong also has an article on the path integral. In fact, they have a whole series of articles on physics. You should check it out.

Finally, I really suggest you all watch Feynman’s lectures on physics. As I said, Feynman has a gift for exposition, and all his videos for non-physicists are available to watch online. If you’re interested in learning some elementary physics from the master, he also taught an introductory college physics class, and Microsoft found the videos of his lectures and put them online.