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I know this is a broad question, but I hope someone can help me understand this. I have looked at lectures and read many articles, but I find it hard to understand them (I could understand the rest of what they said).

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    $\begingroup$ Plan for you: 1) Understand why causality means that fields commute when evaluated in spacelike-separated points (and why we do not require the transition amplitude to vanish) Hint: think about the notion of the measurement. 2) Take a look at the calculation for the quantised fields and make sure that the fields indeed behave as expected. Try Peskin and/or other books on QFT. $\endgroup$ – mavzolej Nov 16 '16 at 3:37
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Basically, you need the commutator of field operators to vanish for spacelike intervals. That is $$ [\phi(x),\phi(y)]=0 $$ for spacelike $x$ and $y$. This means the field operators are uncorrelated outside the lightcone, which is causal.

Moreover, the implication of this is that the transition amplitude $$ \langle0\rvert \phi(x)\phi(y)\lvert0\rangle $$ is antisymmetric in $x$ and $y$. This means, if a particle can travel beyond the lightcone, the antiparticle travels backward in the path so that the net amplitude is zero.

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