All Paths at Once: Quantum Mechanics Through Path Integrals

Here’s the thing about quantum mechanics: the whole mystery of the double-slit experiment dissolves the moment you stop asking the wrong question.

We fire electrons one at a time through two narrow slits. Each electron arrives at the detector as a single point—click, there it is. A particle. But when we wait and collect thousands of these clicks, they don’t pile up behind the slits like bullets would. Instead they form an interference pattern—bright and dark bands that only make sense if each electron somehow went through both slits simultaneously, like a wave spreading through water.

Now, your classical intuition screams at this. It says: “Wait a minute! The electron is a particle. It must go through one slit or the other. We just don’t know which.” So you put a detector at the slits to check. And sure enough, each time you look, the electron goes through exactly one slit. Problem solved, right?

Wrong. The moment you add that detector, the interference pattern vanishes. You get two boring piles behind the two slits, exactly what you’d expect from particles. The universe seems to be playing games—behaving like a wave when you’re not looking, collapsing to a particle when you are.

This isn’t philosophy. This is what happens. The question is: how do we calculate it?

The Particle That Goes Through Both Slits

Look, here’s where we get stuck if we insist on classical thinking. Classical mechanics says a particle has a definite position at every moment. It follows one trajectory through space and time—a single path from point A to point B. Either the electron goes left through slit one, or it goes right through slit two. There’s no third option.

But if the electron takes only one path, where does the interference come from? Interference requires at least two waves overlapping—one from each slit—with peaks and troughs either reinforcing or canceling each other. You can’t get interference from a single trajectory.

So we’re forced into a contradiction. The final pattern demands that information from both slits contributes to where each electron lands. But when we check which slit the electron actually used, we always find it at exactly one location. How can the electron “know about” both slits without physically going through both?

The standard response is wave-particle duality: “Sometimes it’s a wave, sometimes it’s a particle.” That’s not an explanation—that’s just naming the mystery. You’ve got a mathematical formalism that works beautifully for calculating probabilities, but the conceptual picture is mud. The electron is somehow both a particle with a definite position and a spread-out wave at the same time. What does that even mean?

The classical assumption—one particle, one path—is what’s killing us here. We’re trying to cram quantum behavior into a classical box, and it doesn’t fit.

What If It Takes Every Path?

Here’s the move that changes everything. Stop assuming the particle takes one path. Instead, assume it takes all possible paths simultaneously.

I mean that literally. The electron doesn’t choose between going through slit one or slit two. It explores every conceivable trajectory from the source to the detector: straight lines, gentle curves, wild zigzags, paths that loop back in time if you let the math run wild. Infinitely many paths, all at once.

Each path contributes a complex number—an amplitude—that depends on the action along that path. In classical mechanics, action is the integral of the Lagrangian along a trajectory, essentially the accumulated difference between kinetic and potential energy. For quantum mechanics, you take this classical action $S$, divide by Planck’s constant $\hbar$, and compute $e^{iS/\hbar}$. That’s the amplitude for that particular path.

Now here’s the crucial part: you sum up the amplitudes from all paths. Literally all of them. The total amplitude for the particle to go from A to B is the superposition of every conceivable route it could take. And the probability of actually detecting the particle at B is the squared magnitude of that total amplitude: $P = |\sum \text{amplitudes}|^2$.

This is the path integral formulation. It’s not a metaphor. It’s a computational recipe. You can sit down, write out the sum over paths, evaluate it, and get numbers that match experiments to absurd precision.

Adding Up Infinity

You might think adding up infinitely many paths makes the calculation impossible. But here’s the beautiful part: the paths don’t contribute equally.

Each amplitude is a complex number with magnitude and phase. The phase rotates around the complex plane as the action changes. For most wild, zigzagging paths, the action is huge compared to $\hbar$, so the phase spins rapidly. Adjacent crazy paths have wildly different phases—some point up, some down, some sideways. When you add them together, they interfere destructively. They cancel out.

