Instantons, as I understand it, are mathematical constructions in Euclidean spacetime. Does it imply that instantons do not exist in real spacetime or the instanton tunneling effects does not have observable consequences in Minkowsian spacetime?

How does an instanton solution "look like" in real time? In imaginary time, the instanton solution interpolates between two vacua. What happens in real space?

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    $\begingroup$ The same. The whole point of the Euclidean theory is to compute every amplitude in Euclidean time where everything converges and then analytically continue back to Minkowski space. (To prove the well-definedness and existence of the analytic continuation is, in turn, the point of axiomatic approaches to QFT) So an instanton tunneling amplitude stays an instanton tunelling amplitude. $\endgroup$ – ACuriousMind Oct 22 '15 at 13:42
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    $\begingroup$ @ ACuriousMind- In Minkowski space, the coordinates $(x,y,z,t)$ are real numbers. What does it physically mean to take $t$ to imaginary value, $t\rightarrow -i\tau$? $\endgroup$ – SRS Oct 22 '15 at 14:58
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    $\begingroup$ As I said, that's analytic continuation. It has no physical meaning, and it is intricate to prove (meaning it has not rigorously been done) for arbitrary theories that this Wick rotation/analytic continuation really allows us to use the Euclidean results in the Minkowski case just by resubstituting $\tau\mapsto \mathrm{i}t$. The sole purpose of the switch to Euclidean space is that many integrals are better behaved there. $\endgroup$ – ACuriousMind Oct 22 '15 at 15:02
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/80889/2451 , physics.stackexchange.com/q/323456/2451 and links therein. $\endgroup$ – Qmechanic Mar 7 '18 at 19:27

This question has been studied in some detail for instantons in QM, for example the standard instanton in the potential $V=(x^2-x_0^2)^2$. Naive analytic continuation $\tau\to it$ gives a complex trajectory which does not have direct physical meaning, but corresponds to a saddle of the path integral in the same way that ordinary real integrals have complex saddle points.

This does not mean that the instanton does not have physical significance. The instanton calculation provides a semi-classical approximation to the path integral, and there is a one-to-one map to the WKB approximation for the Schroedinger equation. In particular, we can write down the WKB wave function for a particle tunneling from one minimum to the other, and this solution corresponds to instanton contribution in euclidean path integral.


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