Incorporating action and reaction into a particle interpretation for quantum mechanics -- Dirac case

Roderick Sutherland

REVIEW 4 major objections 3 minor 21 references

Incorporating action and reaction into a particle interpretation for quantum mechanics -- Dirac case

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A pilot-wave field can both guide the particle and be sourced by it, while still reproducing the Dirac equation.

desk verdict A promising action-reaction Lagrangian for pilot-wave theory, but the Dirac equation is recovered by assuming the guidance condition, not deriving it. read the letter →

arxiv 1908.04897 v2 pith:B63FVTNT submitted 2019-08-14 quant-ph

Roderick Sutherland This is my paper · ORCID

classification quant-ph

keywords actionandreactionpilot-waveinterpretationDiracequationLagrangiandensityBornruleHamilton-Jacobitheorygaugetransformationquantumtrajectories

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 paper claims that the one-way pilot-wave picture, where a guiding field acts on a particle but the particle never acts back, can be replaced by a two-way interaction without disturbing the empirical content of quantum mechanics. Working with a Dirac spinor, the author postulates a Lagrangian density whose particle term couples the hidden particle's four-velocity to the Dirac current, with the current magnitude $\rho_0$ playing the role that the rest mass plays in electromagnetism. A statistical average over possible particle positions, using the Born rule as the position distribution, turns the resulting non-linear field equation into the standard Dirac equation after a gauge-like redefinition of the field. If this is correct, the fields-only appearance of standard quantum mechanics is a statistical artifact, gauge transformations acquire a mechanical meaning through the particle's action $S$, and the pilot-wave program gains a back-reaction that standard quantum predictions cannot detect.

What carries the argument

The load-bearing object is the combined Lagrangian density $\mathcal{L} = \mathcal{L}_{\mathrm{field}} + \sigma_0 L$, with $L = -k\rho_0\sqrt{u_\alpha u^\alpha} + k u_\alpha j^\alpha$, where $j^\alpha = \bar{\psi}\gamma^\alpha\psi$ is the Dirac current, $\rho_0 = \sqrt{j_\alpha j^\alpha}$, $u^\alpha$ is the hidden particle's four-velocity, and $\sigma_0$ is a delta-function rest-density locating the particle. Its role is to replace the rest mass in an electromagnetic-type coupling with $k\rho_0$, so that the particle and field act as sources for each other. The argument then leans on three further moves: statistical averaging over particle positions with $P = \bar{\psi}\gamma^0\psi$, the Hamilton-Jacobi identity $p_\alpha = -\partial_\alpha S = k(\rho_0 u_\alpha + j_\alpha)$, and the gauge-like change of field variable $\Psi = e^{-iS}\psi$. The gauge phase $S$ is the action integral along the hidden trajectory, and the change of variable is what removes the source term in the averaged equation, leaving the Dirac equation.

What would settle it

Numerically solve the non-averaged field equation $\gamma^\alpha\partial_\alpha\phi - m\phi = \sigma_0 (k u^\alpha + k j^\alpha/\rho_0)\gamma_\alpha\phi$ together with the particle's equation of motion for a single, spherically emitted particle, then average over the Born distribution $P = \bar{\psi}\gamma^0\psi$. If the averaged field is not equivalent to a solution of $i\gamma^\alpha\partial_\alpha\Psi - m\Psi = 0$ up to the gauge factor $e^{-iS}$, the claimed consistency fails.

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Extended reading notes

Core claim

The central claim is that the Lagrangian density $\mathcal{L} = \mathcal{L}_{\mathrm{field}} + \sigma_0 L$ with $L = -k\rho_0\sqrt{u_\alpha u^\alpha} + k u_\alpha j^\alpha$ produces the Dirac equation while allowing full action and reaction between the particle and its guiding field. The non-statistical field equation contains a delta-function source, but after averaging over particle positions with the Born distribution and redefining the field as $\Psi = e^{-iS}\psi$, the source term cancels and $i\gamma^\alpha\partial_\alpha\Psi - m\Psi = 0$ results. The same Hamilton-Jacobi machinery identifies the generalized momentum as $p_\alpha = k(\rho_0 u_\alpha + j_\alpha) = -\partial_\alpha S$, making the phase of the gauge transformation the action of the hidden particle. The guidance relation $u^\alpha = \bar{\Psi}\gamma^\alpha\Psi/\rho_0$ persists, so the field still steers the particle while the particle acts back as its source. The paper concludes that conserved exchanges of energy and momentum between particle and field are present and that the fields-only appearance of ordinary quantum mechanics is an artifact of statistical averaging.

