The Non-Relativistic Limit of Keldysh Spinors

A. Jourjine

REVIEW 3 major objections 5 minor 7 references

The Non-Relativistic Limit of Keldysh Spinors

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Keldysh spinors, whose action is the negative of the Dirac action, still have a non-relativistic limit governed by the positive-definite Pauli Hamiltonian.

desk verdict A correct standard reduction whose central claim needs restating: the positive Pauli Hamiltonian governs the charge-conjugate of the Keldysh component, not the component itself. read the letter →

arxiv 2501.04514 v1 pith:NTBIEZZS submitted 2025-01-08 hep-ph quant-ph

A. Jourjine This is my paper · ORCID

classification hep-phquant-ph

keywords Keldyshspinorsnegative-actionfermionsnon-relativisticlimitPauliHamiltonianDiracequationFoldy-Wouthuysentransformationpositive-definiteelectromagneticcoupling

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

The paper asks whether fermionic fields whose action is the negative of the Dirac action—called Keldysh spinors—can have a sensible quantum-mechanical limit. It argues yes: interacting with a classical electromagnetic field, their non-relativistic limit is the Schrödinger equation with the positive-definite Pauli Hamiltonian, just as for ordinary Dirac spinors. The only difference is the sign of the electromagnetic coupling. Because negative-action spinors show up in non-equilibrium quantum field theory, flavor-mixing models, and black-hole information studies, showing that they still produce ordinary quantum mechanics is a useful consistency check for all those settings.

What carries the argument

The central objects are Keldysh spinors: spinor fields that obey the Dirac equation but have the negative of the Dirac action and Hamiltonian. The load-bearing mechanism is the phase choice $e^{+imt}$ for Keldysh positive-energy states, together with the creation/annihilation operator swap taken from earlier work, which turns the formally negative Hamiltonian into a positive-definite one. The reduction of $(\vec{\sigma}\cdot\vec{\pi})^{2}$ then converts both the Dirac and Keldysh two-component equations into the same Pauli Hamiltonian, with the charge sign as the only distinction.

What would settle it

Solve the exact four-component equation with the negative-action sign for a constant magnetic field and compare its Landau levels with the Pauli-level prediction of Eq. (8); a mismatch in the level spacing would falsify the claimed non-relativistic limit.

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Extended reading notes

Core claim

The central claim is that the non-relativistic limit of Keldysh spinors coupled to a classical U(1) electromagnetic field is described by the Schrödinger equation with the same positive-definite Pauli Hamiltonian as the Dirac case. The derivation parallels the usual Dirac one, with one crucial difference: for Keldysh spinors the fast vacuum phase is $e^{+imt}$ rather than $e^{-imt}$, because their positive-energy states are the negative-energy solutions of the Dirac equation. After eliminating the small components and reducing $(\vec{\sigma}\cdot\vec{\pi})^{2}$, both cases yield the Pauli Hamiltonian, differing only in the sign of the charge coupling. The paper therefore concludes that Dirac and Keldysh spinors are locally indistinguishable in low-energy experiments, and that bringing the two species together gives a vanishing scalar coupling, a vanishing electric vector coupling, and only elastic magnetic scattering with no energy transfer.

Load-bearing premise

Everything rests on treating the positive-energy states of the negative-action spinor as the Dirac equation's negative-energy states; if that identification is wrong, the non-relativistic limit would yield a negative-definite Hamiltonian.

Editorial extensions

If this is right

In the non-relativistic limit, Dirac and Keldysh spinors obey the same positive-definite Pauli Hamiltonian, so low-energy local experiments cannot tell them apart.
The Keldysh Hamiltonian is obtained from the Dirac Hamiltonian by flipping the sign of the electromagnetic coupling; the apparent negative sign becomes just a redefinition of the charge.
Mixed Dirac–Keldysh scalar coupling vanishes and the electric part of the vector coupling vanishes, so static electric fields cannot couple the two species.
The surviving vector coupling permits only elastic magnetic scattering, with no energy transfer between Dirac and Keldysh fields.
The Foldy–Wouthuysen expansion gives the same power series for both types, reinforcing that they are locally indistinguishable at low energies.

