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What makes the distinctions between the soliton objects in Minkowski or in Euclidean spacetime?

It looks that usually, the Euclidean path integral is easier to be performed in many cases. In fact, most of the lattice simulations as far as I concern seem to start with the Euclidean lattice path integral.

In terms of the general soliton objects (including monopoles, hedgehog, domain walls, etc), we quite often to think of them in the Euclidean spacetime.

So,

(1) the soliton objects in Minkowski spacetime, and

(2) the soliton objects in Euclidean spacetime,

  • how do (1) and (2) relate to each other?

  • If the soliton objects in Euclidean spacetime affect the dynamics of the underlying theory [e.g. phase transitions driven by topological soliton objects], whether the same dynamics will be expected to occur for the Wick rotating the theory from Euclidean spacetime to Minkowski spacetime? If yes, so, how can we assure ourselves with the proof? If not, what are counterexamples?

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    $\begingroup$ Related question specialized to instantons here. I've only ever seen Wick rotation applied to instantons, not other solitons. Speaking without much experience, I don't think it would make a difference for other things, because only the instantons are inherently time-dependent. And indeed there is no claim that instantons in Euclidean time correspond to anything real or topologically stable in real time; they're just a calculational tool. $\endgroup$ – knzhou Aug 15 '18 at 22:50

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