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REVIEW 2 major objections 4 minor 62 references

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

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Pith's one-line read This paper argues that the spin term in the electron probability current is an intrinsically non-relativistic effect, derivable directly from the Schrödinger-Pauli and Lévy-Leblond equations.

desk verdict Useful teaching review, but the spin current's 'no additional assumptions' derivation is oversold: the continuity equation alone does not force the spin term. read the letter →

arxiv 1908.03276 v2 pith:MTILELDR submitted 2019-08-08 quant-ph

James M. Wilkes This is my paper · ORCID
classification quant-ph
keywords PauliequationLévy-Leblondspincurrentdensitynon-relativisticprobabilityGordondecompositioncontinuitytrajectoryinterpretation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that the spin current density $\mathbf{J}_{\mathrm{spin}} = (\hbar/2m)\nabla\times(\psi^{\dagger}\boldsymbol{\sigma}\psi)$ is not a relativistic leftover from the Dirac equation but an intrinsic feature of non-relativistic spin-1/2 quantum mechanics. The author shows that the term appears directly in the current density of the Schrödinger-Pauli equation and of the Lévy-Leblond equation, a linearization of the Schrödinger equation, without extra assumptions. If the argument is right, textbook presentations of the Pauli current density are incomplete, and the extra term should be included whenever probability or charge currents for electrons are computed. The author also points to proposed time-of-flight experiments where the spin term would leave a measurable signature.

What carries the argument

The mechanism is the linearization of the non-relativistic Schrödinger equation by an auxiliary spinor. Defining $\chi = -(1/2mc)(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi$ converts the second-order equation into a pair of first-order equations, the Lévy-Leblond equation, and computing the continuity equation from this pair before eliminating $\chi$ exposes a surface term $\nabla\cdot(\psi^{\dagger}\boldsymbol{\sigma}\chi + \chi^{\dagger}\boldsymbol{\sigma}\psi)$; substituting for $\chi$ and using $\sigma_i\sigma_j = \delta_{ij}+i\epsilon_{ijk}\sigma_k$ turns that surface term into the curl $(\hbar/2m)\nabla\times(\psi^{\dagger}\boldsymbol{\sigma}\psi)$. The auxiliary spinor is the device that keeps the spin degree of freedom in the current rather than hiding it in the squared Hamiltonian.

What would settle it

Compute or measure the arrival-time distribution of spin-polarized electrons in a cylindrical waveguide for the specific initial state discussed in the paper: if the data match the prediction that ignores $\mathbf{J}_{\mathrm{spin}}$ and show no spin-dependent cutoff, the claim that the spin current is part of the non-relativistic probability current is falsified. Alternatively, a Galilean-covariant non-relativistic derivation producing a different conserved current satisfying the same continuity equation would break the uniqueness premise.

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Extended reading notes

Core claim

The central discovery is that the spin current $\mathbf{J}_{\mathrm{spin}}$ is derivable from non-relativistic wave equations alone. Working from the free-particle Schrödinger-Pauli equation $(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^2\psi = 2m\hat{E}\psi$ and introducing an auxiliary spinor $\chi = -(1/2mc)(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi$, the conserved current obtained before any expansion of $(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^2$ has the form $\mathbf{J} = -(i\hbar/2m)[\psi^{\dagger}\nabla\psi - (\nabla\psi)^{\dagger}\psi] + (\hbar/2m)\nabla\times(\psi^{\dagger}\boldsymbol{\sigma}\psi)$; the curl term emerges from the Pauli matrix identity $\sigma_i\sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k$. The same result follows from the Lévy-Leblond equation, whose bispinor structure yields $\mathbf{J} = -c(\psi^{\dagger}\boldsymbol{\sigma}\chi + \chi^{\dagger}\boldsymbol{\sigma}\psi)$ and then the same final current, including the interaction term when minimal coupling is used. In the author's reading, Lévy-Leblond's linearization had already established this in 1967, and later derivations that invoked magnetization or other physical pictures were unnecessary.

Load-bearing premise

The argument depends on the premise that the non-relativistic current is the unique limit of the relativistic Dirac current; without that imported uniqueness, the non-relativistic continuity equation alone would allow adding an arbitrary divergenceless term, so the derivation would show consistency rather than necessity.

