Rest frame and center of mass

In a classical nonrelativistic Galilei invariant system, the uniform motion of the system is eliminated by setting the total momentum to zero. This defines the rest frame and constitutes the separation of center of mass motion. (This has nothing to do with positions, though the conservation laws imply that the center of mass will have in such a frame a time-independent position. Which position depends on the rest frame used.)

In a classical relativistic Poincare invariant system, the notion of a center of mass is dubious, except close to a nonrelativistic limit; see, e.g.,

But without any notion of a center of mass, one can still eliminate the uniform motion of the system by setting the total 3-momentum to zero. This defines the rest frame; the zero component of the total 4-momentum in the rest frame is the total mass of the system times c^2. (This reduces the 3D mass hyperboloids in 4D momentum space to an ordinary mass spectrum in 1D.)

In the quantum case (relativistic or not), position loses its meaning in the center of mass frame, since the position operator does not commute with the 3-momentum.

Arnold Neumaier (
A theoretical physics FAQ