Feynman Path Integral

K(b,a) = ∫ D[x(t)] e^(iS[x]/ℏ) — sum over all paths, classical emerges from stationary phase

Potential
Each path contributes e^(iS/ℏ)

Classical path: δS=0
Nearby paths: phases cancel
→ stationary phase approx

Small ℏ: quantum→classical
Large ℏ: all paths matter

S[x] = ∫[½mẋ²−V(x)]dt

Color = phase angle
Brightness = weight
Arrow = phasor e^(iS/ℏ)
Amplitude: computing...