Load-bearing premise

The load-bearing premise is that the particle's four-velocity is locked to the spinor current by $j^\alpha = \rho_0 u^\alpha$. The paper presents this as following from Eq. (27), but Eq. (27) only defines $u^\alpha$ from the Hamilton-Jacobi relation while $j^\alpha$ is fixed by the spinor, so the equality is an extra guidance condition; if it fails, the particle source term survives and the standard Dirac equation is not recovered.

Editorial extensions

If this is right

The pilot-wave picture becomes causally two-way: the particle is both steered by the field and contributes a local source to it, while Dirac predictions are unchanged.
Individual events are described by a non-linear field equation containing a $\delta$-function source; the linear Dirac equation is the statistical version obtained after averaging over the particle's unknown position.
Energy and momentum are conserved for the combined system because the total energy-momentum tensor has zero four-divergence, even though field and particle separately exchange energy and momentum.
The phase of a gauge transformation is identified with the hidden particle's action, giving a mechanical reading of gauge freedom that arises from the model rather than being imposed.
When an external four-potential is present, the extra source term can be absorbed into the potential by a gauge-like shift, indicating the mechanism extends beyond free-space Dirac fields.

Reading between the lines

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

Because the paper never solves the non-linear event-level equation, an immediate testable extension is to compute its predictions for interference or near-field setups; any correction to the averaged Dirac result would mark where standard quantum mechanics and this model diverge.
The unproven equality $j^\alpha = \rho_0 u^\alpha$ suggests a family of nearby models: relaxing it while keeping the action/reaction Lagrangian would produce non-Dirac source terms, and precision measurements could bound how much freedom remains.
If the fields-only appearance is really an averaging effect, then measurement and entanglement should be re-examined in this model, since the event-level equation may produce correlations that the linear Dirac equation alone cannot describe.
The same Lagrangian device could be applied to Dirac fields on curved backgrounds, where the gauge phase $S$ would be tied to the particle's geodesic action and the back-reaction would express local energy-momentum conservation.

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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

Colleague,

The paper tries to restore action-reaction in a de Broglie-Bohm picture for Dirac particles and claims a by-product: gauge transformations emerge with the gauge phase being the particle's action. That's a genuinely interesting idea. But the central derivation has a load-bearing gap. Eq. (28) — j^α = ρ0 u^α — is not a consequence of Eq. (27). Uniqueness of u^α at each x tells you there's a well-defined vector field; it says nothing about whether the spinor current j^α is proportional to that vector. The proportionality is exactly the de Broglie-Bohm guidance condition, and it is being inserted rather than derived. Without it, the source term in Eq. (22) does not become -(∂^α S)γ_α ψ, and the subsequent phase transformation just moves the problem around. So the Dirac equation is recovered only by assuming the substantive quantum ingredient.

The statistical averaging in Sec. 5 is also shaky. Replacing σ0 by P/u^0 is asserted by analogy, and the u^0 factor is not controlled. Footnote 11 actually concedes that if there is a spread of u^α at a point, only a mean velocity can be extracted from j^α; the paper never shows the distribution is a delta. Combined with the Born-rule postulate in Sec. 6, this means the statistical treatment is where the quantum predictions enter. The non-linear individual field equation (8) is never solved, so we have no independent check that the model's dynamics preserve the Born rule.

What's genuinely new: the negative sign in the interaction term, the coupling to ρ0 rather than m, and the identification of the gauge phase with the action. These are worth thinking about. The conservation-law section is a routine Noether exercise and appears consistent. The paper is clearly written and the author knows the pilot-wave literature.

But the central claim — that Eq. (4) leads to the Dirac equation with action and reaction intact — is not supported. The gap is internal to the derivation, not a matter of philosophical taste. This doesn't mean the program is dead; it means the paper needs a real proof that the guidance condition emerges from the Lagrangian, or a derivation of the Born rule from the individual dynamics. Right now it's input equals output.

For a referee: send it. The question is important enough and the proposal concrete enough to deserve a careful referee report. But recommend reject in its current form; the author should be given the chance to fix the gap. I would not cite it as a result; I might cite the gauge-action observation as a curiosity.

