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An instanton describes a finite-action trajectory in imaginary time that crosses between classical vacua. It accounts for quantum tunnelling effects exponentially small in $\hslash$, beyond an ordinary power series around one minimum. A symmetric double well makes the connection between the Euclidean path integral, fluctuation theory and the low-energy spectrum particularly explicit.

For a particle of mass $\mu$, use a Wick rotation $t=-i\tau$ in the kernel. The Euclidean action and its classical equation areThe latter is Newtonian motion in the inverted potential $-V$. Subtract the common minimum energy so $V(\pm a)=0$. A finite-action path approaching $\pm a$ at infinite Euclidean times has conserved Euclidean energy $\mu {\dot{q}}^2/2-V=0$. ThereforeThis gives both the classical trajectory by a quadrature and its tunnelling exponent. The plus solution is the instanton; reversal is the anti-instanton. A single crossing has action $S_I$, not $2S_I$, although a periodic closed path necessarily has equal numbers of crossings in both directions.

An explicit double well. Take $V(q)=\mu {\omega }^2(q^2-a^2{)}^2/(8a^2)$, whose local harmonic frequency at either minimum is $\omega$. Integrating the zero-energy equation gives the quartic double-well instantonThe arbitrary center ${\tau }_0$ is a collective coordinate. The tunnelling regime is $S_I/\hslash \gg 1$, so the crossing width ${\omega }^{-1}$ is short compared with the typical separation between rare crossings.

The exponential alone does not give a dimensionally complete amplitude. Expand $q=q_I+\eta$; the quadratic Euclidean action is $\mu \int \eta L_I\eta d\tau /2$, whereThe derivative ${\dot{q}}_I$ is a normalizable translation instanton translation zero mode. The factorization of this one-dimensional fluctuation operator shows it has no negative modes. In $z=\omega (\tau -{\tau }_0)/2$, its two discrete eigenfunctions are proportional to ${\mathrm{sech}}^{2}z$ and $\mathrm{sech}z\tanh z$, with eigenvalues zero and $3{\omega }^2/4$; the continuum begins at ${\omega }^2$. Treating its zero eigenvalue as an ordinary Gaussian integral would give a divergent determinant. Instead integrate over ${\tau }_0$: since $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, the Jacobian supplies $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero fluctuations supply the determinant ratio. With common endpoint regularization,This is the instanton fluctuation prefactor, with $K$ and $\kappa$ having frequency units. The prime omits precisely the translation mode.

For this quartic potential the determinant can be evaluated explicitly. Set $z=\omega (\tau -{\tau }_0)/2$ and add a small positive spectral regulator $\epsilon$ to both operators. With $r=2\sqrt{1+\epsilon /{\omega }^2}$, the dimensionless operator is $-{\partial }_z^2+r^2-6{\mathrm{sech}}^{2}z$. Applying successively $-{\partial }_z+\tanh z$ and $-{\partial }_z+2\tanh z$ to the free decaying solution $e^{-rz}$, and normalizing at $z\to +\infty$, gives a growing coefficient at $z\to -\infty$ equal to $(r-1)(r-2)/[(r+1)(r+2)]$. The common Dirichlet functional determinant ratio has this coefficient. Hence the quartic double-well fluctuation determinant isThe frequency factor cannot be discarded when deleting the zero eigenvalue.

Summing rare crossings. Consider Euclidean duration $T$ much longer than a crossing width. In the dilute instanton gas, $n$ ordered centers have integration volume $T^n/n!$, and the amplitude per crossing is $\kappa$. Starting in the left well, directions alternate: an even number returns to the left well, and an odd number ends in the right well. If $E_{\mathrm{well}}=\hslash \omega /2+$ local anharmonic corrections, the leading kernels, with a common endpoint normalization $\mathcal{A}$, areThe even/odd sums must not be replaced by counting all possible crossing directions independently. Comparing their exponential components with the spectral representation givesFor the quartic example, $\Delta E=2\hslash \omega \sqrt{6S_I/(\pi \hslash )}{e}^{-S_I/\hslash }$ to leading semiclassical order. This is the double-well tunneling splitting: the even state is lower because the symmetric wavefunction has no node. The equivalent low-energy Hamiltonian in localized left/right states is $E_{\mathrm{well}}I-\hslash \kappa {\sigma }_x$. An initially localized state tunnels coherently, with probability ${\sin }^2(\Delta Et/(2\hslash ))$ of occupying the other well. Tunnelling does not imply irreversible decay in this closed symmetric system.

The instanton approach thus yields the same leading forbidden-region action as the WKB approximation, while identifying the zero-mode measure, fluctuation normalization and multiple tunnelling events systematically. Perturbation theory about either separate minimum cannot generate $e^{-S_I/\hslash }$. Multi-instanton interactions and higher-loop fluctuations correct the leading dilute-gas result; at finite temperature one instead uses periodic Euclidean boundary conditions, whose saddles may differ when the period is comparable to the crossing width. A false-vacuum bounce returning to the same metastable minimum has a negative fluctuation mode and describes a decay rate. It should not be confused with the stable-vacuum connecting instanton above, which has a translation zero mode but no negative mode.

