Dilute instanton gas ($\kappa =K{e}^{-S_I/\hslash }\ll \omega$)

When individual instantons are much narrower than their typical separation, multi-crossing paths factor at leading semiclassical order. Integrating $n$ ordered centers over time $\tau$ contributes ${\tau }^n/n!$. In a double well, crossings alternate direction and even/odd counts yield hyperbolic cosine/sine amplitudes. This approximation requires large action relative to $\hslash$, matched local well structure and control of corrections from interactions and local anharmonicity.