We added the phases of two paths. So what if there were countless paths? ── Feynman's bold and beautiful answer
In Episode 2, adding the wave functions that went via two paths made the phase difference appear as interference. So then ── what if there were not two paths but countless? From start A to finish B, the electron may take any path. Straight, the long way around, zigzag. Feynman's answer is bold enough to be anticlimactic ── nature assigns to "every possible path" from A to B its own phase \(e^{iS/\hbar}\), and adds them all up. The total of countless turning arrows. Usually, since the action \(S\) is all over the place from path to path, the phases point in every direction and cancel. But only near the path where the action is stationary (where its change is zero ＝ least action) is the phase almost the same as the neighboring path's, so they line up and survive. And in the classical world of \(S\gg\hbar\), that single one remains and ── Newton's trajectory appears. This is the episode where the principle of least action seen in "Force That Clicks" rises up as the shadow of the quantum.
The "two paths" of the double slit are not, in fact, special. Add a third slit, a fourth, and the paths multiply. Remove the wall entirely, and ── from A to B there are infinitely many paths. Quantum theory treats all of these infinitely many on an equal footing. The very question "which path did the electron take?" is old-fashioned ── the electron travels all paths "at once," and appears at B as the result of adding their phases. We push Episode 2's superposition of \(\psi\) out over the whole set of paths.
To each path from A to B, assign a phase \(e^{iS/\hbar}\) (a turning arrow of length 1) set by that path's action \(S\), and add them all. This is just Episode 2's "addition of two paths" extended to infinitely many. What counts is, again, how many ℏ's of action ── \(S/\hbar\).
Add up the infinitely many, and most of them vanish. The reason is how the phase turns. A wild roundabout path changes its action \(S\) a lot for even a small change of route, so its phase \(S/\hbar\) spins round and round ── neighbors face away from each other and cancel. But near the path where the action is stationary (\(\delta S=0\), a path where a small change of route does not change \(S\) to first order), the neighboring paths have almost the same phase. The arrows line up and reinforce, and survive.
This \(\delta S=0\) is exactly the principle of least action seen in "Force That Clicks." Here is the answer to "why does classical mechanics choose the path of least action?" ── it is not choosing. Add up all the paths, and only the neighborhood of the least-action path survives.
And the size of \(\hbar\) matters. For \(S\gg\hbar\) (classical, macroscopic), the phase \(S/\hbar\) spins ferociously even for a tiny change of path, so the cancellation is violent ── everything is wiped out except a very thin single strand around the stationary path. The surviving strand is the classical trajectory. Conversely for \(S\sim\hbar\) (quantum), the phase turns only slowly, and a broad bundle around the stationary path survives ── this is the quantum spread, interference, and the true nature of tunneling (even paths impassable classically survive a little).
In the figure below, the left is a family of paths from A to B (the dark line in the middle is the path where the action is stationary ＝ least action). The right is a chain of phasors (the total) formed by appending each path's arrow \(e^{iS/\hbar}\) in turn. Near the stationary path (the middle of the chain) the arrows line up and stretch out straight, while for far-off paths (the two ends) the phase spins round and curls into a spiral and cancels. The thick arrow from the origin to the tip of the chain is the summed amplitude.
Raise the slider "classicality \(S/\hbar\)" (make \(\hbar\) small ＝ macroscopic), and the spirals at both ends coil tighter, so what survives is only the straight middle part ── the amplitude is essentially set by the contribution from the stationary path (the classical trajectory). Lower it (toward quantum), and the whole chain is gentle, with a broad range of paths contributing. You can see with your own eyes that the classical trajectory is the "core" left behind by the quantum sum over all paths as \(S/\hbar\to\infty\).
With this, the true nature of the principle of least action ── which looked handed-down-from-above in "Force That Clicks" ── becomes clear. Nature is not "choosing" the least-action path ── add up all the paths, and in the world of \(S\gg\hbar\) only the neighborhood of the stationary path survives, so as a result only the least-action path is seen. Classical mechanics (Newton's trajectory, least action) was the shadow cast by the \(S/\hbar\to\infty\) limit of quantum theory's sum over paths. This is the concrete mechanism behind Episode 1's "classical ＝ \(\hbar\to0\)."
Conversely, in the quantum world of \(S\sim\hbar\), the bundle around the stationary path survives, so the particle has no single trajectory but spreads out, multiple paths interfere, and it slips a little through walls it could never cross classically (tunneling). A single path (classical) or a bundle (quantum) ── that dividing line, too, is held by the action ratio \(S/\hbar\). The action \(S\), which has units, is stage machinery; all that counts is how many ℏ's it amounts to.
The path integral (that the sum of \(e^{iS/\hbar}\) over all paths gives the amplitude, Feynman), that by the stationary-phase approximation the \(\delta S=0\) (least action) path gives the leading contribution and the classical trajectory is recovered as \(S/\hbar\to\infty\), and the spread, interference, and tunneling of paths in the quantum regime ── all are established physics, and the path integral is a standard tool of quantum field theory.
However. ① The "sum over all paths" is mathematically delicate in the definition of how the paths are "counted" (the measure), and a rigorous formulation requires care (the physical predictions come out correctly). ② Strictly, it is not "least" action but stationary action (\(\delta S=0\); including minima, maxima, and saddle points). ③ The figure's phasor chain (a Cornu spiral) is a schematic that represents the paths by a single parameter (the bulge of the route) and approximates the action to second order around the stationary point. In reality the paths are infinite-dimensional and the action differs from system to system. ④ "The electron travels all paths at once" is a picture belonging to the path-integral method of calculation, and it is safest not to push it too far as a naive claim about reality.
Extending the two-path interference (Episode 2) to infinitely many gives the path integral ── nature assigns to every path from A→B the phase \(e^{iS/\hbar}\), and adds them all (\(\psi=\sum e^{iS/\hbar}\)). Wild paths spin their phases round and cancel, and only the neighborhood of the path where the action is stationary (\(\delta S=0\) ＝ least action) survives with phases aligned. The least action of "Force That Clicks" rises up this way from the sum over paths ── nature does not choose the path; when you add them up, that is what remains.
For \(S\gg\hbar\) (classical) the phase spins ferociously, and only a very thin single strand around the stationary path remains, so Newton's trajectory appears. For \(S\sim\hbar\) (quantum) the bundle around the stationary path remains, giving spread, interference, and tunneling. Classical mechanics is the shadow cast by the quantum sum over all paths as \(S/\hbar\to\infty\) ── the concrete mechanism behind Episode 1's "classical ＝ \(\hbar\to0\)." What counts is, again, only the action ratio \(S/\hbar\).
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, raising "classicality S/ℏ" makes both ends of the phasor chain spiral, leaving only the stationary path (the classical trajectory). "See the answer" opens each solution.