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Taken from here

Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/ℏ$

But I was under the impression position and time were both parameters on equal footing in QFT? If this is so, why does the Wick rotation in position space correspond to something else*?

*I am uncertain what it corresponds to but I'm sure it's not statistical mechanics?

I'm quite confused on the source of this seemingly asymmetry (if it is real as well)? Where is my understanding failing me?

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In relativistic QFT, location and time are on the same footing in the same sense that they're on the same footing in classical special relativity. They're on the same footing in the sense that they are both used to parameterize the field operators (in the Heisenberg picture) and can be mixed with each other by symmetries, but the signature is still Lorentzian, so timelike directions are still distinguished among all directions in spacetime.

Wick rotation to Euclidean signature makes them even more on the same footing, in the sense that timelike directions are no longer distinguished among all directions after Wick rotation. One way to answer your question, then, is to think of things the other way around: start in Euclidean signature, then Wick rotate to Lorentzian signature. From this perspective, Wick rotation is what introduces a set of distinguished directions (the timelike ones).

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  • $\begingroup$ In the "One way to answer your question" the asymmetry comes from not wick rotating the spatial coordinates. Right? Like I mean, the notion of co-ordinate invariance (general covariance) seems broken to me $\endgroup$ – More Anonymous Oct 29 at 16:07
  • $\begingroup$ @MoreAnonymous Maybe I missed the point of your question. Wick rotation of a spatial coordinate doesn't accomplish anything useful because the spatial momentum operator, which generates spatial translations, doesn't have a lower-bounded spectrum. The Hamiltonian (energy operator), which generates time translations, does have a lower-bounded spectrum. For that reason, Wick rotation of the time coordinate converts an oscillating integrand $e^{iS}$ in the path integral to a rapidly-converging integrand of the form $e^{-S_E}$, where $S_E$ is the positive "Euclidean" action. That's why it's useful. $\endgroup$ – Chiral Anomaly Oct 30 at 1:46
  • $\begingroup$ So the asymmetry comes from the dynamics as opposed to the kinematics? (where the Hamiltonian gets involved). Does this mean I can have a meaningful connection in periodic paths to statistical mechanics (but a different entropy perhaps)? $\endgroup$ – More Anonymous Oct 30 at 2:37
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    $\begingroup$ @MoreAnonymous The spectrum condition on the Hamiltonian is the reason why applying Wick rotation to a time coordinate is useful and applying it to a space coordinate is not. QFT requires this condition on the Hamiltonian (energy operator) and not on the spatial momentum operators. This might seem asymmetric, but the full condition is actually Lorentz-symmetric: relativistic QFT requires that the spectrum of the set of energy-momentum operators be confined to the forward light-cone. Timelike directions are still distinguished from spacelike ones, but it's Lorentz-symmetric. $\endgroup$ – Chiral Anomaly Oct 30 at 2:59
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    $\begingroup$ Perhaps this would be the right question for that? physics.stackexchange.com/questions/510976/… $\endgroup$ – More Anonymous Oct 30 at 5:58
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When we say that space and time are on equal footing in QFT, we mean that they are both treated as continuous parameters, as opposed to in quantum mechanics where $t$ is a parameter, but $X$ is an operator.

Treating them on equal footing does not mean that they must be exactly the same: space and time in QFT are related to each other the same way space and time in classical special relativity are related to each other. Indeed, the Lorentz transformations of special relativity act on quantum fields the same way they act on the coordinates of spacetime in the classical setting - this is why QFT hinges on representation theory of the Lorentz and Poincare groups.

Wick rotation changes this situation, by replacing $t$ by $i \tau$ we turn $ -dt^2 + dx^2 + dy^2 + dz^2$ into $d\tau^2 + dx^2 + dy^2 + dz^2$, and time is now treated in an exactly symmetrical fashion with space. The Lorentz group $SO(3,1)$ becomes the Eucliean rotation group $SO(4)$.

Remember that Wick rotation is a computational trick used to improve the mathematical properties of the formalism.

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  • $\begingroup$ Can you elaborate why "Wick rotation is a computational trick"? I know of no such trick with position :( $\endgroup$ – More Anonymous Oct 29 at 12:12
  • $\begingroup$ I mean it doesn't matter whether you use Wick rotation when doing your calculations and definitions; it doesn't have any physical significance. Anything done in the Euclidean theory can in principle be done in the Lorentzian theory. $\endgroup$ – Harry Wilson Oct 29 at 12:22
  • $\begingroup$ @HaryWilson but then the asymmetry comes from not wick rotating the spatial coordinates. Right? Like I mean, the notion of co-ordinate invariance (general covariance) seems broken to me. $\endgroup$ – More Anonymous Oct 29 at 16:03
  • $\begingroup$ After some more assimilating are you saying this is a trick that will only work in lorentzian manifold and hence, it's okay not to have such a trick with position as it "does not mean that they must be exactly the same"? In which case that makes sense and I'd only wish you'd elaborate on what you mean by "computational trick" or even reference it? $\endgroup$ – More Anonymous Oct 29 at 17:12

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