The angular momentum 4-tensor has 6 independent components, three angular momentum components and three new guys. Some call these new guys the 'boosts', but since they are the conjugate momentum of the Lorentz Transformation (aka the actual boosts) I think 'boost momentum' is a better name.

These quantities are related to energy and momentum by (up to a sign, depending on the text)

$\vec K=\frac{E}{c} \vec x-ct \vec p$

While conservation of angular momentum is pretty useful for a lot of problems, conservation of the boost momentum seems to be a loose end.

It seems like conservation of the boost momentum only serves to ensure continuity of motion. A particle undergoing a discontinuous jump (hopping from one point to another in an unphysical way) could conserve energy, momentum, and angular momentum, but it wouldn't conserve boost momentum.

Are there physical problems where conservation of boost momentum could be as useful as conservation of angular momentum?