Reading between the lines

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

If the central claim is right, negative-action spinor sectors could sit beside ordinary matter while staying invisible to low-energy mass and charge probes, since every local measurement sees the same Pauli Hamiltonian.
A natural extension the paper does not carry out is an explicit $1/m^{2}$ Foldy–Wouthuysen calculation; writing down the spin-orbit and Darwin terms for Keldysh spinors would turn the asserted indistinguishability into a checkable computation.
The vanishing scalar and electric vector couplings suggest a selection rule that a full second-quantized calculation of Dirac–Keldysh scattering could test, by verifying that elastic magnetic scattering is the only open channel.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the derivation is standard and the algebra is fine, but the paper's main claim is overstated. Eq. (7) is the actual non-relativistic equation for the Keldysh lower component, and it has a negative kinetic term. The positive Pauli Hamiltonian in Eq. (8) comes from complex-conjugating that component, which flips the charge. So the result is a charge-conjugation equivalence, not a new positive-definite dynamics for Keldysh spinors.

What's new: the explicit reduction and the decoupling identities in Section 3. The reduction is a textbook Foldy-Wouthuysen calculation; the only new element is applying it to the negative-action Keldysh field. The decoupling identities (scalar and vector currents vanish between Dirac and Keldysh in the non-relativistic limit) are correctly derived and might be useful, though they are a kinematic consequence of the phase choices.

The paper's own text admits the time dependence is different (exp(+iE t) vs exp(-iE t)) but treats it as a convention. The stress-test note is right: switching the sign of the phase is equivalent to complex conjugation. If you define η = ξ_K^*, Eq. (7) becomes the Dirac Pauli equation with reversed charge. The spectrum of Eq. (7) as written is unbounded below, so the claimed positive-definite Hamiltonian does not govern the component whose limit was derived. The paper should restate the result: the non-relativistic limit of Keldysh spinors yields the charge-conjugate Pauli wavefunction. That is still a consistent picture, consistent with ref. [6]'s operator swap, but it is not what the abstract says.

Minor issues: the derivation is terse, sign conventions are hard to track, and there are typos. The positivity argument is imported from the author's own ref. [6]; in a note that is acceptable, but the reader cannot verify it without going to that paper.

The decoupling discussion is framed as 'curious,' but it is largely forced by the identification of Keldysh positive energy with Dirac negative energy. I would not call it an independent result.

If I were editing, I would send this to a referee. The calculation is correct and the corrected interpretation is worth stating. It is too niche to be a big deal, but it is not wrong in the way that matters most once fixed. I would ask for a rewritten abstract and conclusion that make the charge-conjugation nature of the limit explicit, and for a note about the unbounded-below spectrum of Eq. (7) in the original variable.

Recommendation: worthwhile for a serious referee; not acceptable as is.

Referee Report

3 major / 5 minor

Summary. The manuscript considers spinors that obey the Dirac equation but have the negative of the Dirac action and Hamiltonian, called Keldysh spinors. Coupling them to a classical U(1) gauge field, the author performs the standard two-component non-relativistic reduction for both Dirac and Keldysh fields. The central claim is that the Keldysh spinor has a well-defined non-relativistic limit described by the Schrödinger equation with the same positive-definite Pauli Hamiltonian as the Dirac spinor, up to a change in the sign of the electromagnetic coupling. The paper further discusses couplings between non-relativistic Dirac and Keldysh fields, reporting a 'curious decoupling' in scalar and vector interactions.

