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Closed loop in scalar quantum field or QED

Physics Asked on August 19, 2020

I want to know how to write the propagation amplitude of a 2 point closed loop in scalar quantum field or QED (i think it should be even possible as a scattering process in RQM). As an example, in Greiner QED (ch. 2, page 53) pair creation/aniquilation in terms of propagators are described, and the closed loop processs (see fig. 2.4 in page 41) is mentioned but never written down. I think this amplitude is

begin{equation}
int d^{4}x_{1}d^{4}x_{2}G_{-}(x_{2};x_{1})V(x_{2})G_{+}(x_{2};x_{1})V(x_{1})
end{equation}

where $G_{+}$ describes the propagation of a particle with positive energy and $G_{-}$ the propagation of antiparticle (negative energy particle backward in time) [the QED process should have the same structure but using the electron/positron propagators]
enter image description here

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