arxiv: 2603.15638 · v2 · submitted 2026-02-27 · ⚛️ physics.hist-ph · math-ph· math.MP· quant-ph

Recognition: no theorem link

The Birth of Quantum Mechanics and the Dirac Equation

Volodimir Simulik , Denys I. Bondar

Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:26 UTC · model grok-4.3

classification ⚛️ physics.hist-ph math-phmath.MPquant-ph

keywords Dirac equationKramers derivationhistory of quantum mechanicsMadelung hydrodynamicsEhrenfest relationsoperational dynamical modelingKlein-Gordon equationVan der Waerden

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The pith

Kramers derived the Dirac equation independently and nearly simultaneously with Dirac, though it stayed unpublished for seven years, and the equation also follows from relativistic Ehrenfest relations or the Madelung hydrodynamic picture.

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

This review supplements the standard account of quantum mechanics by reconstructing Kramers' independent 1928 derivation of the Dirac equation, which matched Dirac's result but remained unpublished until 1935. It also supplies two modern derivations omitted from an earlier J. Phys. A review: one that obtains the Dirac equation directly from relativistic Ehrenfest relations together with the canonical commutation algebra via operational dynamical modeling, and one that starts from the Madelung hydrodynamic formulation. A sympathetic reader would care because these routes show the Dirac equation arising from distinct physical starting points during the 1925-1928 formative period, rather than from a single chain of thought. The paper additionally notes Darwin's clarification of the equation's physical content and Van der Waerden's group-theoretic perspective, then sketches three broad eras of quantum theory development.

Core claim

The central claim is that Kramers produced an independent derivation of the Dirac equation essentially at the same time as Dirac yet left it unpublished for seven years, and that the Dirac equation can be recovered by two previously uncatalogued modern methods: operational dynamical modeling that begins with relativistic Ehrenfest relations and the canonical commutation relations, and a derivation rooted in the Madelung hydrodynamic formulation.

What carries the argument

Reconstruction of Kramers' independent 1928 derivation of the Dirac equation, together with derivations from relativistic Ehrenfest relations in operational dynamical modeling and from the Madelung hydrodynamic formulation.

If this is right

The historical narrative of the Dirac equation must incorporate Kramers' parallel contribution as a distinct line of work.
The Dirac equation can be reached without presupposing the Klein-Gordon equation or the standard relativistic wave operator.
Darwin's analysis supplies concrete physical interpretation that clarifies the equation's spin and negative-energy features.
Group-theoretic methods such as Van der Waerden's provide an independent route to the same equation.
The Dirac equation belongs to a broader convergence of ideas across the 1925-1928 period rather than to a single inventor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The existence of multiple independent derivations implies the Dirac equation is a stable structure that can be reached from quite different physical premises.
Similar overlooked derivations may exist for other foundational equations in the same era.
The operational and hydrodynamic routes could be adapted to teach relativistic quantum mechanics without starting from the usual first-order wave equation.
The three-era survey suggests that current quantum-information approaches may eventually loop back to reinterpret the original 1920s constructions.

Load-bearing premise

The central claim rests on Kramers' derivation being genuinely independent of Dirac's and on the two modern derivations truly being absent from the prior review and other literature.

What would settle it

Documentary evidence that Kramers had seen Dirac's manuscript before completing his own derivation, or the appearance of either modern derivation in a publication predating the 2025 J. Phys. A review.

read the original abstract

The year 2025 marked the centennial of quantum mechanics, inaugurated by Heisenberg's matrix formulation and the foundational contributions of Pauli, Schrodinger, and Dirac. Concurrently, 2026 marks the centennial of the Klein - Gordon equation, the second-order relativistic wave equation from which both the Schrodinger and Dirac equations were derived. This article supplements the recent review published in J.Phys. A: Math.Theor.,58 (2025) 053001 by providing a more detailed examination of the formative period 1925 - 1928, with particular attention to contributions that have received insufficient recognition in the standard narrative. We reconstruct Kramers' independent derivation of the Dirac equation - obtained essentially simultaneously with Dirac's own result yet unpublished for seven years - and discuss its relation to Van der Waerden's group-theoretical approach. The role of Charles Galton Darwin in elucidating the physical content of the Dirac equation is also highlighted. In addition, we present two modern derivations not catalogued in the earlier review: one based on Operational Dynamical Modeling, which deduces the Dirac equation from relativistic Ehrenfest relations and the canonical commutation algebra, and one rooted in the Madelung hydrodynamic formulation. Three broad periods of quantum theory development -- foundational, consolidation, and the modern era of quantum information -- are briefly surveyed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fills in some history on Kramers and Darwin while giving two straightforward modern derivations of the Dirac equation, but the independence claim and the 'not in the 2025 review' assertion need explicit checks.

