Gauge invariant generalizations of the Proca equation and the Yang-Mills-Proca equation

Nikolay Marchuk

arxiv: 2606.20828 · v1 · pith:QOQONLGZnew · submitted 2026-06-18 · 🧮 math-ph · math.MP

Gauge invariant generalizations of the Proca equation and the Yang-Mills-Proca equation

Nikolay Marchuk This is my paper

Pith reviewed 2026-06-26 15:04 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Proca equationgauge invarianceYang-Mills-Proca equationvector bosonsnon-Abelian gauge symmetrymassive vector fieldsgauge transformations

0 comments

The pith

Adding an extra vector field renders the Proca equations gauge invariant.

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

The paper demonstrates that the Proca equations for massive vector bosons, which break gauge invariance, can be made gauge invariant by introducing a second vector field. This construction avoids the Stueckelberg approach and extends directly to the Yang-Mills-Proca equations to produce versions with non-Abelian gauge symmetry. A reader would care because gauge invariance is required for consistent quantization and for eliminating unphysical modes in vector field theories. The method keeps the mass term intact while restoring the symmetry.

Core claim

By introducing an additional vector field into the Proca equation, a gauge invariant generalization is obtained; the results extend to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

What carries the argument

An additional vector field introduced into the Proca equation to restore gauge invariance.

If this is right

The generalized equations describe massive spin-1 particles while remaining invariant under gauge transformations.
The same construction yields gauge-invariant Yang-Mills-Proca equations for non-Abelian groups.
The mass term for the vector field is preserved in the gauge-invariant formulation.
Physical predictions for the vector bosons remain those of the standard Proca theory.

Where Pith is reading between the lines

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

The construction might allow consistent effective theories of massive gauge bosons without spontaneous symmetry breaking.
Quantization procedures developed for gauge theories could be applied directly to these massive generalizations.
The extra field could be interpreted as an auxiliary variable that decouples in the physical sector.

Load-bearing premise

The added vector field restores gauge invariance without introducing extra unphysical degrees of freedom or changing the physical spectrum of the original Proca theory.

What would settle it

A direct count of propagating degrees of freedom in the new equations that differs from the two transverse polarizations of a massive vector boson in the original Proca theory would falsify the claim.

read the original abstract

The Proca equations (1936) are used in quantum field theory to describe vector bosons (spin 1) with a nonzero mass. The Proca equations are not gauge invariant. In contrast to Stueckelberg's approach (1938), this article presents a gauge invariant generalization of the Proca equation by introducing an additional vector field into the Proca equation. The results are extended to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

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 claims a gauge-invariant Proca generalization via an added vector field (distinct from Stueckelberg), extended to the non-Abelian case, but supplies no equations or constraint analysis in the abstract.

read the letter

The central claim is that coupling the Proca field to a second vector restores gauge invariance while preserving the massive vector content, and the same trick works for Yang-Mills-Proca. That is the one concrete thing the abstract offers beyond the 1936-1938 literature.

What the work does is straightforward: it replaces the usual mass term with an interaction involving the extra field and asserts that the resulting system is gauge invariant under a joint transformation of both vectors. The extension to the non-Abelian case is stated without extra machinery, which at least keeps the proposal compact.

The soft spot is exactly the one the stress-test flags. A free vector adds three polarizations; nothing in the abstract shows how the coupled equations eliminate the extra three modes, enforce the original dispersion relation, or avoid ghosts. No Lagrangian is written, no constraint count is given, and no reduction to the Stueckelberg case or to ordinary Proca is exhibited. Without those steps the physical equivalence remains unverified.

This is a short mathematical-physics note aimed at readers who already work on alternative massive gauge formulations. It will be useful to them only if the full text contains an explicit Hamiltonian or BRST analysis that closes the degrees-of-freedom count. On present evidence the construction is too schematic to cite or to build on.

I would send it to referees rather than desk-reject, but the first round of reports should focus narrowly on whether the extra field is truly auxiliary.

Referee Report

1 major / 0 minor

Summary. The paper claims to obtain a gauge-invariant generalization of the Proca equation (which lacks gauge invariance) by introducing an additional vector field, in contrast to the Stueckelberg approach that uses a scalar field. The construction is extended to the Yang-Mills-Proca equations to produce equations possessing non-Abelian gauge symmetry.

