# The Pauli and $\text{L\'{e}vy-Leblond}$ Equations, and the Spin Current Density

## TL;DR

This paper reviews the derivation of the Pauli and Lévy-Leblond equations, emphasizing that the spin and spin current are inherently non-relativistic, and discusses the physical implications of the spin-dependent current density.

## Contribution

It clarifies that the spin current term in the non-relativistic limit is a fundamental feature of Lévy-Leblond's equation, not an ad hoc addition, and provides a simplified derivation.

## Key findings

- The spin-dependent term in the current density is non-relativistic in origin.
- Lévy-Leblond's work conclusively shows the non-relativistic nature of spin.
- Potential experimental implications of the spin current are discussed.

## 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 $\text{L\'{e}vy-Leblond}$ from a linearization of the $\text{Schr\"{o}dinger}$ equation. It was established conclusively by $\text{L\'{e}vy-Leblond}$'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 $\text{L\'{e}vy-Leblond}$ 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 $\text{L\'{e}vy-Leblond}$ equation and its current density, commenting on possibilities for experimental work that might indicate measurable consequences of the spin term in the current density.