Why do we have to sum the expansions around all the action's stationary points?

This is in some sense a follow-up question to my previous question Why is it OK to keep the quadratic term in the small $\hbar$ approximation?. I understand how we can expand the action around a stationary point. This way we obtain a semiclassical expansion. I have read it in Coleman's Aspects of Symmetry that "If there are several stationary points, in general one has to sum over all of them".

I do not see why. I mean, we can always make the expansio around the stationary point of our choosing, isn't it? why do we have to sum all the expansions?

• The path integral is taken over all paths. If you throw away the parts that are not stationary as not contributing, you are left with a sum over stationary paths. – By Symmetry Aug 8 '16 at 17:10

1. A path integral is in principle a weighted sum over all possible histories. Its precise mathematical definition is in general an open problem. A path integral is typically evaluated as an instanton sum $$Z~=~\sum_j Z_j ,\tag{1}$$ where each instanton $\phi_j$ has a fluctuation path integral $$Z_j~:=~\int\!{\cal D}\eta~ \exp\left\{\frac{i}{\hbar}S[\phi_j\!+\!\eta]\right\} \tag{2}$$ that is viewed as a perturbative expansion in $\hbar$.