Sketch of a Gauge Model of Gravity with SU(2) Symmetry in Minkowski space

Nikolay Marchuk

arxiv: 2604.18280 · v1 · submitted 2026-04-20 · 🧮 math-ph · math.MP

Sketch of a Gauge Model of Gravity with SU(2) Symmetry in Minkowski space

Nikolay Marchuk This is my paper

Pith reviewed 2026-05-10 03:31 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords SU(2) gauge symmetrygravitational interactionMinkowski spaceDirac-type equationYang-Mills fieldfundamental fermionsClifford analysisgauge model of gravity

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The pith

A 2002 Dirac-type equation supplies an SU(2) gauge symmetry whose Yang-Mills field is identified as gravity for fermions in flat Minkowski space

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

The paper proposes replacing the ordinary Dirac equation with a 2002 Dirac-type equation that carries an extra SU(2) gauge symmetry. The Yang-Mills field generated by this symmetry is then interpreted as the gravitational interaction between fundamental fermions. All of this occurs inside ordinary Minkowski spacetime rather than a curved manifold. If the identification holds, gravity would be described by the same kind of gauge theory already used for the weak and strong forces.

Core claim

The Yang-Mills field that corresponds to the additional SU(2) gauge symmetry present in the 2002 Dirac-type equation is identified with the gravitational field of interacted fundamental fermions, yielding a gauge model of gravity that remains inside Minkowski space.

What carries the argument

The 2002 Dirac-type equation, which introduces an SU(2) gauge symmetry absent from the standard Dirac equation and whose associated Yang-Mills field is taken to be gravity.

If this is right

Gravitational effects on fermions are governed by the standard Yang-Mills equations for an SU(2) connection in flat space.
Gravity enters the same gauge-theoretic language used for electroweak and strong interactions.
No geometric curvature of spacetime is needed to account for the interaction.
Elements of Clifford analysis become available for handling the field equations.

Where Pith is reading between the lines

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

The model opens a route to quantize gravity with ordinary gauge-field techniques.
It raises the possibility that some phenomena currently attributed to spacetime curvature might instead be reinterpreted as SU(2) gauge effects.
Differences from general relativity could appear in high-energy or strongly quantum regimes.

Load-bearing premise

That the extra SU(2) symmetry of the 2002 Dirac-type equation can be consistently read as the gravitational interaction without producing contradictions with observations or requiring spacetime curvature.

What would settle it

A precise calculation or measurement of the force or spin precession between two fermions that deviates from the predictions of the SU(2) Yang-Mills equations in flat Minkowski space.

read the original abstract

We propose a gauge model with the SU(2) symmetry, which describes a gravitational interaction of fundamental fermions (leptons and quarks) in the Minkowski space. In the Standard Model one uses a Dirac-Yang-Mills system of equations with U(2) gauge symmetry for electroweak interactions and with SU(3) gauge symmetry for QCD interactions. A key idea of the model is to use the Dirac-type equation (invented in 2002) instead of the standard Dirac equation. This Dirac-type equation has an additional SU(2) gauge symmetry. The Yang-Mills field, which corresponds to this SU(2) symmetry, we identify with the gravitational field of interacted fundamental fermions. Some elements of Clifford analysis are used in the model.

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

This sketch identifies the SU(2) Yang-Mills field from Marchuk's 2002 Dirac-type equation with gravity in flat Minkowski space but supplies no derivations or limits to support the claim.