Referee Report

4 major / 3 minor

Summary. The manuscript presents a Lagrangian formulation for a de Broglie–Bohm-type particle coupled to a Dirac spinor field. The proposed Lagrangian density (4) includes a particle source term of the form -σ0 k ρ0 (uα uα)^(1/2) - σ0 k uα jα, with the field variable φ distinct from the statistical wavefunction ψ. The author argues that after averaging the Lagrangian over particle positions with the Born-rule distribution and applying a local phase transformation generated by the Hamilton–Jacobi action S, the standard Dirac equation is recovered. The paper also derives the generalized momentum, establishes a Hamilton–Jacobi relation, and constructs an energy–momentum tensor with zero divergence. Sections 8 and 9 discuss conservation laws and possible many-particle generalizations.

Significance. The question addressed—whether pilot-wave models can include action and reaction without contradicting quantum predictions—is a legitimate and interesting one. The manuscript is explicit about its postulates and contains a detailed, self-contained derivation of the energy–momentum tensor. However, the central claim is not established: the reduction to the Dirac equation relies on the unjustified identity jα = ρ0 uα, which is precisely the de Broglie–Bohm guidance condition, and on a statistical averaging step that is an ansatz rather than a derivation. These are load-bearing gaps, not presentation issues. If the derivation were repaired, the paper could be significant; in its present form, the main result is not supported.

major comments (4)

[Sec. 7, Eqs. (27)–(28)] The inference from single-valuedness of u^α to the identity j^α = ρ0 u^α is not valid: Eq. (27) defines u^α algebraically in terms of ∂^α S and j^α, while j^α is fixed by the spinor via Eq. (25). A single value of u^α at each x is compatible with j^α not being parallel to u^α. Equation (28) is exactly the guidance condition stated in Eq. (9), which the manuscript itself identifies as the de Broglie–Bohm guidance condition. The subsequent reduction of Eq. (22) to Eq. (30) and the cancellation of the source term in the gauge-transformed equation depend entirely on this assumed identity. Footnote 11 acknowledges that a distribution of u^α would require averaging but provides no reason that the distribution is concentrated at a single value. Thus the central derivation assumes the very relation that was to be obtained, and the claim that the Lagrangian (4) leads to the Dirac equation is not established.
[Sec. 5, Eqs. (20)–(22)] The statistical averaging step is not a derivation. The original Lagrangian density (4) depends on the particle position through σ0. Multiplying by P(x_p) and integrating over x_p gives an averaged Lagrangian (21), but the field equation (22) is then obtained by varying this averaged Lagrangian while treating P and u^0 as independent of the field. This is inconsistent with the later postulate P(x) = j^0(x) in Eq. (26), because j^0 depends on the field; a proper variation with P = j^0 would produce additional terms from the variation of P. Furthermore, the map from the individual field equation (18) to the 'statistical' field equation (22) is made 'by analogy' (replacing σ0 with P/u^0 and φ with ψ), not by any well-defined averaging of the nonlinear dynamics. Equation (22) is therefore an ansatz, not a consequence of the Lagrangian density (4).
[Sec. 7, Eq. (31) and Sec. 10] The final step is a field redefinition. Even if Eq. (30) were derived, the transformation Ψ = e^{-iS}ψ is a change of variables that makes Eq. (30) look like the free Dirac equation; it does not remove the need to derive Eq. (30) from the Lagrangian. The paper's conclusion that the Lagrangian 'leads, via a statistical treatment together with a change of notation, to the correct wave equation' overstates the case because the substantive quantum ingredients—the Born-rule postulate (26) and the guidance condition (28)—are assumed, not derived. The negative sign of the source term in Eq. (4) is chosen so that the source cancels under the phase transformation, which is a construction rather than an explanation.
[Secs. 6–7] The manuscript does not prove that the postulated probability distribution P = j^0 is preserved by the non-linear individual dynamics. The continuity equation (24) is derived from the statistical field equation (22), not from the coupled system consisting of Eq. (8) and the particle equation of motion. Without a demonstration that the Born-rule distribution is an equilibrium (or otherwise stable) distribution of the individual process, the claim that the standard quantum predictions are recovered is incomplete.

minor comments (3)

