The Classical Limit of the Path Integral

The quantum propagator can formally be written as a sum over all paths:

$$K(q_f, t_f; q_i, t_i) = \int \mathcal{D}q \, \exp\!\left(\frac{\mathrm{i}}{\hbar} S[q]\right).$$

As $\hbar \to 0$, the phase oscillates rapidly, and contributions from neighboring paths largely cancel. The phase remains locally stationary only where the first variation of the action vanishes.

The Stationary-Phase Condition

Write $q = q_{\mathrm{cl}} + \eta$, with $\eta = 0$ at the endpoints. Expanding the action gives

$$S[q] = S[q_{\mathrm{cl}}] + \delta S[q_{\mathrm{cl}}; \eta] + \frac{1}{2} \delta^2 S[q_{\mathrm{cl}}; \eta] + \cdots .$$

Classical trajectories are selected by $\delta S = 0$. The classical limit therefore does not mean “keeping only one path”; it arises from coherent addition of quantum amplitudes near stationary-phase configurations.

Semiclassical Structure

Quadratic fluctuations determine the Van Vleck determinant and carry information about the stability of the classical flow into the propagator. The semiclassical approximation thus does more than remove quantum effects: it organizes them into a hierarchy of fluctuations around classical trajectories.