read the letter

The main thing to know is that Marchuk proposes treating the extra SU(2) gauge field in his earlier Dirac-type equation as the gravitational field for fermions, all while working strictly in flat space. The setup reuses that 2002 equation, notes its additional symmetry beyond the standard Dirac form, and applies ordinary gauge identification to link the corresponding Yang-Mills field to gravity. A few Clifford analysis elements are brought in to handle the algebra. The paper is explicit that this is only a sketch and connects the idea to how the Standard Model already uses gauge symmetries for other forces. That reuse of prior work is the main concrete step shown here. The proposal stays internally consistent on its own terms and avoids unnecessary complications by remaining in Minkowski space. The central weakness is the complete absence of supporting calculations. The abstract and sketch state the identification but do not derive the Newtonian limit for a static source, extract post-Newtonian corrections, or show that test particle motion reproduces geodesic behavior or the equivalence principle. Without those steps the claim stays an assertion rather than a checked result. The model also defines gravity by construction as the SU(2) field, leaving no independent benchmark against observations. Readers already following Marchuk's sequence of papers on modified Dirac equations or flat-space gauge gravity might find the outline useful as a starting point for further development. Most others working in general relativity or quantum gravity would see little immediate value until the missing limits are supplied. The work shows clear engagement with its own prior results and the gauge literature, but the evidential gaps are large enough that it does not yet warrant peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript sketches a gauge model of gravitational interaction for fundamental fermions (leptons and quarks) in flat Minkowski space. It replaces the standard Dirac equation with a 2002 Dirac-type equation possessing an extra SU(2) gauge symmetry and identifies the associated Yang-Mills field with gravity, while retaining the U(2) and SU(3) symmetries of the Standard Model and employing elements of Clifford analysis.

Significance. If the central identification were substantiated by explicit derivations, the model would constitute a flat-space, gauge-theoretic alternative to general relativity that could be embedded within the Standard Model framework. The reuse of the author's prior 2002 Dirac-type equation provides a concrete starting point, but the absence of any limit calculations or consistency checks leaves the significance prospective rather than demonstrated.

major comments (2)

[Abstract] Abstract: The statement that 'the Yang-Mills field, which corresponds to this SU(2) symmetry, we identify with the gravitational field' is presented as the key idea without any derivation of the coupled field equations, the non-relativistic limit for a static source, or the effective geodesic motion of test fermions. This identification is load-bearing for the entire claim yet remains an assertion.
[Model description] The model section (following the abstract): The gravitational field is introduced by direct identification with the SU(2) Yang-Mills field of the 2002 equation, with no independent external benchmark, no computation of the Newtonian potential, and no verification that the equivalence principle emerges from the flat-space dynamics. This renders the construction circular by design.

minor comments (2)

[Notation] The manuscript would benefit from explicit notation distinguishing the new SU(2) gauge field from the electroweak and QCD fields already present in the Standard Model.
[Discussion] A brief comparison table or paragraph contrasting the proposed field equations with the Einstein equations in the weak-field limit would improve clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our sketch of a gauge model of gravity. We address the major points below, emphasizing that the work is a conceptual proposal based on the 2002 Dirac-type equation rather than a complete derivation of all limits and consistency checks.

read point-by-point responses

Referee: [Abstract] The statement that 'the Yang-Mills field, which corresponds to this SU(2) symmetry, we identify with the gravitational field' is presented as the key idea without any derivation of the coupled field equations, the non-relativistic limit for a static source, or the effective geodesic motion of test fermions. This identification is load-bearing for the entire claim yet remains an assertion.

Authors: We acknowledge that the central identification is proposed without the requested derivations. The manuscript is explicitly framed as a sketch that adopts the independently derived 2002 Dirac-type equation (which carries the extra SU(2) symmetry) and posits that its Yang-Mills field corresponds to gravity for fundamental fermions in Minkowski space, while retaining Standard Model gauge symmetries. This provides a flat-space gauge-theoretic alternative whose full dynamical consequences, including coupled equations and limits, lie beyond the present scope. We will revise the abstract and introduction to state more explicitly that the identification is a defining postulate of the model to be explored further. revision: partial
Referee: [Model description] The gravitational field is introduced by direct identification with the SU(2) Yang-Mills field of the 2002 equation, with no independent external benchmark, no computation of the Newtonian potential, and no verification that the equivalence principle emerges from the flat-space dynamics. This renders the construction circular by design.

Authors: The construction begins from the 2002 Dirac-type equation, which was obtained independently of gravitational considerations and already possesses the additional SU(2) gauge symmetry. The subsequent identification of the associated Yang-Mills field with gravity defines the model we wish to study; it is not derived from an external benchmark but is instead the hypothesis under investigation. We agree that explicit calculations of the Newtonian potential or emergence of the equivalence principle are absent and would be valuable, yet their absence does not render the proposal circular. The use of Clifford analysis elements supports the mathematical consistency of the symmetry structure within flat space. revision: no

standing simulated objections not resolved

Derivation of the coupled field equations, non-relativistic limit for a static source, effective geodesic motion, Newtonian potential, and verification that the equivalence principle emerges from the flat-space dynamics.