[Eq. (5)] The definition of ρ0 as (jα jα)^(1/2) is ambiguous for the chosen signature (+---); the author should specify that the positive root is taken, with the understanding that the Dirac current is timelike.
[Sec. 5, Eq. (20)] In Eq. (20), the field point and the integration variable are both denoted x in places, which is confusing; the integral kernel should display the field variable explicitly, for instance by writing L(x) = ∫ d^3x_p P(x_p) L(x, x_p).
[Sec. 5] The manuscript assumes ∫P(x_p)d^3x_p = 1, but the later postulate P(x) = j^0(x) in Eq. (26) is only normalized if the Dirac wavefunction is normalized; this normalization condition is not stated.

Circularity Check

1 steps flagged · score 7.0 of 10

Eq. (28) silently reintroduces the de Broglie-Bohm guidance condition, which then reappears as the derived relation Eq. (35); the Dirac-equation derivation is therefore partially circular.

fitted input called prediction [Section 7, equations (27), (28), (29), and (35); compare with Section 3, equations (9) and (10).]
"In previous work, the field equation (8) was simply reduced to the standard Dirac equation by making the extra assumption that the bracket on the right hand side was zero ... which is the same as the guidance equation of the de Broglie-Bohm model ... From Eq. (27) it is clear that u^α is a function of x, as is usual in a Hamilton-Jacobi formulation. Therefore this equation implies there will be only a single value of u^α for each position, which then allows the 4-current density j^α to be written in the following product form: j^α = ρ0 u^α (28)."

Equation (27) only defines u^α through the Hamilton-Jacobi relation; single-valuedness of u^α does not imply that the independently defined spinor current j^α is parallel to u^α. The equality j^α = ρ0 u^α is exactly the de Broglie-Bohm guidance equation, which the paper itself initially labels an "extra assumption" when it appears as Eq. (9). This unproved equality supplies P = ρ0 u0, cancels the source term in Eq. (22), and thereby enables the gauge step that yields the Dirac equation. The same equality is later re-derived as Eq. (35) and presented as evidence that the field influences the particle's 4-velocity. Thus the central action/reaction result assumes, at Eq. (28), the very guidance relation it purports to recover.

full rationale

The main circularity is internal to Section 7. Equation (27) determines u^α from the Hamilton-Jacobi relation, but nothing in that equation forces the spinor current j^α to equal ρ0 u^α. A unique value of u^α at each x is insufficient; as the paper's own footnote notes, a distribution of u^α values would only give a mean velocity. The jump to Eq. (28) is therefore an extra dynamical postulate, equivalent to the dBB guidance equation previously called an extra assumption in Section 3. That postulate is what makes P = ρ0 u0 and removes the source term, so the derived Dirac equation is not a consequence of the Lagrangian alone. The later Eq. (35) merely restates Eq. (28) after the gauge redefinition, so presenting it as showing field-to-particle influence is circular. The Born rule in Section 6 is explicitly and honestly postulated, so it is not hidden circularity, but it further confirms that the Lagrangian does not by itself generate the standard quantum equation. The gauge transformation itself is a genuine algebraic identity and is not the defect. Self-citations to the author's earlier papers [3,4] are present but are not the load-bearing circular element here. Overall, the central derivation reduces to an assumed guidance condition, warranting a score of 7.

Assumptions & free parameters 1 free parameters · 5 assumptions · 3 invented entities

The model rests on the ad hoc Lagrangian and three added assumptions (Born rule, statistical averaging, guidance condition), which together build in the quantum predictions. No new falsifiable entity is introduced.

free parameters (1)

k
Arbitrary constant with dimensions ML^3 introduced in Eq. (4); cancels in the final reduction to the Dirac equation, so no numerical value is needed.

