# Non-vanishing amplitude outside light cone doesn't violate causality? [duplicate]

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.

• Does this answer your question? Is causality violated in QFT? Nov 17, 2023 at 23:26
• Not quite. It is somewhat similar, but I am trying to understand the train of thought here. Why bother calculating the first two expressions just to shift to the commutator? Nov 18, 2023 at 3:28
• Does this answer your question? In QFT, why does a vanishing commutator ensure causality? Nov 18, 2023 at 13:30

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