  A general characterization of lower and upper threshold configurations for
two particle reactions is determined under the assumptions that the single
particle dispersion relations E(p) are rotationally invariant and monotonic in
p, and that energy and momentum are conserved and additive for multiple
particles. It is found that at a threshold the final particle momenta are
always parallel and the initial momenta are always anti-parallel. The
occurrence of new phenomena not occurring in a Lorentz invariant setting, such
as upper thresholds and asymmetric pair production thresholds, is explained,
and an illustrative example is given.
