  Proposed in 1954 by Gell-Mann, Goldberger, and Thirring, crossing symmetry
postulates that particles are indistinguishable from anti-particles traveling
back in time. Its elusive proof amounts to demonstrating that scattering
matrices in different crossing channels are boundary values of the same
analytic function, as a consequence of physical axioms such as causality,
locality, or unitarity. In this work we report on the progress in proving
crossing symmetry on-shell within the framework of perturbative quantum field
theory. We derive bounds on internal masses above which scattering amplitudes
are crossing-symmetric to all loop orders. They are valid for four- and
five-point processes, or to all multiplicity if one allows deformations of
momenta into higher dimensions at intermediate steps.
