We implement the worldline formalism in phase space to compute scattering amplitudes. First, the Feynman rules exhibit several useful universal features, reflecting elements of the symplectic geometry of the phase space target. Next, noncompact worldline topologies automate LSZ reductions in accordance with the boundary conditions available in phase space, provided correct identifications of the associated moduli spaces. Further, employing noncanonical coordinates cubicizes the Feynman rules and manifests gauge invariance. As a result, our phase space implementation could provide a framework optimized for computing scattering amplitudes while retaining the nice features of the original formalism. For an explicit demonstration, we compute the multi-photon Compton amplitudes up to six points in the classical limit. Compton amplitudes in Yang-Mills theory and gravity are also computed in a uniform fashion by supposing the backgrounds of nonlinearly superposed plane waves.
Joon-Hwi Kim (Sun,) studied this question.