← Atlas·Path integral

The Path Integral

How does a particle get from A to B? Feynman's answer: it takes every path at once. Each history contributes a little arrow (a complex phase e^(iS/ℏ) set by its action S), and you add all the arrows. Wild paths have wildly spinning phases that cancel; the paths near the classical trajectory all point the same way and survive. Watch the phasors add — and watch the classical path emerge as ℏ shrinks.

ℏ (quantum-ness) 0.40

paths sampled 160

spread 1.4

classical action S_cl

—

|amplitude| (arrow sum)

—

regime

quantum

stationary path

δS = 0

why nature looks classical even though it's quantum

Feynman's sum over histories (1948) reformulated quantum mechanics. The amplitude to go from A to B is:

⟨B|A⟩ = Σ_paths e^(iS[path]/ℏ)

where S = ∫(½m v² − V) dt is the action of each path. Every conceivable trajectory — straight, looping, absurd — contributes a unit-length arrow whose angle is S/ℏ. The probability is the squared length of the total sum.

The visualization: the upper panel shows sampled paths from A to B colored by their phase; the lower-right is the phasor sum — each path's arrow added tip-to-tail (a "Cornu spiral"). Two things to try:
• Crank up the spread: wild paths far from classical have rapidly varying S, so their arrows spin fast and curl into tight useless loops at the spiral's ends — they cancel.
• The arrows that add coherently (the straight part of the spiral) come from paths near where the action is stationary, δS = 0. That stationary path is exactly the classical trajectory — Newton's laws are the principle of least action!

The ℏ → 0 limit: shrink ℏ and S/ℏ becomes enormous, so even tiny deviations from the classical path spin the phase through many turns and cancel. Only the stationary path survives. This is why the macroscopic world is classical: ℏ is just very small compared to everyday actions, so the path integral collapses onto Newton. Quantum effects (interference, tunneling) appear precisely when S is comparable to ℏ.

Reach: the path integral is the language of modern physics. Quantum field theory sums over all field configurations; e^(iS/ℏ) becomes e^(−S_E/ℏ) in imaginary time, connecting quantum mechanics to statistical mechanics (the partition function is a path integral). Feynman diagrams are its perturbative expansion. Lattice QCD computes it numerically. It underlies string theory, quantum gravity, and the worldline formalism. "Sum over everything, let the phases sort it out" turned out to be how the universe computes.