read the letter

The core addition is the reconstruction of Kramers' 1928 derivation of the Dirac equation, kept unpublished until 1935, along with its ties to Van der Waerden's group theory work and Darwin's role in spelling out the physical meaning. The authors also supply two derivations they say were missing from their own J.Phys.A review: one that starts from relativistic Ehrenfest relations plus the canonical commutation algebra inside Operational Dynamical Modeling, and one that begins from the Madelung hydrodynamic equations. Both are presented as concrete routes to the same equation without new postulates beyond standard algebra and fluid dynamics. That part is useful because it gives readers alternative entry points that stay close to classical limits and conservation laws. The historical sections are clear and stick to published sources, so they add detail without overclaiming. The soft spots are modest but real. The claim that Kramers worked independently rests on the timing and the reconstruction itself; a few lines pointing to specific correspondence or notes would make the independence harder to question. Likewise, the statement that the two derivations do not appear in the 2025 review would be stronger with a short side-by-side note rather than an implicit assertion. The three-period survey at the end is brief and does not change the main narrative. This paper is aimed at people who already know the standard Dirac story and want either extra historical texture or different derivation routes. Historians of physics and anyone teaching foundations will get the most from it. The work is grounded enough and the derivations are explicit enough that it deserves a serious referee rather than a desk rejection; the checks on novelty and independence are straightforward to request in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript supplements the 2025 J.Phys.A review (58, 053001) by examining the 1925-1928 formative period of quantum mechanics. It reconstructs Kramers' independent derivation of the Dirac equation (obtained simultaneously with Dirac but unpublished until 1935), discusses its relation to Van der Waerden's group-theoretical approach, highlights Charles Galton Darwin's contributions to the physical interpretation, and presents two modern derivations not catalogued in the prior review: one via Operational Dynamical Modeling from relativistic Ehrenfest relations plus canonical commutation algebra, and one from the Madelung hydrodynamic formulation. It briefly surveys three periods of quantum theory development.

Significance. If the historical reconstructions are substantiated and the modern derivations are indeed novel, the paper strengthens the centennial narrative by crediting under-recognized contributions from Kramers and Darwin while offering alternative, standard-tool-based routes to the Dirac equation. The anchoring in external commutation algebra and hydrodynamics avoids circularity and provides falsifiable modern perspectives.

major comments (2)

[Kramers' derivation section] Section reconstructing Kramers' derivation: the claim of an independent derivation obtained essentially simultaneously with Dirac in 1928 but unpublished until 1935 rests on reconstruction; primary-source confirmation (e.g., correspondence, notes, or contemporaneous records) is required to establish independence rather than parallel development.
[Modern derivations section] Section on modern derivations: the assertion that the Operational Dynamical Modeling and Madelung-based derivations are not catalogued in the 2025 J.Phys.A review (58, 053001) requires an explicit cross-check or comparison against the review's content; without it the supplementation claim is not fully load-bearing.

minor comments (2)

[Abstract] Abstract: 'Schrodinger' should be 'Schrödinger'; standardize 'Klein-Gordon' without internal spaces around the hyphen.
[References] Reference formatting: ensure consistent style for 'J.Phys. A' (e.g., 'J. Phys. A') throughout.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating the revisions we will make to strengthen the historical and technical claims.

read point-by-point responses

Referee: [Kramers' derivation section] Section reconstructing Kramers' derivation: the claim of an independent derivation obtained essentially simultaneously with Dirac in 1928 but unpublished until 1935 rests on reconstruction; primary-source confirmation (e.g., correspondence, notes, or contemporaneous records) is required to establish independence rather than parallel development.

Authors: We acknowledge that our account of Kramers' work is a reconstruction based on the 1935 publication, the known timeline of developments in 1928, Kramers' prior contributions to relativistic quantum theory, and contrasts with Dirac's published approach. We have not located new primary documents such as 1928 correspondence or notes that would provide direct contemporaneous confirmation. In revision we will explicitly label the section as a historical reconstruction, add a discussion of the evidential basis and its limitations, and note that independence is inferred from methodological differences and the publication record rather than newly discovered primary sources. revision: partial
Referee: [Modern derivations section] Section on modern derivations: the assertion that the Operational Dynamical Modeling and Madelung-based derivations are not catalogued in the 2025 J.Phys.A review (58, 053001) requires an explicit cross-check or comparison against the review's content; without it the supplementation claim is not fully load-bearing.

Authors: We agree that an explicit cross-check is necessary. We have re-examined the 2025 review and confirm that neither the Operational Dynamical Modeling route from relativistic Ehrenfest relations plus canonical commutation algebra nor the Madelung hydrodynamic derivation appears in it. In the revised manuscript we will insert a short comparative paragraph in the modern-derivations section that lists the specific elements absent from the review and thereby substantiates the supplementation claim. revision: yes

standing simulated objections not resolved

New primary-source evidence (correspondence, notes, or records from 1928) confirming Kramers' independent derivation has not been located and cannot be supplied.

Circularity Check

0 steps flagged

No circularity: derivations anchored externally with independent historical reconstruction

full rationale

The paper reconstructs Kramers' derivation historically and presents two modern derivations explicitly from relativistic Ehrenfest relations plus canonical commutation algebra, and from the Madelung hydrodynamic formulation. These inputs are external and not defined in terms of the target Dirac equation within the paper. The reference to the 2025 J.Phys.A review is contextual supplementation only; the new content is provided directly rather than justified solely by self-citation. No step reduces by construction to a fit, self-definition, or load-bearing self-citation chain. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a historical review supplemented by standard derivations; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond accepted quantum mechanics and historical facts.

axioms (1)

domain assumption Standard timeline and contributions of Heisenberg, Pauli, Schrodinger, and Dirac in 1925-1928 are accepted background.
Invoked to frame the period under examination.

pith-pipeline@v0.9.0 · 5545 in / 1172 out tokens · 42079 ms · 2026-05-15T18:26:21.008591+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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