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I am following Peskin & Schroeder's QFT book. And on equation 2.51, we get an expression for the free Klein-Gordon propagator for timelike intervals $x^0-y^0=t$, $x-y=0$: $$D(x-y) \sim e^{-imt}\tag{2.51}$$ as $t$ goes to infinity. He then goes on to the spacelike case, where we get: $$D(x-y) \sim e^{-mr}\tag{2.52}$$ as $r$ goes to infinity. I understand how we got to those results, but not what he means by (p.28):

"Outside the light cone, the propagation amplitude is exponentially vanishing but nonzero".

Isn't this an issue? Don't we want this to be zero outside the light cone in order to preserve causality?

He then claims that what we really have to do is calculate $[\phi(x),\phi(y)]$. But then, what was the point of the previous calculation? What is my takeaway here? I am really confused about this.

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A propagator like

$\langle 0 | \phi(x) \phi(y)|0 \rangle$

(or a time-ordered correlation) is an expression of the correlation of field values at different space-time points. Nonzero correlations over spacelike distances can exist in quantum mechanics respectively quantum field theory. The most famous example is the so called Einstein-Podolsky-Rosen paradox where two particles with spin are emitted from a location without spin and angular momentum. After a certain time one decides to measure the spin of one of both emitted particles, and immediately one knows the spin of the other emitted particle. This observation is not problematic, because it only expresses a correlation of observables at different points in a (space-like) distance.

However, if the commutator of two field operators $[\phi(x), \phi(y)]$ were non-zero at a space-like distance, then a measurement carried out at $x$ could affect another measurement carried out at another point $y$ (where $x$ and $y$ are separated by a space-like distance) then indeed causality would be violated.

Peskin & Schroeder show on p.28-29 of their QFT book that the commutator of two field operators is only non-zero between points which have time-like distance. At a space-like distance the commutator is zero. Therefore causality is preserved.

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The point of the first calculation is to show that our naive idea of what makes a QFT causal needs to be replaced with the idea that field operators commute at spacelike distances. If you look a little further you'll see that the commutator is the difference between two such "propagators", one going from $x$ to $y$, the other going form $y$ to $x$, an interpretation for this in terms of particles and anti-particles is then given, and is more clearly demonstrated in the case of the complex scalar field, as Peskin & Schroeder go on to describe.

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