We know that parallel to scalar QED, a common formalism that describes a massive particle coupled to electromagnetism is through a relativistic worldline formalism, which writes

$$\mathcal{S}=\int\ ds\ \left\{-m\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}}+eA_{\mu}\frac{dx^{\mu}}{ds}\right\}.$$

My question is how one reproduces the amplitudes for processes such as the scalar Compton scattering.

Below is my attempted solution, taking Compton scattering as an example.

In this case the massive particle is treated as a source, and to the lowest order, by assuming a larger mass, the 4-velocity of the point particle can be taken as constant. Then the states are specified solely by the photon momenta. Then the lowest order amplitude should just be the matrix element of the coupling term between in and out states (which are essentially specified by $k$ and $k'$ of incoming and outgoing photons. Is this correct, and how does one evaluate the scattering amplitude in this formalism in general?



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