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Wikipedia distinguishes between three kinds of propagators for a scalar field:

  • The Retarded propagator's contours have $\mathrm{Im}(k^0)>0$ on both poles, so its limit is completely in the first and second quadrants.
  • The Advanced propagator's contours have $\mathrm{Im}(k^0)<0$ on both poles, so its limit is completely in the third and fourth quadrants.
  • The Feynman propagator's contour has $\mathrm{Im}(k^0)<0$ at the $k_0=-\sqrt{\vec{k}^2+m^2}$ pole and $\mathrm{Im}(k^0)>0$ at the $k_0=+\sqrt{\vec{k}^2+m^2}$ pole, so its limit lies completely in the first and third quadrant.

What's the relation between the choice of contours and the causality/light cones of the propagators? In other words, why are the first two propagators retarded and advanced, and what's their connection to the time-ordered operator product?

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