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My question is exactly this one from Physics Forums, but I don't see any duplicates on SE and it doesn't seem to have gotten a clear answer there.

If the Wick rotation switches out the complex $e^{iS/\hbar}$ for the positive real $e^{-S/\hbar}$ in the path integral, then how can there be any points with zero amplitude in the double slit experiment? The only way for $e^{-S/\hbar}$ to be zero is if $S = \infty$; does something about the Wick rotation make the action infinite for all paths that arrive at the minimum on the detector screen?

Adam Herbst
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  • "The only way for $e^{-S/\hbar}$to be zero is if..." Why do you think this matters? Neither $e^{iS/\hbar}$ nor $e^{-S/\hbar}$ can be zero for any finite value of S... – hft Mar 08 '23 at 20:58
  • @hft Right, but even though $e^{iS/\hbar}$ can't be zero, it is complex and therefore can cancel out when integrated. $e^{-S/\hbar}$ is always nonnegative, so cannot cancel out, unless it was zero to begin with. – Adam Herbst Mar 08 '23 at 21:17
  • I guess you are assuming that the action has to be real? – hft Mar 08 '23 at 23:14
  • @hft Are you saying $S$ is imaginary after the Wick rotation due to $dt$ becoming $i dt$ in the integral of the Lagrangian? If so, why do people make a big deal about the parallel between the Wick rotation and statistical physics? If the $i$ is merely absorbed into $S$, what’s the difference? – Adam Herbst Mar 08 '23 at 23:25
  • I'm not really saying anything, I'm really just trying to understand what you are asking. Your question starts off with a conditional premise: "If the Wick rotation switches out..." that I am not even sure makes sense. Wick rotation is usually just accomplished by deforming a contour in a complex integral. If your time is complex, then I don't see why your action shouldn't be complex, since it is an integral over time... – hft Mar 08 '23 at 23:30
  • @hft Oh okay, well I guess I'm confused about that premise too. But I asked about it here and got no response. And posts like this one made me think that the premise was true after all (as well as the Physics Forums post I linked in my question here) – Adam Herbst Mar 09 '23 at 01:11
  • I think the upshot is: (1) you do need interference to happen; (2) you do get interference when you evaluate $e^{iS}$ for different paths, e.g. two paths, $e^{iS_1}+e^{iS_2}$; (3) evaluating the action for a specific path can be done with whatever mathematical trick you like (e.g., Wick rotation) and it does not stop the alternative paths from interfering. – hft Mar 09 '23 at 02:27

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Thanks to the discussion with @hft and thinking more carefully about the example given by Wikipedia, I think I understand now, but if I'm wrong then maybe an expert will correct me.

I think the point is that, after doing the Wick rotation, the equations and solutions that you get in Euclidean space do not have to exhibit the same features as the original ones in Minkowski space. If you interpret the Wick-rotated solution as describing a hypothetical physical world (a world with only space, no time), then that world need not even have the same qualitative phenomena as the world we perceive. That's why, in the Wikipedia example, a ball in free fall can turn into a hanging spring fixed at both ends, when Wick-rotated. Granted, certain generalities may have to be shared between the two, but in particular, and quite surprisingly, interference does not!

It doesn't matter, because the Wick-rotated Euclidean world is merely a staging ground in which the integral can be performed more easily; hence it need not look like our world. As A. Neumaier says in the original post I linked above, after you've done the integral you still have to Wick-rotate the solution back to Minkowski space to find out what phenomena we will actually perceive.

After all, even the most fundamental thing, the metric, is already dramatically different between the two spaces; a curve of constant interval is perceived as a circle in one, and a hyperbola in the other! So why shouldn't other features be dramatically different as well?

But I still can't help wondering, given that the Wick-rotated world seems so much more intuitive in the sense of having a simple Euclidean metric and simple additive probability, whether the Wick-rotated universe could somehow be the real one, and our perceptual mechanisms the true agents of rotation...

Adam Herbst
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