Deriving the anomalous Lorentz transformation and side jump using a trace method

Abstract

The side jump in the anomalous Lorentz transformation, arising from the spin-orbit interactions, plays important roles in various intriguing physics, such as chiral vortical effects and spin polarization. In this work, the side jump of the spin-half particles is visualized and evaluated ab initio within Dirac theory, through a comprehensive study of the density ${\psi }^{†}\psi$ and position $\int d^{3}x{\psi }^{†}\mathbf{x}\psi$ of a wave packet under Lorentz transformation. A compact gamma-matrix representation of the Lorentz boost matrix of a Dirac spinor, $S(\mathbf{\eta })=h_{\mathbf{\eta }}^{\mu }{\gamma }_{\mu }{\check{u}}^{\nu }{\gamma }_{\nu }$, has been found. Based on this expression, a trace method has been developed to derive the anomalous Lorentz transformation of the center of the probability density of a spin-half fermion wave packet, and a compact analytical expression for the side jump has been obtained through a transparent and tractable derivation. The momentum width plays an important role in the side jump. For example, for Gaussian wave packets, increasing the width suppresses the displacement, while the side jump at fixed width depends nonmonotonically on mass, reaching a maximum at a finite mass. In the zero-width limit, the side jump takes a compact form perpendicular to both the Pauli-Lubanski vector and the boost direction, recovering the known result for a chiral particle.

arXiv comment9 pages, 3 figures; major revision with a new trace-method derivation and expanded results

Cited by in corpus (1)

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