  In particular, if we insist that our particle is a point particle, then if its trajectory should happen to intersect P it will simply disappear. What could cause the extremely fortuitous result that the trajectory strikes precisely at P? The emergence of some other charged particle, with just the right charge and trajectory, from P (on L-). And we are no longer bound by any conservation laws: the bare singularity can both swallow and produce whatever mass or change or momentum we like. So if we insist on point particles, then we have to take account of the singularity, and that again saves the day.