 Solution

The exponent is $i{S}_N/\hslash$, whereis the classical action of the piecewise-linear interpolating path. For smooth paths its limit is $S={\int }_0^{T}m{\dot{x}}^{2}dt/2$. The discretization defines the quantum integral; it does not assert that typical quantum paths are differentiable. Weighting histories by their action phase is the central formulation of the original path-integral paper.^[ref]

When action differences on the experimental scales are large compared with $\hslash$, nearby nonstationary phases cancel. stationary phase selects $\delta S=0$ at fixed endpoints, hence $m\ddot{x}=0$. The surviving free classical trajectory is $x_{\mathrm{cl}}(t)=x_0+(x_T-x_0)t/T$, with action $m(x_T-x_0{)}^2/(2T)$. This is the semiclassical mechanism; it does not mean each individual path becomes a classical path.