I'm studying QFT, and Peskin is his book takes a couple of paragraphs to talk about causality in QFT, using the Klein-Gordon field as an example. The book says on p. 28:
To really discuss causality, however, we should ask whether [...] a measurement performed at one point can affect a measurement at another point whose separation from the first is spacelike. [...] we should compute the commutator; if this commutator vanishes, one measurement cannot affect the other.
and then proceeds to calculate the said commutator for both spacelike and timelike intervals, the first one is zero, the second one is nonzero, so
no measurement in the Klein-Gordon theory can affect another measurement outside the light-cone.
All the calculations in Peskin's book are correct, but I'm not understanding his claim: if a commutator is zero, shouldn't this mean that the measurements can be done simultaneously? If the said measurements are spins of entangled particles, doesn't this mean that my measurement is affecting the other one on a spacelike interval, therefore violating causality?