Circularity Check

2 steps flagged

SU(2) Yang-Mills field identified with gravity by direct assertion, without deriving Newtonian or geodesic limits

specific steps

self definitional [Abstract]
"The Yang-Mills field, which corresponds to this SU(2) symmetry, we identify with the gravitational field of interacted fundamental fermions."

The model is introduced as a 'gauge model of gravity,' yet the only link supplied is the explicit identification of the pre-existing SU(2) Yang-Mills field with gravity. Because no dynamical derivation (e.g., post-Newtonian expansion or geodesic equation) is performed to confirm that this field reproduces gravitational phenomenology, the claim reduces to a re-labeling of the input field.
self citation load bearing [Abstract]
"A key idea of the model is to use the Dirac-type equation (invented in 2002) instead of the standard Dirac equation. This Dirac-type equation has an additional SU(2) gauge symmetry."

The extra SU(2) symmetry is imported from the author's prior 2002 work; the present manuscript treats that symmetry as given and immediately equates its gauge field with gravity. No external benchmark or independent derivation is supplied to justify why this particular gauge field must be gravitational.

full rationale

The paper's central claim rests on adopting the author's 2002 Dirac-type equation (which already contains an extra SU(2) gauge symmetry by construction) and then declaring that the associated Yang-Mills field is the gravitational field. No subsequent derivation computes the weak-field limit, static-source potential, or test-particle motion to show equivalence with Newtonian gravity or the equivalence principle. The identification therefore functions as a definitional renaming rather than an independent result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The model rests on the properties of the 2002 Dirac-type equation and the direct identification of its gauge field with gravity; no independent evidence for the identification is provided.

axioms (2)

domain assumption The 2002 Dirac-type equation possesses an additional SU(2) gauge symmetry beyond the standard Dirac equation.
Invoked in the abstract as the key idea enabling the model.
ad hoc to paper The Yang-Mills field of this SU(2) symmetry can be identified with the gravitational field.
Stated directly in the abstract without further justification.

invented entities (1)

SU(2) gravitational Yang-Mills field no independent evidence
purpose: To represent gravity acting on fundamental fermions
Postulated by identifying the gauge field of the modified Dirac equation with gravity; no independent falsifiable handle given.

pith-pipeline@v0.9.0 · 5421 in / 1405 out tokens · 55871 ms · 2026-05-10T03:31:35.441626+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages · 1 internal anchor

[1]

IOP Publishing, Bristol (2002)

Blagojevic, M.: Gravitation and Gauge Symmetries. IOP Publishing, Bristol (2002)

2002
[2]

World Scientific, (2009)

Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Advanced Classical Field Theory. World Scientific, (2009). doi:10.1142/7189

work page doi:10.1142/7189 2009
[3]

: Gauge theory of gravitation - A unified formulation of Poincare and (Anti-)de Sitter gauge theories

Hayashi, K., Shirafuji, T. : Gauge theory of gravitation - A unified formulation of Poincare and (Anti-)de Sitter gauge theories. Progr. Theor. Phys. 80 (4), 711-730 (1988). doi:10.1143/PTP.80.711

work page doi:10.1143/ptp.80.711 1988
[4]

Hehl, F., McCrea, J., Mielke, E., Neeman, Y.: Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance. Phys. Rep. 258 (1-2), 1-171 (1995). doi:10.1016/0370-1573(94)00111-F

work page doi:10.1016/0370-1573(94)00111-f 1995
[5]

Amazon, CreateSpace Independent Publishing Platform, (2012)

Marchuk, N.: Field Theory Equations. Amazon, CreateSpace Independent Publishing Platform, (2012)

2012
[6]

Marchuk N.G., Shirokov D.S., General solutions of one class of field equations , Rep. Math. Phys., 78:3 (2016), 305-326, arXiv: 1406.6665

work page arXiv 2016
[7]