Significance. If the central claim were correct, it would imply that Keldysh spinors are locally indistinguishable from Dirac spinors of opposite charge in low-energy experiments, which would be relevant to the Keldysh non-equilibrium QFT and to other contexts where negative-action fermions appear. The two-component reduction and the Pauli equation for the Dirac case are standard and correctly reproduced. However, as argued below, the positive-definite Pauli Hamiltonian in Eq. (8) is not the Hamiltonian of the Keldysh component derived in Eqs. (5)-(7), but of its charge-conjugate field. The 'curious decoupling' in Section 3 is then a direct consequence of the prior identification of Keldysh positive-energy states with Dirac negative-energy states, rather than an independent result. The manuscript's main claim is therefore not established for the original field variable.

major comments (3)

[Section 2, Eqs. (7)-(8)] The derivation correctly yields Eq. (7), i∂_t ξ_K = [−(σ·π)^2/(2m) + eA0] ξ_K, which is a Schrödinger equation with a negative-definite kinetic term. The paper then asserts that because positive-energy eigenvalue solutions have the time dependence exp(+iE_k t) instead of exp(−iE_k t), one can write the positive Pauli Hamiltonian of Eq. (8). This is not a harmless convention: changing the time-frequency convention is equivalent to complex conjugating the wavefunction. Defining η = ξ_K^* converts Eq. (7) into i∂_t η = [(σ·(p+eA))^2/(2m) − eA0 − (e/2m)σ·B] η (up to the standard spin rotation in the charge-conjugate spinor), which is the Pauli Hamiltonian for the charge-conjugate field, not for the original Keldysh component. The spectrum of Eq. (7) itself is unbounded below, so the abstract's claim that Keldysh spinors have a non-relativistic limit with a positive-definite Pauli Hamiltonian is not supported for the field variable whose limit was derived in Eqs. (5)-(7).
[Section 2, positive-energy identification] The paper states that for the Keldysh case the positive-energy solutions are the negative-energy solutions of the Dirac case, encoded in the phase e^{+imt}. This identification is the entire basis for choosing that phase in Eq. (5). Consequently, the non-relativistic limit obtained here is essentially a charge-conjugation statement: the positive-definite Hamiltonian describes the conjugate of the Keldysh component, not a new positive-definite dynamics for the original field. The 'curious decoupling' in Section 3 is likewise a standard property of positive- and negative-energy Dirac solutions in the non-relativistic limit, where Dirac spinors have only upper components and Keldysh spinors only lower components. The paper should either provide an independent derivation of positivity that does not rely on this identification, or explicitly frame the result as an equivalence under charge conjugation.
[Section 2, positivity from ref. [6]] The manuscript imports the key premise that the quantum Keldysh Hamiltonian H_K can be made positive definite from the author's earlier publication [6] without reproducing the relevant construction. Since the choice of e^{+imt} as the positive-energy phase depends on this operator swap, the paper should at least state the precise creation/annihilation operator assignment and the definition of the vacuum, or provide the argument in an appendix. Without this, the positivity claim is essentially an assumption carried over from a cited paper, and the reader cannot assess whether the non-relativistic result is independent of that construction.

minor comments (5)

[Section 2, notation] The definitions of the two-component spinors φ, χ, φ_D, ξ_D, φ_K, ξ_K and the phase factors are not explicitly displayed as column vectors and equations, which makes the derivation difficult to follow; please present the steps with clear notation and define every symbol.
[Introduction / Section 3] The Introduction states that 'Section 3 is the Summary', but the section is titled 'Discussion'; please align the labels.
[References] Reference [5] lists 'K. Sravan Kumara João Marto' without a comma between the two authors; the citation should be corrected.
[Section 2, text] The phrase 'The next step is to factor to factor out from φ, χ the fast vacuum phase' contains a redundant 'to factor'; this should be corrected.
[Section 3, energy-transfer claim] The claim that a static A_k 'cannot transfer energy' and hence 'there is no energy transfer between the two fields' is too terse; a static vector potential can mediate elastic scattering with momentum transfer even if no energy is exchanged, so the statement should be justified from the interaction Hamiltonian rather than asserted.

Circularity Check

2 steps flagged · score 7.0 of 10

The positive-definite Pauli Hamiltonian for Keldysh spinors is the charge-conjugate Dirac Hamiltonian by the paper's own time-convention; the central claim reduces to the definition of Keldysh positive energy.

self definitional [Section 2, paragraph defining positive-energy solutions, before Eqs. (4)-(5)]
"Obviously, for Keldysh case these are the negative energy solutions of the Dirac case. For convenience we derive the limit for both cases in parallel."

