  After reviewing the work of Pryce on Center-of-Mass (CoM) definitions in
special relativity, and that of Jordan and Mukunda on position operators for
relativistic particles with spin, we propose two new criteria for a CoM
candidate: associativity, and compatibility with the Poisson bracket structure.
We find that they are not satisfied by all of Pryce's definitions, and they
also rule out Dixon's CoM generalization to the curved spacetime case. We also
emphasize that the various components of the CoM position do not commute among
themselves, in the general case, and thus provide a natural entry point to the
arena of noncommutative spacetime, without the ad-hoc assumptions of the
standard paradigm.
