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The convolution $G_{ret}*f$ of the retarded propagator $G_{ret}$ with a source term $f$ which vanishes sufficiently far in the past is the unique solution of the inhomogeneous Klein-Gordon equation with source term $f$ which vanishes in the far past. It is necessarily a nontrivial superposition of positive and negative energy solutions at all times when it ...

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Conventionally in QFT, particles and antiparticles are defined with positive energy $k^0\geq 0$ only. (Recall that would-be negative energy states are reinterpreted as matter/antimatter of the opposite kind in order to make the vacuum stable.)

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In principle, all the information of the QFT is encoded in the $n$-point functions. This means that, once you know the correlation functions, you know the Hilbert space, the fields and their algebra (modulo a unitary transformation). This is known as the Wightman reconstruction theorem, though AFAIK, it is not known whether the theorem holds for Yang-Mills ...

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