In the path integral, each trajectory contributes a phase

$$e^{iS/\hbar}.$$

When the action changes by much more than $\hbar$ between nearby paths, their phases rotate rapidly and tend to cancel when summed.

Paths near stationary action

Near a path for which

$$\delta S=0,$$

the action changes only to second order under small variations. Neighboring path amplitudes therefore remain more nearly in phase and reinforce one another.

This is the stationary-phase mechanism behind the emergence of the classical trajectory from the quantum sum over histories.

Classical limit

For macroscopic actions, typically

$$S\gg\hbar,$$

phase cancellation away from stationary-action paths is extremely strong. Classical mechanics then becomes an excellent approximation even though the underlying quantum formulation still sums amplitudes over alternatives.

Not literally one path

The classical limit does not mean all nonclassical paths cease to exist in the mathematical sum. Their combined contribution becomes small through destructive interference relative to the neighborhood of stationary action.

This connects Feynman's quantum formulation directly to the classical principle of stationary action and explains why the same action functional appears in both theories.