Nuovo Cimento Soc

Marchuk, N.: A concept of Dirac-type tensor equations. Nuovo Cimento Soc. Ital. Fis. B, 117 (12), 1357-1385 (2002)

2002
[8]

Cambridge University Press, (2009)

Oriti, D.: Approaches to quantum gravity: Toward a new understanding of space, time and matter. Cambridge University Press, (2009). doi:10.1017/CBO9780511575549

work page doi:10.1017/cbo9780511575549 2009
[9]

10 Congres Math

Riesz, M.: In: C.R. 10 Congres Math. Scandinaves, Copenhagen, 1946. Jul. Gjellerups Forlag, Copenhagen, pp.123-148 (1947). Reprinted in G rding, L., H\"ermander, L. (eds.): Marcel Reisz, Collected Papers. Springer, Berlin, 814-832 (1988)

1946
[10]

Rovelli, C.: Notes for a brief history of quantum gravity. (2000). arXiv:gr-qc/0006061

work page internal anchor Pith review arXiv 2000
[11]

Utiyama, R.: Invariant theoretical interpretation of interaction. Phys. Rev. 101 , 1597 (1956). doi:10.1103/PhysRev.101.1597

work page doi:10.1103/physrev.101.1597 1956

[1] [1]

IOP Publishing, Bristol (2002)

Blagojevic, M.: Gravitation and Gauge Symmetries. IOP Publishing, Bristol (2002)

2002

[2] [2]

World Scientific, (2009)

Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Advanced Classical Field Theory. World Scientific, (2009). doi:10.1142/7189

work page doi:10.1142/7189 2009

[3] [3]

: Gauge theory of gravitation - A unified formulation of Poincare and (Anti-)de Sitter gauge theories

Hayashi, K., Shirafuji, T. : Gauge theory of gravitation - A unified formulation of Poincare and (Anti-)de Sitter gauge theories. Progr. Theor. Phys. 80 (4), 711-730 (1988). doi:10.1143/PTP.80.711

work page doi:10.1143/ptp.80.711 1988

[4] [4]

Hehl, F., McCrea, J., Mielke, E., Neeman, Y.: Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance. Phys. Rep. 258 (1-2), 1-171 (1995). doi:10.1016/0370-1573(94)00111-F

work page doi:10.1016/0370-1573(94)00111-f 1995

[5] [5]

Amazon, CreateSpace Independent Publishing Platform, (2012)

Marchuk, N.: Field Theory Equations. Amazon, CreateSpace Independent Publishing Platform, (2012)

2012

[6] [6]

Marchuk N.G., Shirokov D.S., General solutions of one class of field equations , Rep. Math. Phys., 78:3 (2016), 305-326, arXiv: 1406.6665

work page arXiv 2016

[7] [7]

Nuovo Cimento Soc

Marchuk, N.: A concept of Dirac-type tensor equations. Nuovo Cimento Soc. Ital. Fis. B, 117 (12), 1357-1385 (2002)

2002

[8] [8]

Cambridge University Press, (2009)

Oriti, D.: Approaches to quantum gravity: Toward a new understanding of space, time and matter. Cambridge University Press, (2009). doi:10.1017/CBO9780511575549

work page doi:10.1017/cbo9780511575549 2009

[9] [9]

10 Congres Math

Riesz, M.: In: C.R. 10 Congres Math. Scandinaves, Copenhagen, 1946. Jul. Gjellerups Forlag, Copenhagen, pp.123-148 (1947). Reprinted in G rding, L., H\"ermander, L. (eds.): Marcel Reisz, Collected Papers. Springer, Berlin, 814-832 (1988)

1946

[10] [10]

Rovelli, C.: Notes for a brief history of quantum gravity. (2000). arXiv:gr-qc/0006061

work page internal anchor Pith review arXiv 2000

[11] [11]

Utiyama, R.: Invariant theoretical interpretation of interaction. Phys. Rev. 101 , 1597 (1956). doi:10.1103/PhysRev.101.1597

work page doi:10.1103/physrev.101.1597 1956