 Solution

The real-time path integral sums $e^{iS/\hslash }$ over paths. In a classically forbidden region, the dominant semiclassical contribution is clearer after Wick rotation: the weight becomes $e^{-S_E/\hslash }$ and stationary paths obey $m\ddot{q}=V^{\prime }(q)$, equivalent to motion in the inverted potential. A finite-action path joining degenerate minima is an instanton.

For $V(q)=\lambda (q^2-a^2{)}^2$, $\lambda >0$, zero Euclidean mechanical energy gives $m{\dot{q}}^2/2=V$. ThusThe reversed trajectory is an anti-instanton.^[ref] Expanding $q=q_I+\eta$ produces the fluctuation operator $-{\partial }_{{\tau }_E}^2+V^{\prime \prime }(q_I)/m$. Its translation zero mode is handled by integrating the collective centre ${\tau }_0$, not by dividing by a zero determinant. The remaining determinant and Jacobian form an attempt-frequency prefactor $A$.

When $S_0/\hslash \gg 1$, well-separated events form a dilute gas with rate scale $\kappa =A{e}^{-S_0/\hslash }$. Paths returning to the same well have an even number of events, while paths ending in the other well have an odd number. The sums yield $\cosh (\kappa \mathcal{T})$ and $\sinh (\kappa \mathcal{T})$, givingThe ground state is the lower symmetric combination of localized well states: tunnelling lifts the apparent classical degeneracy. This splitting is nonperturbative in $\hslash$, beyond a power series about either minimum. In a metastable well a bounce, with a negative fluctuation mode, describes decay rather than the stable doublet splitting. Finite temperature, overlapping events or a low barrier require improvements beyond the dilute approximation.