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Three-body scattering and quantization conditions from $S$ matrix unitarity

Two methodologies have been presented in the literature which connect relativistic three-particle scattering amplitudes with lattice QCD spectra -- the ``relativistic effective field theory'' approach and the ``finite-volume unitarity'' method. While both methods have been shown to be equivalent in various works, it has not been shown how to arrive at the relativistic effective field theory results directly from $S$ matrix unitarity. In this work, we provide a simple proof of the relativistic effective field theory form of the scattering equations directly from unitarity. Motivated by the finite-volume unitarity approach, we then postulate a set of quantization conditions which relate the finite-volume energy spectra to the $K$ matrices which drive the short-distance physics in the scattering equations, obtaining all previously known results for three identical particles. This work also presents new relations which provide a new pathway to generalize the results to arbitrary systems.