Significance. If the added vector field restores gauge invariance while exactly preserving the original Proca spectrum (three physical polarizations, dispersion relation ω² = k² + m², no ghosts or extra modes), the result would supply an alternative to the Stueckelberg mechanism and could be useful for formulating massive vector bosons in both Abelian and non-Abelian settings. The non-Abelian extension would be of interest if it maintains consistency with the original Yang-Mills-Proca dynamics.

major comments (1)

[Abstract and main construction] The central construction adds a second vector field to restore gauge invariance, yet the manuscript supplies no constraint count, Hamiltonian analysis, or explicit reduction showing that the extra three polarizations are eliminated by algebraic relations or Lagrange multipliers. Without this, it is impossible to verify that the physical spectrum remains identical to the original Proca theory (three degrees of freedom, no ghosts).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need to verify the physical spectrum explicitly. We address the major comment below and will incorporate the requested analysis in a revised manuscript.

read point-by-point responses

Referee: [Abstract and main construction] The central construction adds a second vector field to restore gauge invariance, yet the manuscript supplies no constraint count, Hamiltonian analysis, or explicit reduction showing that the extra three polarizations are eliminated by algebraic relations or Lagrange multipliers. Without this, it is impossible to verify that the physical spectrum remains identical to the original Proca theory (three degrees of freedom, no ghosts).

Authors: We agree that an explicit constraint analysis is required to confirm that the additional vector field does not alter the physical spectrum. The current manuscript focuses on the construction of the gauge-invariant equations but does not include a Hamiltonian formulation or degree-of-freedom count. In the revised version we will add a new section performing a constraint analysis (following the standard Dirac procedure) that demonstrates the extra polarizations are eliminated by second-class constraints, leaving precisely three physical degrees of freedom with the Proca dispersion relation and no ghosts. The same analysis will be extended to the non-Abelian case. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction via added vector field, no reduction to inputs or self-citations

full rationale

The paper presents a gauge-invariant generalization of the Proca equation explicitly by introducing an additional vector field, extending the same idea to the Yang-Mills-Proca case. No derivation chain, fitted parameters, uniqueness theorems, or self-citations are invoked in the abstract or described approach. The central step is a constructive ansatz rather than a claim that reduces by construction to prior inputs or external results. This is a standard non-circular presentation of a new Lagrangian modification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract; the central construction rests on the existence and properties of the added vector field, which is introduced without independent evidence supplied in the visible text.

invented entities (1)

additional vector field no independent evidence
purpose: to restore gauge invariance to the Proca equation while retaining the mass term
Explicitly introduced in the abstract as the mechanism for the generalization; no independent evidence or falsifiable prediction is stated.

pith-pipeline@v0.9.1-grok · 5596 in / 1092 out tokens · 20114 ms · 2026-06-26T15:04:00.197914+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages · 2 internal anchors

[1]

Steklov Mathematical Institute of Russian Academy of Sciences
[2]

Later, a number of modifications and generalizations of the Proca equation were proposed (see, for example, reviews [6], [7])

National Research University Higher School of Economics, orcid.org/0000-0001-6185-434X 1 arXiv:2606.20828v1 [math-ph] 18 Jun 2026 In 1938, Stueckelberg [2, 3, 4, 5] considered a gauge-invariant modification of the Proca equation obtained by introducing an additional scalar field into the Proca equation. Later, a number of modifications and generalizations...

Pith/arXiv arXiv 2026
[3]

Proca, A., Sur la th´ eorie ondulatoire des ´ electrons positifs et n´ egatifs [On the wave theory of positive and negative electrons], J. de Phys. et le Radium 7, pp. 347-353, (1936)

1936
[4]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (I) [The interaction forces in electrodynamics and in the field theory of nuclear forces (I)], Helv. Phys. Acta 11, 225-244, (1938)

1938
[5]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (II) [The interaction forces in electrodynamics and in the field theory of nuclear forces (II)], Helv. Phys. Acta 11, 299-312, (1938)

1938
[6]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (III) [The interaction forces in electrodynamics and in the field theory of nuclear forces (III)], Helv. Phys. Acta 11, 312-328, (1938)

1938
[7]

Henri Ruegg, Marti Ruiz-Altaba, The Stueckelberg Field, Int.J.Mod.Phys. Vol. 19, No. 20, pp. 3265-3348, (2004), https://doi.org/10.1142/S0217751X04019755, arXiv:hep-th/0304245

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x04019755 2004
[8]

(https://www.sciencedirect.com/science/article/pii/S0370269317301806)

Jose Beltr´ an Jim´ enez, Lavinia Heisenberg, Generalized multi- Proca fields, Physics Letters B, Volume 770, 2017, Pages 16- 26, ISSN 0370-2693, https://doi.org/10.1016/j.physletb.2017.03.002. (https://www.sciencedirect.com/science/article/pii/S0370269317301806)

work page doi:10.1016/j.physletb.2017.03.002 2017
[9]

Erwan Allys, Patrick Peter, Yeinzon Rodr´ ıguez, Generalized SU(2) Proca theory, Phys. Rev. D 94, 084041, (2016), arXiv:1609.05870 DOI: https://doi.org/10.1103/PhysRevD.94.084041

