The Pauli and $\text{Leˊvy-Leblond}$ Equations, and the Spin Current Density

Abstract

We review the literature on the Pauli equation and its current density, discussing the progression from the original phenomenological version of Pauli to its derivation by $\text{Leˊvy-Leblond}$ from a linearization of the $\text{Schro¨dinger}$ equation. It was established conclusively by $\text{Leˊvy-Leblond}$'s work that the spin of a spin-1/2 particle such as an electron is non-relativistic in nature, contrary to what was often stated following Dirac's derivation of a relativistic wave equation, and his subsequent demonstration that Pauli's spin interaction term appeared in the non-relativistic limit. In this limit, the Gordon decomposition of the associated probability current density was found to contain a spin-dependent term. Such a term does not follow, however, from the usual derivation of the current density from the Pauli equation, although various physically motivated but otherwise ad hoc explanations were put forward to account for it. We comment on the only exception to these of which we are aware implying the spin term in the current was in fact non-relativistic in nature. However, the earlier work of $\text{Leˊvy-Leblond}$ had already shown, with no additional assumptions, that this term was a prominent feature of the current density derived from his equation. Hence, just as with the spin itself, the spin current was non-relativistic, claims to the contrary notwithstanding. We present a somewhat simplified derivation of the $\text{Leˊvy-Leblond}$ equation and its current density, commenting on possibilities for experimental work that might indicate measurable consequences of the spin term in the current density.

11 pages, no figures, corrected typos, separated section I into two sections, improved derivations in section IV ( results unchanged), references added

References in corpus (4)

• Relativistic electron wave packets carrying angular momentum
• Exotic Bohmian arrival times of spin-1/2 particles I-An analytical treatment
• On some applications of Galilean electrodynamics of moving bodies
• Twisted Localized Solutions of the Dirac Equation: Hopfion-like States of Relativistic Electrons

Cited by in corpus (9)

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• Nucleon relativistic polarization and magnetization distributions
• Times of Arrival and Gauge Invariance
• On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space
• Mechanical form factors and densities of non-relativistic fermions
• Time-dependent Dunkl-Pauli Oscillator
• Lévy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics
• The outcomes of measurements in the de Broglie-Bohm theory-II
• Enlargement of symmetry groups in physics: a practitioner's guide