= Dilute instanton gas
{title2=$\kappa=Ke^{-S_I/\hbar}\ll\omega$}
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 $\hbar$, matched local well structure and control of corrections from interactions and local anharmonicity.
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