Editorial extensions

If this is right

  • The Pauli probability current for an electron should read $\mathbf{J} = -(i\hbar/2m)[\psi^{\dagger}\nabla\psi - (\nabla\psi)^{\dagger}\psi] - (q/m)\mathbf{A}\psi^{\dagger}\psi + (\hbar/2m)\nabla\times(\psi^{\dagger}\boldsymbol{\sigma}\psi)$, not just the first two terms.
  • Because the spin term is a pure curl, it does not affect $\nabla\cdot\mathbf{J}$ or the continuity equation, so its consequences show up mainly in non-local or trajectory-sensitive measurements.
  • For charged particles the spin term contributes to the charge current and is in principle observable in interference, time-of-flight, or transport experiments.
  • In trajectory-based interpretations of quantum mechanics where particle velocity is proportional to the current density, the spin term changes predicted trajectories and arrival-time distributions for spin-polarized electrons.
  • The Lévy-Leblond equation is a valid non-relativistic starting point from which the Pauli equation follows by minimal coupling, so both spin and spin current need no relativistic justification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the spin current is genuinely non-relativistic, transport calculations that omit it are missing a term of the same order as the usual paramagnetic current; this may matter for spintronics or conductivity estimates, though the paper only speculates on that connection.
  • The paper's reliance on relativistic uniqueness to fix the non-relativistic current suggests that a purely non-relativistic uniqueness proof, based for example on Galilean covariance and locality, would settle the matter without importing Dirac theory; the paper does not provide one.
  • An experiment with electrons in a cylindrical waveguide measuring arrival-time cutoffs for spin states perpendicular to the axis would test the physical status of the spin current; the paper discusses this possibility but does not perform the experiment.
  • The same curl structure suggests a rotation or angular-momentum interpretation of the spin current that might link it to spin angular momentum transport; the author reports such an interpretation but does not endorse it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper reviews the history of the Pauli and Lévy-Leblond equations and argues that the spin contribution to the probability current density, J_spin = (ℏ/2m)∇×(ψ†σψ), is an intrinsically non-relativistic phenomenon. The author provides a derivation of this spin current from the free-particle Schrödinger-Pauli equation using an auxiliary spinor, a derivation of the Lévy-Leblond equation via linearization of the Schrödinger equation, and a derivation of the spin current from the coupled Lévy-Leblond equations with electromagnetic couplings. The paper concludes with a discussion of potential experimental consequences of the spin current in Bohmian mechanics and time-of-flight experiments. The central claim is that this term follows 'with no additional assumptions' from the non-relativistic theory.

Significance. If the claim were fully established, the paper would be a valuable pedagogical contribution, showing that the spin current is not an artifact of the Dirac equation's non-relativistic limit. The derivations are clear, self-contained, and correct in outline; in particular, the linearization of the Schrödinger equation to obtain the Lévy-Leblond equation is a nice piece of exposition, and the final expression (50) is the known result. However, the paper's stronger interpretive claim—that the non-relativistic equations alone fix the spin current uniquely—is not supported by the derivations as they stand. The continuity equation determines only the divergence of the current, and the paper itself notes the freedom to add divergenceless terms. Thus the significance of the work is real but narrower than claimed; the algebraic results are correct and useful, but the central assertion of uniqueness requires substantial qualification.

major comments (2)
  1. [Abstract; Sections II and VII] The claim that the spin current follows 'with no additional assumptions' is not supported by the derivations presented. Section II explicitly notes (following Eq. (8)) that a divergenceless term can always be added to a conserved current. The derivation in Section III, Eqs. (16)-(20), exhibits an expression whose divergence equals -∂ρ/∂t, but the ordinary current J0 = -(iℏ/2m)[ψ†∇ψ - (∇ψ)†ψ] already satisfies the same continuity equation for the free Schrödinger-Pauli equation (5), since (σ·p)^2 = p^2. The derivation therefore shows consistency, not uniqueness. The only uniqueness argument cited is Holland's relativistic result (Refs. 22-24), which the text describes in Section II as inherited from the Dirac limit. Because that uniqueness is imported from the relativistic theory, the non-relativistic derivation does not stand alone. This is load-bearing for the abstract's central claim; please either supply a non-relativistic uniqueness criterion (e.g., from a Noether symmetry of the non-relativistic action) or revise the claim to state that the spin current is a natural, but not uniquely forced, consequence of the non-relativistic equations.
  2. [Section IV C, Eq. (38)] In Eq. (38), the factorization of the linearized equation is algebraically incorrect. From the preceding line, the matrix is [σ·p, 2bmc; (2a/c)E, -σ·p]. Writing it as (1/c) times a matrix with entries (c σ·p, 2bmc, 2aE, -c σ·p) gives, upon multiplying back by c, off-diagonal entries (2bmc, 2aE) instead of (2bmc, (2a/c)E). With a = -1/2, b = 1, the first row then yields σ·p ψ + 2m χ = 0 after division by c, which conflicts with Eq. (41a), σ·p ψ + 2mc χ = 0. The top-right entry in the factored form should be 2bmc^2, or equivalently the factor 1/c should apply only to the energy term. Since Eqs. (41) and (42) are used in the later derivation of the spin current, this typo should be corrected.
minor comments (4)
  1. [References] Reference [43] lists the year as '20067'; this should read '2007'.
  2. [Eq. (20)] In Eq. (20), the first line contains a stray parenthesis: '[(∇iψ)†)ψ' should be '(∇iψ)†ψ'.
  3. [Section IV A] The text says that the five matrices Bμ satisfy the Dirac algebra; since the conventional Dirac algebra is often presented with four gamma matrices plus γ5, a brief remark about the counting would help readers.
  4. [Section II] The assertion that the spin current 'must be a feature of the current density for any spin-1/2 particle described by the Dirac equation' would benefit from an explicit citation to the Gordon decomposition or to a standard textbook treatment, rather than a passing reference to Sakurai's discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spin current derivation is algebraically self-contained, though uniqueness is imported from relativistic theory.

