Instanton

An instanton is a localized finite-action classical solution in Euclidean spacetime, used in a semiclassical description of quantum tunnelling and other nonperturbative effects. In one-dimensional quantum mechanics it can connect distinct degenerate potential minima over infinite Euclidean time. A quartic double-well instanton is an elementary example; a Yang-Mills instanton is a gauge-field example with different geometric structure.

The orientation-reversed Euclidean trajectory of an instanton. In a symmetric double well it goes from the right vacuum to the left vacuum and has the same positive action as the forward crossing. Its topological orientation changes sign, not its Euclidean action.

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.

For a one-dimensional quantum tunnelling instanton, the prefactor combines nonzero Gaussian fluctuation eigenvalues with the translation zero-mode Jacobian. Here $L_I=-{\partial }_t^2+V^{\prime \prime }(q_I)/m$, $L_0=-{\partial }_t^2+{\omega }^2$, and the prime excludes the zero mode. Common endpoint regularization and reference normalization define the determinant ratio. The resulting $K$ has frequency units, while $K{e}^{-S_I/\hslash }$ is the dilute crossing rate. This determinant/collective-coordinate structure is also discussed in Dunne, Sulejmanpasic and Ünsal's treatment; unequal neighbouring well frequencies require care and need not share the symmetric-double-well result.

Differentiating the Euclidean classical equation for an instanton gives a zero eigenfunction proportional to its time derivative. Its norm satisfies $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, so replacing the Gaussian zero-mode integral by the center coordinate gives $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero modes belong in the primed functional determinant. This removes the false divergence without deleting the center's integration volume.

For $V(q)=m{\omega }^2(q^2-a^2{)}^2/(8a^2)$, the zero Euclidean-energy condition is $m{\dot{q}}^2/2=V(q)$. Integrating gives the displayed instanton from $-a$ to $a$. Its action is ${\int }_{-a}^a\sqrt{2mV(q)}dq=2m\omega a^2/3$. The arbitrary center $t_0$ supplies a translation zero mode, treated as a collective coordinate in the instanton fluctuation prefactor.

The quartic double-well instanton has $L_I=-{\partial }_{\tau }^2+{\omega }^2[1-3{\mathrm{sech}}^2(\omega \tau /2)/2]$. For a positive regulator $\epsilon$ and $r=2\sqrt{1+\epsilon /{\omega }^2}$, the common infinite-line determinant ratio is $(r-1)(r-2)/[(r+1)(r+2)]$. Its zero-mode factor is $\epsilon /(12{\omega }^2)$ to leading order, giving the displayed primed ratio. Therefore the instanton fluctuation prefactor is $K=\omega \sqrt{6S_I/(\pi \hslash )}$ and the double-well tunneling splitting is $2\hslash K{e}^{-S_I/\hslash }$.