What does this equation mean?

The Newton–Wigner operator was not built for elegance; it was built to repair a specific, observable pathology in the naive Dirac position operator $\hat{\mathbf x}$ , the coordinate that appears literally in the Dirac equation. Because a free Dirac wavepacket generically mixes positive- and negative-energy plane-wave components, $\hat{\mathbf x}$ 's Heisenberg equation of motion picks up, alongside the ordinary uniform drift, an oscillating term with no classical counterpart: an angular frequency $\omega_Z = \frac{2Mc^2}{\hbar}$. set purely by the rest mass. For an electron this is $\omega_Z$ $\approx$ 1.55$\times$10^{21}\ $\mathrm{rad\,s^{-1}}$ , an oscillation period of roughly four zeptoseconds in the particle’s own rest frame — a boosted lab observer would clock a longer, time-dilated period by the ordinary factor $\gamma$ , coordinate time and proper time parting ways in exactly the usual manner — with an amplitude of order the reduced Compton wavelength $\hbar$/($m_e$c) , the same length scale, twice $\rho_M$ for a spin-half particle, that bounded the Møller disk above. Schrödinger first noticed this trembling motion, zitterbewegung, in the Dirac coordinate’s equations of motion; Foldy and Wouthuysen’s 1950 canonical transformation is what removes it, by transforming to exactly the representation in which the position operator is the Newton–Wigner operator and the oscillating term is absorbed into the difference between $\hat{\mathbf x}$ and $\mathbf X_{\mathrm{NW}}$ [ 3 ] .