What does this equation mean?

The first, the covariant centre of energy, transforms simply as the spatial part of a four-vector under a change of Lorentz frame — exactly the property $X^\mu(u,\Sigma)$ was built with above — but its three spatial components fail to commute with each other once spin is present, by an amount proportional to the spin itself. The second, constructed by Newton and Wigner for exactly this purpose and reproduced in Pryce’s classification as a distinct case, has strictly commuting components, [$X_{\mathrm{NW}}^i$, $X_{\mathrm{NW}}^j$] = 0 , the property any operator called “position” ought to have if it is to support ordinary probability densities and wavefunction localization at all, at the price of no longer transforming as a four-vector: an observer in a different inertial frame does not simply Lorentz-boost the Newton–Wigner operator into the new frame’s Newton–Wigner operator, because the two are built from a boost generator that itself carries spin-dependent, frame-tied structure [ 2 ] . Fleming’s manifestly covariant reformulation makes the trade-off explicit rather than incidental: commuting components and simple four-vector transformation cannot both be had from the same operator whenever spin is nonzero, and every choice in the literature buys one property by sacrificing the other [ 4 ] .