full rationale

The paper derives the spin current density from the Schrödinger-Pauli and Lévy-Leblond equations by computing ∂ρ/∂t and identifying a conserved current. The spin term is not inserted as an input; it emerges from the algebraic relation χ = -(1/2mc)(σ·p)ψ and the Pauli matrix identity. The derivation is therefore not circular in the sense of assuming what it proves. The paper itself notes in Section II that the continuity equation alone permits adding a divergenceless term, so the derivation alone shows consistency, not uniqueness. To claim uniqueness, it imports Holland's theorem that the non-relativistic current is the limit of the unique relativistic Dirac current. That is an external assumption and arguably overstates 'no additional assumptions', but it is not a circular reduction: the non-relativistic equations do not define the spin current in terms of themselves, and no fitted parameters are relabeled as predictions. There are no load-bearing self-citations. The core algebraic derivations in Sections III and V are self-contained. Score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No numbers are fitted to data; the auxiliary spinor and the minimal-coupling assumption are the main ad hoc elements. The uniqueness of the spin term is borrowed from prior relativistic analysis, which is the most significant external input. The Clifford-algebra matrices are standard math, not new postulates.

assumptions (5)
  • domain assumption The Schrödinger equation and the standard quantum-mechanical probability density rho = psi-dagger psi are valid starting points.
    The derivations begin from Eq. (3) and use rho = psi-dagger psi throughout; this is the standard non-relativistic framework.
  • domain assumption The minimal coupling prescription replaces p by p - qA and E by E - qphi.
    Used in Eqs. (6), (42), and (44); the paper itself cautions in the conclusion that these potentials are inherently relativistic, which is a recognized limitation.
  • ad hoc to paper An auxiliary spinor chi is introduced to linearize the second-order equation.
    Section III defines chi = -1/(2mc)(sigma dot p) psi; Section V uses the L-L constraint chi = -1/(2mc) sigma dot (p - qA) psi. The spin term emerges only through this linearization.
  • domain assumption Holland's uniqueness theorem (Refs. 22-24) that the non-relativistic current is the unique limit of the relativistic Dirac current.
    Invoked in Section II to argue the spin term is not ambiguous; without it, the continuity equation permits adding divergenceless terms.
  • standard math The existence of five 4x4 matrices satisfying the Dirac anticommutation relations (Clifford algebra).
    Section IV constructs them from Pauli matrices; this is standard linear algebra.
invented entities (1)
  • Auxiliary spinor chi defined as -1/(2mc) sigma dot (p - qA) psi
    purpose: linearizes the second-order Schrödinger-Pauli and Lévy-Leblond equations so the spin current term appears in the continuity equation
    It has no observable consequences by itself and is a bookkeeping device. Its introduction is the key step that makes the spin current appear, so it is a load-bearing construct used in this paper's derivation.

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Pith. "Pith review of The Pauli and $\text{L\'{e}vy-Leblond}$ Equations, and the Spin Current Density." pith.science (2026). https://pith.science/paper/MTILELDR

@misc{pith2026190803276,
  author       = {Pith},
  title        = {Pith review of: The Pauli and $\textL\'evy-Leblond$ Equations, and the Spin Current Density},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTILELDR}},
  note         = {Machine review of arXiv:1908.03276}
}
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.

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