Path Integrals
Sum-over-histories — interference of classical trajectories yields quantum amplitudes.
Feynman: the propagator from $(q_a, t_a)$ to $(q_b, t_b)$ is
$$K = \int \mathcal{D}[q]\, e^{iS[q]/\hbar}.$$Stationary phase recovers classical mechanics: paths near $\delta S = 0$ interfere constructively, others cancel. Wick rotation $t\to -i\tau$ converts the oscillatory integrand to a Boltzmann weight, turning a $d$-dimensional QFT into a $(d{+}1)$-dimensional statistical mechanics problem. In field theory:
$$Z = \int \mathcal{D}\phi\, e^{i\int d^4x\, \mathcal L[\phi]}.$$Interactive: path interference
Many paths from A to B with random kinks; each contributes $e^{iS/\hbar}$. The classical (straight) path dominates when actions are large in units of $\hbar$.