Physics
Propagator = sum over all paths
K(x_f,t_f;x_i,t_i) = ∫D[x] e^{iS/ℏ}
S[x] = ∫L dt = ∫(½mẋ²−V) dt
Classical path: δS = 0 (stationary)
ℏ→0: only classical path survives
Quantum: all paths interfere
Path Statistics
Paths shown—
Classical action S_cl—
|K|² (prob. density)—
ℏ—
Color = phase e^{iS/ℏ}
Bright = near stationary phase
(near-classical paths)
Sum ≈ amplitude at endpoint
Controls
Stationary phase approximation:
Non-classical paths cancel
via rapid oscillation
Classical path contributes most