Quartic double-well instanton
= Quartic double-well instanton
{title2=$q_I(t)=a\tanh[\omega(t-t_0)/2],\quad S_I=2m\omega a^2/3$}
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>.