In Peskin and Schroeder pg. 27-28, they discuss Klein-Gordon theory and causality. For a spacelike separation $(x-y)^2 < 0$, they show that $$\langle 0| \phi(x)\phi(y) |0\rangle \neq 0$$
They go on to say that this doesn't actually break causality. Rather, what one should be looking at isn't whether particles can propagate over space-like intervals, but whether space-like separated measurements can affect one another. Hence, they argue that to understand the measurements of the field $\phi(x)$, one should be trying to understand the the commutator $[\phi(x),\phi(y)]$.
This last statement is opaque to me. In the QM setting, the ordering of the operators can be physically interpreted as one being applied first in time prior to the second. However, this doesn't make sense in the QFT context because one applies the operator at a specific point in space-time; flipping the order is not equivalent to changing the time-order of making the measurements.
- In the QFT context, what is meant by the "measurement of a field," in analogy to the QM measurement of some operator?
- Why is the commutator the object of choice when wanting to understand causality? What is the physical interpretation of the commutator here, particularly with respect to measurements of $\phi(x)$?