= Solution
The real-time <path integral> sums $e^{iS/\hbar}$ over paths. In a classically forbidden region, the dominant semiclassical contribution is clearer after <Wick rotation>: the weight becomes $e^{-S_E/\hbar}$ and stationary paths obey $m\ddot q=V'(q)$, 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 $m\dot q^2/2=V$. Thus
$$
q_I(\tau_E)=a\tanh\!\left[\frac{\omega_0(\tau_E-\tau_0)}2\right],\quad\omega_0^2=8\lambda a^2/m,\quad\boxed{S_0=\int_{-a}^a\sqrt{2mV(q)}\,dq=\frac43a^3\sqrt{2m\lambda}.}
$$
The reversed trajectory is an anti-instanton.<Yang-Mills Theory by David Tong>[§2.3.2, printed pp56–58: Euclidean double-well saddle]{ref} Expanding $q=q_I+\eta$ produces the fluctuation operator $-\partial_{\tau_E}^2+V''(q_I)/m$. 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 $S_0/\hbar\gg1$, well-separated events form a dilute gas with rate scale $\kappa=Ae^{-S_0/\hbar}$. 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 $\sinh(\kappa\mathcal T)$, giving
$$
\boxed{E_{\rm odd}-E_{\rm even}\simeq2\hbar\kappa.}
$$
The ground state is the lower symmetric combination of localized well states: tunnelling lifts the apparent classical degeneracy. This splitting is nonperturbative in $\hbar$, 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.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1-qcm2014-instanton.png]
{title=An original quartic double well and its Euclidean instanton trajectory}
{source=paper-1-qcm2014-instanton.py}
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