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.084041 2016
[10]

C. N. Yang, R. Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review, 1954. Vol. 96, no. 1, pp. 191-195, doi:10.1103/PhysRev.96.191

work page doi:10.1103/physrev.96.191 1954
[11]

- 2nd ed., revised, M.: MTsNMO, (2014), 584 p., ISBN 978-5-4439-0182-4

Novikov S.P., Taimanov I.A., Modern geometric structures and fields. - 2nd ed., revised, M.: MTsNMO, (2014), 584 p., ISBN 978-5-4439-0182-4

2014
[12]

N. G. Marchuk, D. S. Shirokov, Constant solutions of Yang-Mills equations and generalized Proca equations, J. Geom. Symmetry Phys., 42 (2016), 53-72

2016
[13]

D. S. Shirokov, Hyperbolic Singular Value Decomposition in the Study of Yang–Mills and Yang–Mills–Proca Equations, Computational Mathematics and Mathematical Physics, 62:6 (2022), 1007–1019. 9

2022

[1] [1]

Steklov Mathematical Institute of Russian Academy of Sciences

[2] [2]

Later, a number of modifications and generalizations of the Proca equation were proposed (see, for example, reviews [6], [7])

National Research University Higher School of Economics, orcid.org/0000-0001-6185-434X 1 arXiv:2606.20828v1 [math-ph] 18 Jun 2026 In 1938, Stueckelberg [2, 3, 4, 5] considered a gauge-invariant modification of the Proca equation obtained by introducing an additional scalar field into the Proca equation. Later, a number of modifications and generalizations...

Pith/arXiv arXiv 2026

[3] [3]

Proca, A., Sur la th´ eorie ondulatoire des ´ electrons positifs et n´ egatifs [On the wave theory of positive and negative electrons], J. de Phys. et le Radium 7, pp. 347-353, (1936)

1936

[4] [4]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (I) [The interaction forces in electrodynamics and in the field theory of nuclear forces (I)], Helv. Phys. Acta 11, 225-244, (1938)

1938

[5] [5]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (II) [The interaction forces in electrodynamics and in the field theory of nuclear forces (II)], Helv. Phys. Acta 11, 299-312, (1938)

1938

[6] [6]

Stueckelberg, E. C. G., Die Wechselwirkungs Kraefte in der Elektrodynamik und in der Feldtheorie der Kernkraefte (III) [The interaction forces in electrodynamics and in the field theory of nuclear forces (III)], Helv. Phys. Acta 11, 312-328, (1938)

1938

[7] [7]

Henri Ruegg, Marti Ruiz-Altaba, The Stueckelberg Field, Int.J.Mod.Phys. Vol. 19, No. 20, pp. 3265-3348, (2004), https://doi.org/10.1142/S0217751X04019755, arXiv:hep-th/0304245

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x04019755 2004

[8] [8]

(https://www.sciencedirect.com/science/article/pii/S0370269317301806)

Jose Beltr´ an Jim´ enez, Lavinia Heisenberg, Generalized multi- Proca fields, Physics Letters B, Volume 770, 2017, Pages 16- 26, ISSN 0370-2693, https://doi.org/10.1016/j.physletb.2017.03.002. (https://www.sciencedirect.com/science/article/pii/S0370269317301806)

work page doi:10.1016/j.physletb.2017.03.002 2017

[9] [9]

Erwan Allys, Patrick Peter, Yeinzon Rodr´ ıguez, Generalized SU(2) Proca theory, Phys. Rev. D 94, 084041, (2016), arXiv:1609.05870 DOI: https://doi.org/10.1103/PhysRevD.94.084041

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.084041 2016

[10] [10]

C. N. Yang, R. Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review, 1954. Vol. 96, no. 1, pp. 191-195, doi:10.1103/PhysRev.96.191

work page doi:10.1103/physrev.96.191 1954

[11] [11]

- 2nd ed., revised, M.: MTsNMO, (2014), 584 p., ISBN 978-5-4439-0182-4

Novikov S.P., Taimanov I.A., Modern geometric structures and fields. - 2nd ed., revised, M.: MTsNMO, (2014), 584 p., ISBN 978-5-4439-0182-4

2014

[12] [12]

N. G. Marchuk, D. S. Shirokov, Constant solutions of Yang-Mills equations and generalized Proca equations, J. Geom. Symmetry Phys., 42 (2016), 53-72

2016

[13] [13]

D. S. Shirokov, Hyperbolic Singular Value Decomposition in the Study of Yang–Mills and Yang–Mills–Proca Equations, Computational Mathematics and Mathematical Physics, 62:6 (2022), 1007–1019. 9

2022