Recently I realized the concept of center of mass makes sense in special relativity. Maybe it's explained in the textbooks, but I missed it. However, there's a puzzle regarding the zero mass case

Consider any (classical) relativistic system e.g. a relativistic field theory. Its state can be characterized by the conserved charges associated with Poincare symmetry. Namely, we have a covector $P$ associated with spacetime translation symmetry (the 4-momentum) and a 2-form $M$ associated with Lorentz symmetry. To define the center of mass of this state we seek a state of the free spinning relativistic point particle with the same values of conserved charges. This translates into the equations

$$x \wedge P + s = M$$
$${i_P}s = 0$$

Here $x$ is the spacetime coordinate of the particle and $s$ is a 2-form representing its spin (intrinsic angular momentum). I'm using the spacetime metric $\eta$ implicitely by identifying vectors and covectors

The system is invariant under the transformation

$$x'=x+{\tau}P$$

where $\tau$ is a real parameter

For $P^2 > 0$ and any $M$ these equations yields a unique timelike line in $x$-space, which can be identified with the worldline of the center of mass of the system. However, for $P^2=0$ the rank of the system is lower since it is invariant under the more general transformation

$$x'=x+y$$
$$s'=s-y \wedge P$$

where $y$ satisfies $y \cdot P = 0$

This has two consequences. First, if a solution exists it yields a null hyperplane rather than a line*. Second, a solution only exists if the following constaint holds:

$$i_P (M \wedge P) = 0$$

Are there natural situations in which this constaint is guaranteed to hold? In particular, does it hold for zero mass solutions of common relativistic field theories, for example Yang-Mills theory? I'm considering solutions with finite $P$ and $M$, of course

*For spacetime dimension $D = 3$ a canonical line can be chosen out of this hyperplane by imposing $s = 0$. For $D = 4$ this is in general impossible

This post has been migrated from (A51.SE)