This sentence defines 'Keldysh positive energy' as 'Dirac negative energy'. The paper then factors out e^{+imt} for Keldysh in Eq. (5), which is the phase of a negative-energy Dirac solution. After the non-relativistic reduction, Eq. (7) is i∂_t ξ_K = [-(σ·π)^2/(2m)+eA0]ξ_K, a Hamiltonian with negative kinetic energy. The later claim that positive-energy solutions have time dependence e^{+iE_k t} changes the sign convention; this is equivalent to replacing ξ_K by its complex conjugate η=ξ_K^*, which obeys the standard positive Pauli Hamiltonian with reversed charge. Thus the positive-definite Pauli Hamiltonian of Eq. (8) is not a property derived for the original Keldysh component, but is built into the initial identification of Keldysh positive energy with negative-energy Dirac states.
renaming known result [Section 2, discussion after Eq. (7) and before Eq. (8)]
"Despite the appearance, the eigenvalue equation for (7) is the same as for (6) up to the charge sign. This is because the positive energy eigenvalue problem solutions for the two equations have different time dependence exp(±iE_k t)."

This is not a consequence of Eq. (7) under the usual i∂_t ψ = Hψ convention. For Eq. (7), writing ξ_K ∝ e^{+iE_k t} gives i∂_t ξ_K = -E_k ξ_K, so the equation becomes E_k ξ_K = [(σ·π)^2/(2m)-eA0]ξ_K. That is exactly the standard Schrödinger equation for the complex-conjugated wavefunction ξ_K^*, i.e. the charge-conjugate field. The 'same up to charge sign' is therefore the known charge-conjugation equivalence between negative-energy Dirac solutions and positive-energy anti-particle solutions, relabeled as Keldysh positive energy. Presenting this as the non-relativistic limit of Keldysh spinors renames the known charge-conjugation result rather than deriving a new positive-definite dynamics.

full rationale

The paper's technical manipulation of the Dirac equation is self-contained, but the central interpretive step is definitional. The authors explicitly choose e^{+imt} for Keldysh spinors so that the system has positive-energy states, and identify Keldysh positive energy with Dirac negative energy. The resulting Eq. (7) has a negative kinetic term; the paper then asserts that positive-energy eigenvalue solutions have time dependence e^{+iE_k t}, which reverses the sign convention. That reversal is equivalent to complex conjugation, so the positive-definite Pauli Hamiltonian of Eq. (8) is the Hamiltonian of the charge-conjugate wavefunction, not of the ξ_K component whose equation was derived. In effect, the claimed non-relativistic positive-definite dynamics is the standard Dirac Pauli Hamiltonian after charge conjugation, relabeled as a Keldysh result. The self-citation to [6] for the operator swap is used as motivation but is not the main source of circularity; even if that citation is accepted, the central claim still reduces by construction. The 'curious decoupling' in Section 3 is a straightforward consequence of the same convention and does not independently support the positive-Hamiltonian claim. Overall, the central result is largely forced by the definition of Keldysh positive energy and the chosen time dependence, so the paper is partially circular rather than a fully independent derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim does not depend on any numerically fitted parameters. It rests on the defining negative-action property of Keldysh spinors, the conventional positive-energy assignment imported from ref. [6], and the standard non-relativistic limit approximations. No new particles or forces are introduced.

assumptions (4)

domain assumption Keldysh spinor action is the negative of the Dirac action (Eq. 2) while the equation of motion remains the Dirac equation.
This is the defining property of the model, stated in Section 1.
ad hoc to paper Positive energy states for Keldysh spinors are the negative-energy solutions of the Dirac case, implemented by the phase e^{+imt} (Eq. 5) and by the creation/annihilation operator swap of ref. [6].
This convention is load-bearing: without it, the resulting Hamiltonian would not be positive definite. It is asserted, not derived in this paper.
domain assumption Non-relativistic approximation: eA_0 << m and slow variation of the upper/lower components.
Invoked before Eqs. (6)-(7) to eliminate the small components.
standard math Identity (sigma dot pi)^2 = pi^2 - e sigma dot B for the Pauli reduction.
Standard Pauli algebra used to reach Eq. (8).

