Imagine that a particle with negative mass has been discovered. It is known to obey both the equivalence principle and Newton's second law. This means that the particle is a subject to repulsive gravitational force from ordinary matter but also accelerates towards any repulsive force (resulting in the infamous runaway motion).
Let's also say that the particle is able to interact with ordinary matter in the process of "nullification" (particless of oposite energy get close enough to each other and disappear). The problem is that this interaction would violate the conservation of momentum every time the two nullifying particles didn't have exactly identical velocities, which is practically never the case. And from what I understand, conservation of momentum is pretty important to all of our current theories.
My question is: Is there a mechanism that would allow for nullification but also leave the conservation of momentum unviolated and wouldn't break GR and QM? (Energy conditions may be broken.)
Originally I wanted to post this question to Worldbuilding but I decided it's too technical and generally better suited for this site, even if it's quite hypothetical.
Nullification (as it is defined in the last paragraph of Wikipedia's Runaway motion) is the interaction where particless of equal mass of opposite signs “cancel each other out”, completely disappearing from existence.
However, at least in classic physics, the probability that two particles with exactly opposite energy and momentum will find each other in finite time is zero. Therefore nullification would either never happen, or it would break some of the fundamental symmetries.
The answer should either show that this premise is wrong (eg. “in QM the uncertainty of momentum is large enough to have the particles match exactly with a reasonable probability”), propose an interaction that can cause the energy and momentum of the two particles to align by interactions with the outside world, or show that no such interaction is possible and therefore nullification cannot happen.