# In light of Wick rotations is position and time on the same footing in QFT?

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).

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.