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In the symmetric double well potential, the solutions in $1+1$ static and real $\varphi^4$ theory, are solitons. However, we know that such theories are "dual" to one dimensional real $\varphi^4$ theories, and solutions of those after Wick rotation are instantons, also they have same mathematical form of solitons.

My question is, is this soliton-instanton correspondence just a special case? or applicable to any potential of those dual theories? More generally, Is every soliton has corresponding instanton after Wick rotation?

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  • $\begingroup$ I would think so, since the fact that they satisfy the classical equations of motion should be unaffected by the Wick rotation. $\endgroup$ – Siva Mar 31 '17 at 6:53
  • $\begingroup$ @Siva you mean you think so in Dual-Theories? still I want to know in general, since in QFT we speak about instantons usually, not solitons.. $\endgroup$ – TMS Mar 31 '17 at 11:34
  • $\begingroup$ Depending on your exact nomenclature, instantons are special cases of solitons, or solitons and instantons "correspond" as you say, or something in between. Unless you give your precise definition of "soliton", this question is unanswerable. $\endgroup$ – ACuriousMind Mar 31 '17 at 11:44
  • $\begingroup$ @ACuriousMind are you referring to that solitons are special case of solitary waves? Also I do not really understand how instantons can be special case of solitons, since solitons are observable, while instantons are not physical objects, they are just a tool to explain tunneling in QFT, as many sources states. $\endgroup$ – TMS Mar 31 '17 at 12:40

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