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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?

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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?

I think that you are confusing the notion of "simultaneous measurement" (or "simultaneous diagonalizability") in quantum mechanics with the notion of "simultaneity" in relativity.

A "simultaneous measurement" in QM is a measurement that can in principle be performed simultaneously because measuring one quantity does not affect the outcome of the other (and vice-versa) so there is no need to speak of which measurement came first. This, however, tells you nothing about the chronological order of the measurements.

The point that Peskin & Schröder are trying to make is that even though propagators don't vanish outside the light-cone, that should not alarm us. What is important is that information is not transmitted outside of the light cone. Taking two points A and B to be space-like separated and measuring the fields at those two points, it shouldn't matter whether we measure $\phi(A)$ first and $\phi(B)$ after, because if it did, then one would be able to tell the difference, hence information would be transmitted outside of the light-cone. Since the two commute, no such thing happens.

meaning that other field theories, like the Dirac one, can and will violate causality.

In the next Chapter of Peskin & Schröder (Chapter 3), he shows that causality is preserved for the case of anti-commuting Dirac fermions, negating your statement.

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I'm not entirely sure, because I can't find it anywhere (and this is why I asked in the first place) but I think that no measurement in the Klein-Gordon theory only can affect another measurement outside the light-cone, meaning that other field theories, like the Dirac one, can and will violate causality.

If this is the case, then my previous example is wrong because a pair of entangled spin particles can't be described using a KG field.

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