Translational Inertial Spin Effect with Moving Particles

Abstract

Although the final conclusion of a preceding paper was incorrect, as we shall explain, the main point remains, and should entail the existence of sui generis recoil effects associated with nonzero values of curl $\ensuremath{\sigma}$ ($\ensuremath{\sigma}$ is the spin density). These should be observed by testing, not with solids as was previously proposed, but with the probability fluids associated with moving particles; this more refined type of experiment should be able to select, among the set of integrally equivalent energy-momentum tensors, the one describing locally the true or physical energy-momentum flux. In this paper it is shown, by an explicit calculation, that cylindrical type solutions of the extreme relativistic Dirac equation exist with no $z$ dependence of the wave function (and thus no ${k}_{z}$ component of the momentum) but still with a $z$ component of the Dirac probability current; as this conclusion is reached with a $t$ dependence of the wave function of strictly the form $\mathrm{exp}(\ensuremath{-}\frac{\mathrm{iWt}}{\ensuremath{\hbar}})$, there is no question of having to perform a Foldy-Wouthuysen transformation to extract the positive energy contribution (or equivalently, to use the Newton-Wigner position operator). The "transverse inertial spin effect" we predict is locally described by the flux of the Dirac current per time $\mathrm{dt}$ and surface $d\mathrm{s}$, and corresponds to the local transition probabilities between the dynamical state of the beam and a pointlike localization of the incident particles.