Causality and dimensionality in geometric scattering

The scattering matrix which describes low-energy,
non-relativistic scattering of two species of spin-1/2 fermions
interacting via finite-range potentials can be obtained from a
geometric action principle in which space and time do not appear
explicitly arXiv:2011.01278. In the case of zero-range
forces, constraints imposed by
causality –requiring that the scattered wave not be emitted before
the particles have interacted– translate into non-trivial
geometric constraints on scattering trajectories in the geometric
picture. The dependence of scattering on the number of spatial dimensions
is also considered; by varying from three to two spatial dimensions, the
dependence on spatial dimensionality in the geometric picture is
found to be encoded in the phase of the harmonic potential that appears in
the geometric action.