Wick rotation of real-space trajectories

Question

In CFT 2D, we often like to wick rotate the minkowski theory, to end up with an euclidean theory. Often, we then use complex coordinates to parametrize the plane, to make things easier.

For instance, the accelerated trajectory, $x^2-t^2=1$ is mapped to the circle through the Wick rotation, $t\rightarrow i \tau$, where we have $x^2+\tau^2=1$. If we consider $z=x+i\tau$ as a complex variable, we can write it as $z\bar{z}=1$. In this coordinate system, we can for instance easily see that by using a well-chosen Mobius transformation $z\rightarrow \frac{az+b}{cz+d}$ we can map the circle to the real line. This, in real space, corresponds to a conformal transformation that maps the accelerated trajectory to a static one.

However, let us know consider a slightly different trajectory/locus. Consider the equation

$$ (x+t)^3(x-t)=1 $$

This curves looks very similar to the accelerating one in real space. However, after the mapping it goes to :

$$ z^3 \bar{z}=1 $$

Solving this equation yields only two points: $-1$ and $1$.

This seems very strange, as the topology itself changes from a curve to a set of points. This is not unheard of, as it happens also in other cases (c.f. wick rotating the black hole spacetime). What I find surprising is that here, I could certainly find a change of coordinate in real space that maps my curve to the accelerated one.

But from the point of view of the complex coordinate, there is no way one could find such a transformation, since we would need to map two points to a whole curve.

It is clear here that the wick rotation "trick" breaks down. But why is that so? And how could I predict when I am allowed to use such tools? Usually, people gloss over this trick, and so I'm having trouble understanding where things went wrong.

Answer 1

There is probably a much more nuanced explanation with regards to what goes wrong with Wick rotations in such cases, but to me the issue from a practical standpoint which indicates that the mapping is problematic is that the quantity $(x+t)^{3} (x-t)$ under a Wick rotation becomes complex. Even after a Wick rotation, $x^{2} - t^{2} \rightarrow x^{2} + \tau ^{2}$ is still going to be strictly real. For a general curve such as yours that satisfies:

\begin{equation} \tag{1} f(x,t) = 1 \end{equation}

but does not remain real under a Wick rotation:

\begin{equation} \tag{2} [f(x,it)]^{*} \neq f(x,it) \end{equation}

The reformulation $f(x,it) = 1$ as you implicitly demand in your second equation implies:

\begin{equation} \tag{3} Im[f(x,it)] = 0 \end{equation}

which is in direct contradiction with equation (2). Of course, contradiction in this case means you have arbitrarily set a new constraint on your coordinates, hence understandably further reduces the number of accepted points on the plane that satisfy your equation.