The paths that survive this massive cancellation are the ones where nearby paths have nearly the same phase—where the action is stationary. And which paths have stationary action? Exactly the classical trajectories that satisfy the principle of least action, the ones you’d calculate from Newton’s laws or the Euler-Lagrange equations.

So classical mechanics isn’t wrong—it’s the stationary-phase approximation to quantum mechanics. When $\hbar$ is tiny compared to the action (which it is for everyday objects like baseballs), only the classical path contributes significantly. But for electrons passing through slits, $\hbar$ isn’t negligible. Lots of nearby paths have similar phases, so they interfere constructively. That’s where quantum weirdness lives.

At the double slit, paths through the left opening and paths through the right opening both contribute. They have different phases. When those phases align at certain spots on the detector screen, you get bright fringes. Where they’re opposite, you get dark fringes. The interference pattern emerges automatically from summing over all paths.

You don’t need to ask “which slit?” because that question assumes the electron takes one path. It doesn’t. It takes all of them. The pattern you see is the inevitable result of adding up those amplitudes.

Where Classical Mechanics Hides

Now let’s examine what this means. Does the electron “really” take all those paths? That’s the wrong question—like asking whether the number seven is “really” prime. Seven is prime because it fits the definition of primality. The electron behaves as if it explores all paths because that’s how the amplitude calculates.

Asking for some deeper reality behind the math is demanding that nature conform to your intuitive pictures. Nature doesn’t care about your intuitions. It does what it does, and our job is to calculate it correctly.

Here’s what does matter: the path integral formulation makes quantum field theory possible. When you extend this idea from particles to fields, you sum over all possible field configurations, and out pop Feynman diagrams—those little sketches of particles interacting via virtual photons and gluons. Each diagram represents one class of field histories, each carries an amplitude, and you add them up. That’s how we calculated the magnetic moment of the electron to ten decimal places. It’s not philosophy; it’s engineering.

The classical path doesn’t vanish in this picture—it becomes the dominant contribution when quantum effects are negligible. The principle of least action, that beautiful classical idea that trajectories extremize the action integral, is exactly the condition for constructive interference in the path integral. Classical mechanics is quantum mechanics in the limit where one path overwhelms all the others.

You can see this explicitly. When you work out the path integral for a harmonic oscillator or a free particle, the stationary-phase approximation gives you back Newton’s equations. Classical trajectories aren’t an alternative to quantum mechanics; they’re what quantum mechanics predicts when you stop resolving the interference fringes.

The Least Weird Way to Think About It

So where does this leave us? You’ve got electrons that “go through both slits” but don’t split in half. You’ve got interference without waves in any classical sense. You’ve got probability amplitudes that add like waves but square to give probabilities.

The path integral says: stop trying to picture what the electron is “doing” between measurements. What you measure is the amplitude to go from one state to another, and that amplitude is the sum over all conceivable histories connecting those states. Each history contributes $e^{iS/\hbar}$. You add them up, square the result, and that’s your probability.

Nature isn’t computing trajectories. Nature is computing amplitudes. The path integral is our best mathematical description of that process. It’s not complete—we still don’t have a clean story for why measurement gives definite outcomes—but it’s honest. It doesn’t pretend the electron follows a path when it doesn’t.

And here’s the kicker: this formalism works for everything. Electrons in atoms, photons scattering off electrons, quarks binding into protons, Higgs bosons decaying in colliders. Every quantum field theory calculation starts with “sum over all field configurations,” and every amplitude that comes out matches experiment.

If you can’t explain quantum mechanics simply, it’s because quantum mechanics is simple—simpler than our classical prejudices. Particles don’t take one path; they take all paths. You sum the amplitudes. That’s it. The weirdness isn’t in the math; it’s in our stubborn insistence that particles must behave like tiny billiard balls.

They don’t. They behave like quantum fields. And once you accept that, the double slit isn’t a paradox. It’s just interference doing what interference always does—following from the superposition of amplitudes. All paths at once. Nature figured this out billions of years before we did.