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I'm thinking about the Wick rotation. My question may be similar to this one but I don't think it's a duplicate, though you can judge that.

Suppose we take the Wick rotation as an indication that spacetime truly is a Riemannian, not pseudo-Riemannian, manifold. Purely as a theoretical maneuver, it really feels as though it unravels two "knots" by showing that there really was no knot to begin with, just a looped and twisted string.

The first knot is the hyperbolic character of the spacetime metric. Although this structure was a giant leap forward historically, still, there is this quirk about it, viz., the duality between space and time. In a Lorentzian manifold, not all directions are created equal. There is an inherent discontinuity upon crossing the light cone. The spirit behind relativity was the conviction that the laws of physics should be the same in all reference frames; wouldn't the same spirit lead us to hope even that the metric would be independent of direction?

The second knot is quantum interference. In a Lorentzian theory, each path does not have its own probability, just a geometric contribution of $e^{iS/\hbar}$ to the whole. Whereas in the Euclidean theory, each path has a true independent probability of $e^{-S/\hbar}$, such that we can really hope that quantum theory is the statistical limit of some underlying one.

These are reasons to wish that the Wick rotation might be more than a mathematical trick. So, what issues would this raise? I imagine there could be quite a few, such as the fact that there would then effectively be a priveleged direction in 4D space (the one that time replaces), though maybe you could say the big bang explains that by inducing a natural geodesic flow outward from the singularity. And presumably explaining causality would be difficult; I heard this is dealt with by the Osterwalder-Schrader theorem, but if these axioms refer to the pre-rotated Lorentzian theory, I guess they would be lost if you claim the theory was Euclidean from the get-go? Basically, I'm wondering if there's something that inherently precludes my wishful thinking, or what the theoretical obstacles would be.

(To clarify, I'm not thinking about Euclidean quantum gravity where you integrate over possible metrics -- I'm imagining that the single curved Riemannian metric + QFT could be the statistical limit of a theory that still takes place within a single manifold.)

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    $\begingroup$ Related question on MO.SE: mathoverflow.net/q/165304/13917 $\endgroup$
    – Qmechanic
    Sep 28, 2022 at 17:39
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    $\begingroup$ How does your question relate to classical electrodynamics and the finite propagation speed of light the Lorentz metric was introduced for? $\endgroup$
    – kricheli
    Sep 28, 2022 at 19:02
  • $\begingroup$ @kricheli Yeah, I really don't know, and on the surface it wouldn't seem to respect the speed of light. That's what I meant by causality. Maybe that's the downfall of my line of thinking -- that the Euclidean theory only respects causality if it's derived from a Lorentzian theory to begin with. But I don't know much about that so I was hoping for an expert opinion. $\endgroup$ Sep 29, 2022 at 3:56

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Let me try to untie your first knot by more carefully distinguishing some of the principles going into SR.

The spirit behind relativity was the conviction that the laws of physics should be the same in all reference frames; wouldn't the same spirit lead us to hope even that the metric would be independent of direction?

That the laws of physics should be the same in all reference frames implies an invariance requirement for the equations involved, i.e. we have to use tensors, the quantities we use in our equations have to transform according to certain laws when changing between frames of reference. However, this does not imply that the quantities in our equations, in particular the metric tensor, be it in Lorentzian or Euclidean space, have to be equal in all reference frames.

Compare it to material laws in continuum mechanics or electrodynamics: We are using tensors to fulfill invariance requirements. I.e. in $$ D_i = \varepsilon_{ij} E_j $$ when transforming with an orthogonal tensor $A$ the vectors transform as $D'_i = A_{ij} D_j$, $E'_i = A_{ij} E_j$ and the tensor according to $\varepsilon'_{ij} = A_{ik}\varepsilon_{kl}A_{lj}$ such that the material law is invariant under this transformation. This tells us nothing about what $\varepsilon$ looks like, it could be $\varepsilon = \text{diag}(1,1,3)$ in one frame of reference, $\varepsilon = \text{diag}(3,1,1)$ in another, and even uglier (non-diagonal) in others. The fact that a tensor in a material law is isotropic is a property of the material, obtained from experimental observation that the material looks/behaves the same under all directions, and is not derived from invariance properties of the material law.

Now where are experimental evidence or reasons to put in the additional assumption of a positive sign in the time-component of the metric? The Wick rotation is only a technique for calculation...

As concerns your second point, I don't know about quantum interference, but a non-expert's two cents on the matter: Why bother about that $i$ and one quantity being a probability or not? If this entails not being consistent with Einstein, Lorentz, Minkowski, Michelson-Morley etc., then I think you need to get your priorities in order... :)

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