I What's the deal with instantons?

[pines-demon]

I am reading Blundell and Lancaster and I am still lost to what they want me to understand about instantons.

So I get the following:

• By calculating the propagator of a harmonic oscillator, then in Euclidean space the propagator of a potential with a single minima is $\approx e^{-\omega \tau/2}$, which looks like decay, where $\tau$ is the Euclidean rotated time and $\omega^2=V''(a)$ where $V(x)$ is the potential with minima at #x=a##.
• Then they go to a double well and say it is mostly the same with more semiclassics
• They use an inflaton gas to show that the eingenstates of the double well can be approximated as superposition of wavefunctions localized in each side of the well
• They calculate the decay rate and say that this could apply to the universe.

Can somebody motivate better what I am to learn about it? Is it just a different way to recover semiclassical results? Is it the non-perturbative aspect? Should I think of instantons as particles in any way? Is there anything interesting to the topological nature of instantons? What is a better introduction to instantons?

 

[Demystifier]

What one really wants to compute is the tunneling amplitude between different vacua. Computing it exactly is complicated, so one uses the WKB approximation. But it turns out that the computation of tunneling WKB amplitude can mathematically be reduced to the computation of classical action with imaginary time (see e.g. Ryder's QFT book). This "imaginary time" is just a mathematical trick in a computation, the physical time is of course not imaginary. To compute the classical action you must first find the corresponding classical solutions. Those solutions have zero energy (because you consider the tunneling between different vacua, which have the same energy that can be taken to be zero), so these solutions, with imaginary time, turn out to be instantons. Those instantons don't exist in a physical sense, they are just a computational tool that arises as a part of the WKB method. Their topological properties are interesting because understanding them simplifies the computations.