assumptions (5)

ad hoc to paper The overall Lagrangian density has the form L = L_field - σ0 k ρ0 (u^α u_α)^(1/2) - σ0 k u^α j_α (Eq. 4), with the particle rest mass replaced by k ρ0 and the interaction sign chosen negative.
This is the central construction; the sign choice is made to allow the later cancellation, as the author notes in Sec. 2.
domain assumption The particle position is statistically distributed via the Born rule P(x) = ψ̄ γ^0 ψ (Eq. 26).
Postulated in Sec. 6; the author acknowledges it cannot be derived.
ad hoc to paper The averaged Lagrangian is obtained by multiplying by P(x_p) and integrating; field equations are then varied from this averaged Lagrangian (Eqs. 20-22).
This statistical averaging of a Lagrangian is not a standard procedure and is not justified.
ad hoc to paper There exists a global Hamilton-Jacobi function S(x) satisfying Eq. (27) for all x, with u^α single-valued and j^α = ρ0 u^α.
Sec. 7; the product form is an extra assumption, effectively the guidance condition.
domain assumption The particle's rest density σ0 is a Lorentz-covariant delta function and satisfies the continuity equation ∂_β(σ0 u^β) = 0.
Standard for point particles; used in Appendix 4.

invented entities (3)

Bohmian point particle with 4-velocity u^α and rest density σ0
purpose: Provides the hidden variable and acts as a source for the field, enabling action and reaction.
No new experimental prediction; the model aims to reproduce QM.
Actual pilot field φ(x) distinct from the statistical wavefunction ψ(x)
purpose: The field that guides the particle and is sourced by it; only statistically related to ψ.
The paper introduces φ as the real field in the non-statistical Eq. (18), but no direct detection is proposed.
Action function S(x) as a physical field
purpose: Serves as the phase of the gauge transformation and is identified with the particle's action (Eq. 16).
It is a derived quantity, not directly observable.

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Cite this review

Pith. "Pith review of Incorporating action and reaction into a particle interpretation for quantum mechanics -- Dirac case." pith.science (2026). https://pith.science/paper/B63FVTNT

@misc{pith2026190804897,
  author       = {Pith},
  title        = {Pith review of: Incorporating action and reaction into a particle interpretation for quantum mechanics -- Dirac case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B63FVTNT}},
  note         = {Machine review of arXiv:1908.04897}
}

read the original abstract

A weakness which has previously seemed unavoidable in particle interpretations of quantum mechanics (such as in the de Broglie-Bohm model) is addressed here and a resolution proposed. The weakness in question is the lack of action and reaction occurring between the model's field (or "pilot wave") and the particle. Although the field acts on the particle, the particle does not act back on the field. It is shown here that this rather artificial feature is, in fact, not necessary and can be fully eliminated while remaining consistent with the usual quantum predictions. Mathematically this amounts to demonstrating that there exists a suitable Lagrangian density function which generates equations coinciding with quantum mechanics yet incorporates the desired action and reaction. As a by-product, an appealing explanation emerges to another long-standing question, namely why the mathematical formalism of quantum mechanics seems only to be describing fields when measurements generally detect localised particles. A further bonus is that the hitherto unrelated concept of a gauge transformation is found to arise naturally as an essential part of the formalism. In particular, the phase S of the gauge transformation is seen to be the action function describing the hidden motion of the particle.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

de Broglie: Non-linear wave mechanics

L. de Broglie: Non-linear wave mechanics. Elsevier, Amsterdam (1960)

work page 1960

[2] [2]

Bohm: A suggested interpretation of quantum theory in terms of “hidden” variables

D. Bohm: A suggested interpretation of quantum theory in terms of “hidden” variables. I and II. Physical Review 85, 166-179 and 180-193 (1952)

work page 1952

[3] [3]

R. I. Sutherland: Lagrangian description for particle interpretations of quantum mechanics: single -particle case . Foundations of Physics 45, 1454 -1464 (2015). arXiv:1411.3762

work page Pith review arXiv 2015

[4] [4]

R. I. Sutherland: Lagrangian description for particle interpretations of quantum mechanics: entangled many -particle c ase. Foundations of Physics 47, 174-207 (2017). arXiv:1509.02442

work page Pith review arXiv 2017

[5] [5]

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Sutherland R.I.: A spacetime ontology compatible with quantum m echanics Activitas Nervosa Superior 61, 55-60 (2019). arXiv: 1904.05157

work page Pith review arXiv 2019

[6] [6]

Felsager: Geometry, particles and fields

B. Felsager: Geometry, particles and fields. Springer-Verlag, New York (1998)

work page 1998

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Gold stein, C

H. Gold stein, C. P. Poole and J. Safko: Classical m echanics. 3rd Ed. Addison Wesley (2002). 18

work page 2002

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S. S. Schweber: An Introduction to relativistic quantum field theory, Harper International, New York (1964)

work page 1964