Let $K(X)=|k(X)|>0$ on a smooth interval, with $X=ϵx$. For the WKB approximation for a slowly varying oscillator, writeSubstitution into ${ϵ}^2{y}_{XX}+K(X{)}^{2}y=0$ gives, at the first two orders,Thus $S_0^{\prime }=\pm iK$ and $S_1^{\prime }=-K^{\prime }/(2K)$. The leading WKB approximation isThe $K^{-1/2}$ amplitude is the transport correction accompanying the rapid wave phase. Real solutions are equivalent real sine and cosine combinations.

For validity, $K$ must be smooth, nonzero and slowly varying compared with the local wavelength. In physical $x$ derivatives, sufficient local checks are $|K_x|/K^2\ll 1$ and $|K_{xx}|/K^3\ll 1$. Indeed each displayed branch has relative residualSmooth positive $K(X)$ bounded away from zero has these properties on fixed slow intervals. At a zero of $k$ the WKB approximation fails and a classical turning point requires a different local analysis, often an Airy turning-point connection formula. Rapid variations, coefficient singularities and excessively long accumulation of wave phase error also lie outside this leading approximation.

Here the singular perturbation has an exact conservative structure:The integrating factor for this first-order linear differential equation givesAt $x=1$ and $x=-1$ the integral terms have opposite signs while the exponential factors agree. Equal boundary data therefore force $C=0$, and normalization gives the unique exact solutionThis exposes the Gaussian interior layer of a conservative drift equation. Its central scale is $x=O(\sqrt{ϵ})$, where $x=\sqrt{ϵ}\xi$ gives $Y^{\prime \prime }+\xi Y^{\prime }+Y=0$. The central Gaussian function profile is $e^{-{\xi }^2/2}$ with amplitude $e^{1/(2ϵ)}$.

Away from the center, $|x|\gg \sqrt{ϵ}$, use exponential WKB approximation branches rather than an assumed bounded algebraic outer solution. The reduced ordinary differential equation $x{y}_0^{\prime }+y_0=0$ would give $C/x$; matching both equal positive endpoint values with that algebraic branch is impossible. The exact solution selects its zero coefficient and the exponentially large branch with a Gaussian function profile instead. Near either endpoint, a distance $O(ϵ)$ changes $y$ by order one relative to its endpoint value: with $r=(1-|x|)/ϵ$, $y=e^{r-ϵr^2/2}$. These endpoint scales describe normalization of the same global exponential solution.

Thus a central $O(\sqrt{ϵ})$ turning region, exponential outer regions on both sides, and $O(ϵ)$ endpoint normalization scales give the complete regional description. The peak is exponentially large, so no uniformly bounded regular outer expansion should be presumed. The exact expression already provides a uniform solution without further matching calculations.

Set $s=x+1\in [0,2]$. The drift now vanishes quadratically at the left endpoint, and the outer expansion selected from $s=2$ is proportional to $e^{1/s}$, which becomes singular as $s\downarrow 0$. The oscillatory endpoint with a quadratically vanishing drift is not an ordinary monotone layer of width ${ϵ}^{1/3}$: balancing just diffusion and drift on that scale neglects the zeroth-order $y$ term.

The formal local exponential rates obey $ϵ{\lambda }^2+s^2\lambda +1=0$. For $s\gg {ϵ}^{1/4}$ their slow and fast branches are approximately $-s^{-2}$ and $-s^2/ϵ$. They coalesce when $s^4\sim 4ϵ$. This locates the main transition a distance $O({ϵ}^{1/4})$ from the left endpoint.

A reliable procedure removes the first derivative exactly:Use WKB approximation for this transformed equation on each side of its unique positive zero $s_t=\sqrt{2}{ϵ}^{1/4}+O({ϵ}^{1/2})$. For $s>s_t$ the transformed modes are exponential; for $0<s<s_t$ they are oscillatory. Their physical amplitudes include the displayed exponential prefactor.

At $s_t$, $P^{\prime }(s_t)=O({ϵ}^{3/4})$, so the Airy turning-point connection formula usesThis layer connects the oscillatory and exponential WKB approximation branches. On the transition scale $s={ϵ}^{1/4}S$, the exact equation is ${ϵ}^{1/2}{v}_{SS}=(S^4/4-1+{ϵ}^{1/4}S)v$, displaying a small effective WKB approximation parameter ${ϵ}^{1/4}$.

Close to $s=0$, the local wavelength is $O(\sqrt{ϵ})$: setting $s=\sqrt{ϵ}\xi$ gives $Y_{\xi \xi }+Y+\sqrt{ϵ}{\xi }^2{Y}_{\xi }=0$, whose leading solutions are sine and cosine. This endpoint form imposes $y(-1)=1$. Within the oscillatory region, the prefactor changes by order one over the additional envelope scale $s=O({ϵ}^{1/3})$. The $\sqrt{ϵ}$ wavelength and ${ϵ}^{1/3}$ envelope are subscales of the same oscillatory region, not additional omitted ordinary layers.

Use the outer exponential/regular branches for $s\gg {ϵ}^{1/4}$, oscillatory WKB approximation inside the $O({ϵ}^{1/4})$ transition region, an $O({ϵ}^{5/12})$ Airy neighborhood of $s_t$, and the $O(\sqrt{ϵ})$ local endpoint form to apply the left boundary data. Match their constants and enforce the right boundary data at $s=2$; there is no separate right endpoint layer required. Matching amplitudes need not remain $O(1)$, and parameter values admitting a homogeneous Dirichlet solution require a separate solvability check rather than an assumed uniform algebraic expansion.