Instantons and semiclassical saddles

Take $V(x)=\lambda(x^2-a^2)^2/4$ with positive $m$, $\lambda$, and $a$. The Euclidean action contains positive kinetic and potential terms. A finite-action trajectory approaches $-a$ and $+a$ at opposite ends of imaginary time, with velocity tending to zero. Conservation of the Euclidean first integral then gives $m\dot x^2/2=V(x)$. This is not a real-time particle climbing a classically forbidden barrier; imaginary time changes the saddle equation and the interpretation.

Use the first integral to replace imaginary time by position in the action. The action becomes the integral of the positive square root of $2mV$ across the barrier. The resulting number depends on the entire barrier profile, not just its height. A wider barrier can suppress tunneling more strongly at the same peak energy. The dimensionless ratio controlling the leading exponential is the saddle action divided by the reduced Planck constant, and a reliable semiclassical regime requires that ratio to be large.

Fluctuations around the saddle supply a determinant and collective-coordinate factors. Time-translation invariance produces a zero mode: shifting the center of the instanton gives another solution with the same action. Integrating over that center contributes the observation-time dependence needed to extract an energy splitting. The schematic splitting contains a prefactor with energy dimensions multiplying $e^{-S_0/\hbar}$. Quoting only the exponential cannot determine a numerical splitting in energy units.

A dilute gas of well-separated instantons and anti-instantons can organize repeated tunneling when individual events are rare and their interactions are small. This expansion fails when saddles overlap substantially or other low-action configurations compete. In gauge field theory, instanton size and orientation introduce more collective coordinates, and integration over large sizes can encounter strong coupling. A formally identified instanton solution therefore does not automatically furnish a controlled approximation to every observable in the corresponding theory.

A degenerate double-well instanton connects equivalent-energy minima and contributes to a real energy splitting. A false-vacuum bounce instead returns to the metastable vacuum at both ends of Euclidean time and has a characteristic negative fluctuation mode. Its interpretation concerns an instability and a decay rate. Their boundary conditions and fluctuation structures differ, so replacing one word with the other conceals a different physical calculation.

Semiclassical methods also illuminate the limits of ordinary perturbation theory. Terms exponential in an inverse coupling are invisible to any finite polynomial expansion about zero coupling. Relations among different saddle sectors and divergent perturbative series are subjects of resurgence research, but a generic cancellation or exact reconstruction must be demonstrated for the model in question. Here the established result is the explicit saddle action and its leading suppression structure; a full field-theory transseries or precise decay prediction remains outside the exercise.