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So, the uncertainty principle states that one can not measure momentum and position with accuracy simultaneously. However, we know from relativity that simultaneously is something frame dependent in nature, so how can we reconcile these two in relativistic quantum mechanics or even QFT?

I know that in QFT we define observables in different points in spacetime such that their Lie bracket is zero, but I'm not really sure how to go from this to an answer.

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    $\begingroup$ That is why many authors call it "jointly" measurable (in QFT and QM) if two observables commute. Say on a finite-dimensional Hilbert space, $[A,B]=0$ for hermitian $A,B$ is equivalent to the fact that there is another self-adjoint observable $C$ and two functions $f,g$ s.t. $A=f(C)$ and $B=g(C)$. The operational meaning is then to measure the observable corresponding to $C$ and apply the functions $f,g$ to the outcome. $\endgroup$
    – Tobias Fünke
    Commented Oct 10, 2023 at 13:55
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    $\begingroup$ "the uncertainty principle states that one can not measure momentum and position with accuracy simultaneously" - also this is a misleading statement. The HUP says that it is not possible to prepare a state such that the product of variances (of a series of measurements) in that state is arbitrary small. $\endgroup$
    – Tobias Fünke
    Commented Oct 10, 2023 at 14:01

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I think you are conflating two separate issues. Lack of simultaneity in SR is an effect which increases with distance along the direction of travel between two reference frames moving relative to each other. There is nothing to prevent observers from agreeing that two events are simultaneous if they occur in the same spot, as would be the case if you were trying to measure the momentum and position of an electron, say. It is rather like saying that you cannot face north and south simultaneously- that has nothing to do with simultaneity in SR.

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