# What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. Partly, I think this is due to the fact that some authors make a distinction between these, and others just say 'instantons/anti-instantons' for the 'ordinary' type, and ghosts/complex (i'm not sure if these are the same).

My present understanding is as follows:

When we have a path integral in real/Minkowski time:

$$\int \mathcal{D}[\phi] e^{-\frac{iS[\phi]}{\hbar}}$$ with $$S$$ the classical action, there can be more than one classical solution to the equations of motion. i.e. real functions that extremise the action. These correspond to real instantons.

Then when it comes to complex instantons, I thought these were just stationary points of the action if one analytically continues the domain of integration to complex $$\phi$$ as well? I am struggling with seeing the significance of this in the quantum case. I agree that in the real case, we only seek real $$\phi$$ solutions (it may be that $$\phi(t) = x(t)$$), but in the complex case would be not expect to be integrating over the domain of complex paths $$\phi$$, so there is no 'analytic continuation' involved, so to speak?

If complex instantons are indeed complex $$\phi$$ solutions to make the action stationary, how does this correspond to Wick rotation to Euclidean time? i.e. I have seen authors use the Wick rotated path integral $$\int \mathcal{D}[\phi] e^{\frac{S[\phi]}{\hbar}}$$ with $$S$$ now the Eulcidean action, and refer to stationary points of this action as 'instantons'. How does making time imaginary relate with finding complex $$\phi$$ solutions, and thus with a classification of real or complex instantons? How do these notions relate to the physical interpretations of these contributions to the path integral?

Note: an additional thing that may be related. I have some familiarity with Piccard Lefshetz theory and thimble decompostion of an integration path. I know that the imaginary part of the exponential argument is constant on a downwards flow, so one of the advatages of Wick rotation is that real action solutions can be the

I couldn't find any stack exchange posts answering these questions, but relevant papers that I have been trying (with limited success) to digest are:

Tying up instantons with antiinstantons, N. Nekrasov

Resurgence Theory, Ghost instantons, and the analytic continuation of path integrals, G. Basar, G. Dunne, M. Unsal

• These complex saddles have to do with complexifying field space not time. Most of the discussion of these takes place in Euclidean path integrals. The notion of an ordinary instanton is also in the Euclidean path integral by the way (not Minkowski as you claim) – octonion Mar 21 at 14:17
• The way I saw it was, the solutions that extremise the action in the Minkowski time will conserve energy. But in quantum mechanics, you also want to have paths with tunneling events which doesn't conserve the energy (when the particle cross the potential barrier). To take those paths into account, you do a Wick rotation and now the extremal paths will be those tunneling events. Not sure it answers your question. – E. Bellec Apr 8 at 9:39