  Crossing symmetry asserts that particles are indistinguishable from
anti-particles traveling back in time. In quantum field theory, this statement
translates to the long-standing conjecture that probabilities for observing the
two scenarios in a scattering experiment are described by one and the same
function. Why could we expect it to be true? In this work we examine this
question in a simplified setup and take steps towards illuminating a possible
physical interpretation of crossing symmetry. To be more concrete, we consider
planar scattering amplitudes involving any number of particles with arbitrary
spins and masses to all loop orders in perturbation theory. We show that by
deformations of the external momenta one can smoothly interpolate between pairs
of crossing channels without encountering singularities or violating mass-shell
conditions and momentum conservation. The analytic continuation can be realized
using two types of moves. The first one makes use of an $i\varepsilon$
prescription for avoiding singularities near the physical kinematics and allows
us to adjust the momenta of the external particles relative to one another
within their lightcones. The second, more violent, step involves a rotation of
subsets of particle momenta via their complexified lightcones from the future
to the past and vice versa. We show that any singularity along such a
deformation would have to correspond to two beams of particles scattering off
each other. For planar Feynman diagrams, these kinds of singularities are
absent because of the particular flow of energies through their propagators. We
prescribe a five-step sequence of such moves that combined together proves
crossing symmetry for planar scattering amplitudes in perturbation theory,
paving a way towards settling this question for more general scattering
processes in quantum field theories.