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Cite this review

Pith. "Pith review of The Non-Relativistic Limit of Keldysh Spinors." pith.science (2026). https://pith.science/paper/NTBIEZZS

@misc{pith2026250104514,
  author       = {Pith},
  title        = {Pith review of: The Non-Relativistic Limit of Keldysh Spinors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTBIEZZS}},
  note         = {Machine review of arXiv:2501.04514}
}

read the original abstract

Keldysh spinors obey Dirac equation, but have the negative of the Dirac action and Hamiltonian. In an example of the U(1) EM coupling, we show that, despite the sign changes, they have a well-defined non-relativistic limit resulting in quantum mechanics with the positive-definite Pauli Hamiltonian. When non-relativistic Dirac and Keldysh fields are brought to interact, we observe curious decoupling of the two fields in mass-like and vector couplings.

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Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[6]

Jourjine, The quantum theory of the Lorentzian fermionic differential forms, Theoretical and Mathematical Physics, 202 (2020) 183

A. Jourjine, The quantum theory of the Lorentzian fermionic differential forms, Theoretical and Mathematical Physics, 202 (2020) 183

work page 2020
[1]

L. V. Keldysh, Diagram technique for nonequilibrium processes, Soviet Physics JETP, 20, 1018 (1965)

work page 1965
[2]

Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, 2023

A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, 2023

work page 2023
[3]

Jourjine, Scalar spin of elementar fermions, Phys Lett B 728 (2014) 347

A. Jourjine, Scalar spin of elementar fermions, Phys Lett B 728 (2014) 347

work page 2014
[4]

Time Flow and Flavor Mixing in the Flavor Spin Theory

A. Jourjine, Time flow and flavor fixing in the flavor fpin fheory, arXiv:2404.11914

work page Pith review arXiv
[5]

Sravan Kumara João Marto, Towards a unitary formulation of quantum field theory in curved spacetime: the case of Schwarzschild black hole, arXiv:2307.10345

K. Sravan Kumara João Marto, Towards a unitary formulation of quantum field theory in curved spacetime: the case of Schwarzschild black hole, arXiv:2307.10345

arXiv
[7]

The Negative Action Keldysh Spinors

A. Jourjine, The negative action Keldysh spinors, arxiv:2406.06194

work page Pith review arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [6]

Jourjine, The quantum theory of the Lorentzian fermionic differential forms, Theoretical and Mathematical Physics, 202 (2020) 183

A. Jourjine, The quantum theory of the Lorentzian fermionic differential forms, Theoretical and Mathematical Physics, 202 (2020) 183

work page 2020

[2] [1]

L. V. Keldysh, Diagram technique for nonequilibrium processes, Soviet Physics JETP, 20, 1018 (1965)

work page 1965

[3] [2]

Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, 2023

A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, 2023

work page 2023

[4] [3]

Jourjine, Scalar spin of elementar fermions, Phys Lett B 728 (2014) 347

A. Jourjine, Scalar spin of elementar fermions, Phys Lett B 728 (2014) 347

work page 2014

[5] [4]

Time Flow and Flavor Mixing in the Flavor Spin Theory

A. Jourjine, Time flow and flavor fixing in the flavor fpin fheory, arXiv:2404.11914

work page Pith review arXiv

[6] [5]

Sravan Kumara João Marto, Towards a unitary formulation of quantum field theory in curved spacetime: the case of Schwarzschild black hole, arXiv:2307.10345

K. Sravan Kumara João Marto, Towards a unitary formulation of quantum field theory in curved spacetime: the case of Schwarzschild black hole, arXiv:2307.10345

arXiv

[7] [7]

The Negative Action Keldysh Spinors

A. Jourjine, The negative action Keldysh spinors, arxiv:2406.06194

work page Pith review arXiv