Path Integrals

There are two ways to learn quantum mechanics. The first, which almost everyone encounters first, is Schrodinger's way: you write down a wave equation, solve it for the allowed states, and compute probabilities from the wave function. It works, but it carries a lot of conceptual baggage — "wave-particle duality," "collapse of the wave function," all those arguments about what's "really" going on between measurements.

The second way is Feynman's. It starts from a single, disorienting premise: to find the probability that a particle goes from A to B, you sum up a contribution from every possible path connecting them. Not just the classical path, not just the shortest path — every path, including absurd ones that loop around the moon and back. Each path contributes an amplitude of the same magnitude but different phase, where the phase is proportional to the total time (or more precisely, the action) along the path. And then you add them all up.

The Spinning Clock

Yudkowsky, channeling Feynman's QED, gives the clearest visual picture of this I've seen. Imagine the photon as a tiny spinning clock. The hand of the clock always stays the same length (constant amplitude), but it rotates at a rate set by the photon's frequency. For each possible path from source to detector, you let the clock spin for the total time the path takes, then look at where the hand ends up. That's the path's contribution — a little arrow, a complex number.¹

Now you add up all the little arrows. For most paths, neighboring paths have wildly different total times, so their arrows point in random directions and cancel each other out. But near the path where the total time is stationary — the classical path — neighboring paths have almost the same total time, so their arrows point in nearly the same direction and reinforce each other. The classical path isn't special because the photon "chooses" it. It's special because it's the one neighborhood where the contributions don't self-destruct.

This is how a mirror works: each point on the mirror contributes a path from source to detector, but only the paths near the middle — where the angle of incidence equals the angle of reflection — survive the cancellation. The rest of the mirror might as well not exist. Unless you're clever enough to scrape away strips of the mirror at regular intervals (making a diffraction grating), in which case you delete the out-of-phase contributions and let distant parts of the mirror contribute coherently. The same physics, exploited differently.¹

Why the Classical World Looks Classical

The path integral picture gives an unusually clean explanation of why big objects look classical. The phase of each path is proportional to the action in units of Planck's constant. For a photon of visible light, the wavelength is a few hundred nanometers — the clock spins slowly enough that many neighboring paths stay in phase, giving a fat bundle of non-cancelling paths. For something with the mass of a baseball, the effective wavelength is unimaginably tiny. The clock spins so fast that only an absurdly narrow neighborhood of paths stays in phase. The result is that a baseball follows what appears to be one definite trajectory — the classical path — not because quantum mechanics stops applying, but because all the quantum alternatives have destroyed each other.¹

Yudkowsky calls this the "classical hallucination." I think that's the right way to think about it. The classical world isn't a different regime from the quantum world. It's the quantum world viewed at a scale where the path integral's cancellations are so thorough that only one history survives. There's no mysterious boundary between quantum and classical — just a smooth transition from "many paths matter" to "essentially one path matters," controlled by the ratio of action to Planck's constant.

Feynman's Probabilistic Instincts

What's distinctive about Feynman isn't just the path integral formalism — it's the habit of mind behind it. He consistently treated physics as probability theory with teeth. You see this even when he wanders outside physics entirely.

There's a lovely manuscript, probably from the 1970s or 80s, where Feynman attacked Fermat's Last Theorem — the conjecture that x^n + y^n = z^n has no integer solutions for n > 2. His approach was purely probabilistic. He estimated the probability that a random large number is a perfect n-th power (roughly N^(1/n - 1) for a number near N), then integrated over all possible x, y to get the expected number of solutions for each exponent n. The result: for n = 3 the expected number of solutions is already small, and it drops rapidly with increasing n. Summing over all n > 2, and using Sophie Germain's proof that no solutions exist for small n, Feynman calculated that the total probability of any solution existing is less than a few percent.²

"For my money, Fermat's theorem is true," he concluded. This is, of course, not a proof — Andrew Wiles needed 110 pages of deep algebraic geometry to actually prove it in 1995, fifteen years after Feynman's death. But it's a beautiful example of what probabilistic reasoning can do: it tells you where to put your money, even when you can't prove the theorem. And it's pure Feynman — skip the formalism, estimate the answer, check if the universe cooperates.

The same instinct drove the path integral. Where Schrodinger asked "what wave equation describes this system?", Feynman asked "what's the probability amplitude for each possible history?" Both give the same answers, and in fact you can derive the Schrodinger equation from the path integral. But the path integral starts from a more operational place: here's what could happen, here's how to weight each possibility, now add them up. It's quantum foundations through the back door — instead of mystifying axioms about Hilbert spaces and operators, you get a recipe that a physicist can grab and compute with.

Lattice Path Integrals and the Connection Machine

The path integral also turns out to be one of the most practical tools in computational physics. In quantum chromodynamics — the theory of quarks and gluons inside protons — the standard way to compute properties like the proton's mass is to discretize spacetime into a lattice, assign field configurations to each link, and then sum over all possible configurations weighted by their action. This is literally a path integral, evaluated numerically by Monte Carlo sampling.

When Danny Hillis was building the Connection Machine in the early 1980s — a massively parallel computer with 64,000 processors — Feynman was the one who figured out that lattice QCD was a natural fit for the architecture. Each processor could handle a piece of the lattice, and since the sum over configurations is embarrassingly parallel, all processors could work simultaneously at 100% efficiency. Feynman wrote the QCD program in a parallel version of BASIC that he invented for the purpose, simulating it by hand to estimate performance. The Connection Machine, even without floating-point hardware, outperformed CalTech's purpose-built QCD machine.³

What I find striking here is the continuity between the conceptual insight and the computational method. The path integral isn't just a way to think about quantum mechanics — it's a way to compute quantum mechanics. The same sum-over-histories that gives you a clean understanding of why mirrors work also gives you a practical algorithm for calculating the mass of a proton. That unification of understanding and calculation feels distinctively Feynman.

The path integral connects to quantum computing as well: the quantum Fourier transform in Shor's algorithm exploits interference between computational paths in much the same way that the path integral exploits interference between physical paths. And the information-theoretic view of spacetime and information — where spacetime geometry emerges from entanglement patterns — can be seen as a descendant of the path integral philosophy: reality is the sum of all possible configurations, weighted by their amplitudes.