Understanding multiple instanton contributions to vacuum tunneling in a double potential well

Question

I'm trying to understand the method of calculating the transition probability of one vacuum state to another in a double well potential, using instantons. The reference I am following is Sidney Coleman's book "Aspects of symmetry".

The approach uses the path integral formulation of QM and we look at the semi-classical limit of small $\hbar$. In this limit our path integral is dominated by the stationary points of the (euclidean) action. Solving the equation of motion I find an infinite number of single instanton solutions (each different by a shift of the time origin of the instanton) starting at one vacuum and ending at the other.

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For example, if I took a potential of the form

$V = \lambda(x^2-a^2)^2~~~$ and define $~~~\omega^2 = 8\lambda a^2$

our solution to the E.O.M with $~x(-\infty) = -a~$ and $~x(\infty) = a~$ would be

$x(t) = a\tanh\left(\frac{\omega}{2}(t-t_c)\right)$

where $t_c$, the center of the instanton is arbitrary.

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Coleman then introduces other "approximate stationary points" which are effectively instanton/antiinstanton chains which start at one vacuum and end at the other. Coleman then continues in his analysis by adding up all of these approximate stationary point contributions to the original single instanton stationary point term.

My question is the following: in what sense are these multiple instanton/antiinstanton solutions "approximate stationary points"? (Is there some limit I can take where it becomes exact, or could someone explain why these paths don't have a negligible contribution to the path integral unlike other paths?)

Answer 1

The reason why classical solutions add a "lot" to the path integral is that their action (phase) is stationary i.e. almost the same phase as the action (phase) in their reasonably large vicinity of the configuration space; one gets positive interference as a consequence. More generic paths cancel with the adjacent ones whose phases are different and random.

The exact stationarity of the action means that one has classical solutions. One may also consider approximate solutions – configurations that solve the equations of motion up to relatively small errors – and for these approximate solutions, one still gets positive interference with nearby solutions but a less perfect one. The action is approximately stationary around these approximate solutions; $\delta S$ contains not only terms of order $[\delta x(t)]^2$ and higher powers but also $a\cdot \delta x(t)$ but the coefficients $a$ are very very small so that for certain purposes, it must be OK to approximate $a\sim 0$ and consider these things solutions (or stationary points).

For the instanton/antiinstanton chains, the error in these approximate solutions goes to zero if the instantons and antiinstantons are separated from each other by an infinite distance (by distance, I mean the $t$ direction in your purely-temporal example of kinks). When these chains of instantons and antiinstantons are really widely separated, the solution is almost exact: $x(t)$ almost completely reaches the ultimate stable points $\pm a$.

When the instantons and antiinstantons are closer to each other, the error by which this configuration fails to be a solution gets larger because the configuration forces $x(t)$ to revert the direction well before it reaches $x(t)=\pm a$ which means that some "additional acceleration" (violating the original equations of motion) had to be added. When the distance between the pieces of the chain gets infinite, one instanton in the chain is used almost in its full length and $x(t)$ gets almost as constant as it does for $|t-t_c|\to\infty$ for an antiinstanton which means that one may glue the instanton and antiinstanton.

Clearly, the approximation in which the chains are "solutions" is only good if $\omega\Delta t\gg 1$. We may replace $\gg$ by $\gt$ to define a convention of which configurations we allow. However, we must realize that there's no unique, systematic procedure to deal with these chains-of-instantons contributions. The importance of these chain configuration is that they're there and they surely do contribute something to the path integral while they're missed by all the perturbative terms; and missed by the single-instanton terms, too. So they're important for those processes for which they're important and for which the perturbative and one-instanton terms don't contribute much. The previous sentence is a tautology; but it's also meant as an encouragement to think because when one thinks a little bit, he may determine what kind of processes are sensitive to such things.