Sum Over Histories: Path Integrals and Quantum Interference

Look, I’m going to tell you something about quantum mechanics that most people get backward. Everyone focuses on the weirdness—particles in two places at once, spooky action at a distance, all that jazz. But here’s what actually matters: nature doesn’t choose one path. She explores them all.

All Paths at Once

In classical mechanics, if you throw a ball from point A to point B, it follows one trajectory. You know the drill—Newton’s laws, initial conditions, maybe you invoke the principle of least action if you’re fancy. The ball takes the path that minimizes action, where action is just kinetic energy minus potential energy integrated over time. One path. Deterministic. Predictable.

Quantum mechanics throws that out the window. An electron going from A to B doesn’t take one path. It takes every conceivable path simultaneously—straight lines, zigzags, loops, paths that go backward in time and forward again. Every single possibility. And I mean every one.

Here’s the key: each path contributes a complex amplitude $A = e^{iS/\hbar}$, where $S$ is the action for that path and $\hbar$ is Planck’s constant. The total amplitude for getting from A to B is the sum—really an integral—over all possible paths: $\Psi = \int e^{iS/\hbar} \mathcal{D}[\text{path}]$. That’s the path integral. The probability of finding the particle at B is just $|\Psi|^2$.

Now, why does classical mechanics work at all? Because when action is huge compared to $\hbar$—which it always is for macroscopic objects—the phase $e^{iS/\hbar}$ oscillates like crazy for paths away from the classical trajectory. Neighboring paths have wildly different phases and cancel each other out through destructive interference. The only paths that survive are those near the classical path where the action is stationary, where $\delta S = 0$. Those paths have similar phases and add constructively. Classical mechanics emerges from quantum mechanics through interference.

At quantum scales, where action is comparable to $\hbar$, many paths contribute with similar weight. That’s when you get all the non-classical phenomena—tunneling through barriers, diffraction around obstacles, superposition of states. It’s not magic. It’s interference.

Double-Slit: Quantum Interference in Action

Let’s talk about the double-slit experiment because it’s the perfect example of why you can’t think classically about quantum mechanics. You fire electrons one at a time at a screen with two slits. Behind the screen, you’ve got a detector. What pattern do you see?

If electrons were classical particles, you’d see two blobs—one behind each slit. But that’s not what happens. You see interference fringes: alternating bright and dark bands. The hallmark of waves.

Here’s how the path integral explains it. The electron takes every possible path from the source to the detector. Some paths go through slit 1, some through slit 2, and in principle, some do even weirder things. But let’s focus on the dominant contributions. The amplitude for paths through slit 1 is $A_1 = \sum e^{iS_1/\hbar}$ and through slit 2 is $A_2 = \sum e^{iS_2/\hbar}$.

The total amplitude is the superposition: $A = A_1 + A_2$. When you calculate the probability, you get $P = |A_1 + A_2|^2 = |A_1|^2 + |A_2|^2 + 2\text{Re}(A_1^*A_2)$. That last term—the interference term—is what creates the fringes. It’s positive where the amplitudes add constructively and negative where they cancel destructively.

But here’s the twist. If you measure which slit the electron went through, you force it onto a definite path. The amplitude collapses to either $A_1$ or $A_2$, not both. Now the probability is $P = \frac{1}{2}|A_1|^2 + \frac{1}{2}|A_2|^2$—no interference term. The fringes disappear.

People ask, “Which path did the electron really take?” Wrong question. Before measurement, it took all paths. Measurement changes the situation. Don’t ask what the electron does when you’re not looking. Just calculate the probabilities for what you observe when you do look.

Diagrams: Particles as Paths in Spacetime

I extended this path integral business to quantum field theory, where particles get created and destroyed. That’s when I developed those diagrams everyone associates with my name.

A Feynman diagram is a visual representation of particle interactions. Lines represent particle worldlines in spacetime—straight arrows for electrons, wavy lines for photons. Vertices represent interactions where particles meet, exchange, emit, or absorb other particles.

Take electron-electron scattering. Two electrons approach, exchange a virtual photon—the carrier of the electromagnetic force—and scatter apart. You draw two electron lines, one photon line connecting them, and two vertices. Each element of the diagram corresponds to a mathematical expression. You multiply them together, integrate over internal momenta, and out pops the amplitude for that particular interaction scenario.

But there’s not just one diagram. There are infinitely many. An electron could exchange two photons instead of one. Or the photon could split into an electron-positron pair before recombining. Or a photon could emit another photon. Each additional vertex adds complexity and—crucially—reduces the amplitude by a factor of roughly $\sqrt{\alpha} \approx 0.085$, where $\alpha$ is the fine structure constant.

So you sum the diagrams order by order: tree level (no loops), one-loop corrections, two-loop corrections, and so on. It’s a perturbation series. The simple diagrams give you the bulk of the answer. The complex ones add tiny corrections. This approach—quantum electrodynamics, or QED—predicts the electron’s magnetic moment to ten decimal places. No other theory in physics comes close to that precision.

Just Shut Up and Calculate

Here’s my philosophy: quantum mechanics works. It predicts experimental results with absurd accuracy. Does the electron “really” travel all paths? Is the universe actually splitting into branches? Does the wave function “really” collapse?

I don’t know, and I don’t care. These are questions about interpretation, not physics. The path integral is a mathematical tool that produces the right probabilities. Whether nature literally explores all paths or whether that’s just a convenient fiction—it doesn’t matter for doing physics. Use what works.

That said, the path integral does suggest a perspective: quantum systems explore all possibilities, and interference selects what we observe. Some people take this seriously and propose many-worlds interpretations where every path is real in some branch of a splitting universe. Others focus on decoherence—how entanglement with the environment destroys interference between macroscopic paths, which is why we don’t see cats in superpositions of alive and dead.

These are interesting questions for philosophy. For physics, I say: shut up and calculate. The path integral gives you a recipe. Initial state, final state, sum over all paths weighted by $e^{iS/\hbar}$, square the amplitude, get your probability. It works for quantum mechanics, quantum field theory, even attempts at quantum gravity where you sum over spacetime geometries.

I got a Nobel Prize in 1965 for this stuff, shared with Schwinger and Tomonaga who developed equivalent formulations. I also wrote the Feynman Lectures on Physics, popularized science through books and talks, and—just to keep things interesting—played the bongo drums and cracked safes at Los Alamos. Physics shouldn’t be all serious all the time.

The point is this: quantum mechanics isn’t weird because particles do impossible things. It’s subtle because nature doesn’t commit to one history. She computes with all possible histories, and interference determines what survives to become observable reality. That’s not mysticism. That’s just how the universe calculates probabilities.

And if you don’t like it, well, take it up with nature. She’s been using this method for 13.8 billion years, and it works just fine.