Side jump and anomalous Lorentz transformation of a coherent Dirac wave packet

Abstract

Spin-orbit coupling makes the mean position of a spinning particle transform anomalously under a Lorentz boost, with an additional shift known as a side jump. In this work, we develop a trace method to derive an exact but compact transformation kernel for the probability center of a general three-dimensional Dirac wave packet under an arbitrary finite Lorentz boost. The expression includes the physics of finite momentum width, spin-momentum correlations, and coherent phase structure without a semi-classical expansion. We identify texture-dependent side jumps even at fixed initial mean momentum, spin, and orbital angular momentum, as well as a non-monotonic mass dependence for Gaussian packets at fixed momentum width. The explicit kernels further resolve the spin and orbital angular-momentum contributions for massive and massless fermions, providing a kinematic basis for covariant particle-based spin transport that retains wave-packet information.

The paper studies how the probability centroid of a coherent Dirac wave packet transforms under a finite Lorentz boost, including the “side jump” relative to the Lorentz-transformed original centroid trajectory. It treats three-dimensional positive-energy packets with finite momentum width, momentum-dependent spinors, and coherent phases, without a semiclassical expansion. A gamma-matrix trace method yields an exact momentum-space transformation kernel. The centroid shift separates into an average of the classical Pauli–Lubanski side-jump expression and a correlation between mode velocities and boost-induced time shifts. Phase gradients and spinor Berry connections enter through the latter. Energy weighting of the same kernel gives the energy centroid, and explicit kernels describe the redistribution between spin and orbital angular momentum. Gaussian examples show that finite momentum width $\sigma$ produces a finite massless side-jump scale, whereas the zero-width result scales as $1/m$. At fixed $\sigma$, the shift is nonmonotonic in mass. Helicity and phase-winding packets have probability-centroid shifts with angular dependence beyond the standard total-angular-momentum expression. Opposite spin twists produce opposite spin-parallel shifts despite identical initial mean momentum, energy, spin, and total angular momentum. These extra angular features and spin-parallel shifts disappear for the energy centroid. The results are kinematic: a dynamical spin-transport implementation is left for future work. Detailed derivation steps are deferred to a companion paper. For finite-width packets, the side jump also depends on the reference-worldline and equal-time prescription; boosted centroid trajectories need not be parallel.