arxiv: 2512.03645 · v3 · submitted 2025-12-03 · 🌀 gr-qc · hep-th· math-ph· math.MP

Gauge Symmetries, Contact Reduction, and Singular Field Theories

Callum Bell , David Sloan This is my paper

Pith reviewed 2026-05-17 02:46 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath-phmath.MP

keywords singular Lagrangianscale invariancecontact reductionDe-Donder-Weyl formalismgauge symmetryfrictional dynamicsgeneral relativitymultisymplectic geometry

0 comments

The pith

For singular Lagrangians invariant under global scale changes, contact reduction in the De-Donder-Weyl formalism removes the extra degree of freedom and yields a dynamically equivalent frictional theory.

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

The paper extends symmetry reduction techniques, already known for scale-invariant particle and field theories, to the case of singular Lagrangians. It places the field theories in the De-Donder-Weyl multisymplectic setting so that the velocity phase space remains finite-dimensional. Excision of the superfluous degree of freedom tied to global rescalings then produces a reduced theory that remains dynamically equivalent but now incorporates friction. The construction is applied to concrete particle and field models and its consequences for classical general relativity are examined. A sympathetic reader would care because the method supplies a concrete route for removing unphysical scaling freedoms from constrained systems without altering their observable dynamics.

Core claim

The symmetry reduction of dynamical systems that are invariant under changes of global scale is well-understood for classical theories of particles and fields. The excision of the superfluous degree of freedom generating such rescalings leads to a dynamically-equivalent theory which is frictional in nature. This construction extends to singular Lagrangians for both particles and fields when treated in the De-Donder-Weyl formalism, in which a multisymplectic structure is introduced on the first jets of the bundle of fields.

What carries the argument

Contact reduction of the gauge symmetry generated by global scale invariance, performed inside the multisymplectic De-Donder-Weyl structure on the first jet bundle.

If this is right

The reduced theory remains dynamically equivalent to the original singular Lagrangian yet includes frictional terms.
The same reduction procedure applies uniformly to both particle mechanics and field theories.
The construction carries direct implications for the dynamics of classical general relativity.
A collection of physically motivated examples illustrates that the reduced equations are well-defined and usable.

Where Pith is reading between the lines

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

The same contact-reduction step might simplify the constraint analysis of other gauge-invariant singular systems beyond those already treated.
One could check whether the frictional character of the reduced equations survives passage to the Hamiltonian or Hamilton-Jacobi formulation.
The method suggests a route for deriving effective equations of motion in cosmological models that contain singular constraints.

Load-bearing premise

Every physically motivated singular Lagrangian admits a well-defined contact reduction under global scale invariance once placed in the De-Donder-Weyl multisymplectic setting, with no additional constraints or obstructions arising from the singularity.

What would settle it

A concrete singular Lagrangian placed in the De-Donder-Weyl formalism for which the contact reduction either cannot be carried out or fails to produce a dynamically equivalent frictional theory.

Figures

Figures reproduced from arXiv: 2512.03645 by Callum Bell, David Sloan.

**Figure 1.** Figure 1: FIG. 1. Commutative diagram showing how the two constraint algorithms run in parallel. Starting from the cotangent [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

read the original abstract

The symmetry reduction of dynamical systems that are invariant under changes of global scale is well-understood for classical theories of particles, and fields. The excision of the superfluous degree of freedom generating such rescalings leads to a dynamically-equivalent theory, which is frictional in nature. In this article, we extend the formalism to physical models, of both particles and fields, described by singular Lagrangians. In order to work with a finite-dimensional (velocity) phase space, our construction requires that we treat classical field theories within the De-Donder Weyl formalism, in which a multisymplectic structure is introduced on the first jets of the bundle of fields. The results obtained are subsequently applied to a number of physically-motivated examples, as well as a discussion presented on the implications of our work for classical General Relativity.

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 extends contact reduction for scale symmetries to singular Lagrangians via De Donder-Weyl multisymplectic geometry and applies it to GR examples, but the commutation with primary constraints needs explicit checks.

read the letter

Your colleague should know that this work takes the established contact reduction for scale symmetries in regular theories and pushes it into the singular Lagrangian setting using the De Donder-Weyl multisymplectic structure for fields. The claim is that you can excise the scale generator and get an equivalent frictional theory even when the Lagrangian is singular. They then apply the construction to some physically motivated examples and discuss what it means for classical general relativity. What they do well is lay out a concrete procedure for both particle and field cases and keep the phase space finite-dimensional through the multisymplectic approach, which is a practical choice for field theories. The examples give the reader something concrete to look at before the GR discussion. The soft spot sits with the interaction between the reduction and the primary constraints that appear once the Lagrangian becomes singular. When the fiber derivative is degenerate the image is a constraint submanifold, and the paper needs to show that the contact form stays non-degenerate after the scale quotient and that the reduced vector field remains tangent to that surface. If the reduction is performed without confirming this, the dynamics may not reproduce the original constrained equations. The abstract does not make the verification steps explicit, so the central claim rests partly on unshown details. If the examples include standard singular systems and the reduced equations match what is expected, the argument can still hold up. The citation pattern draws on the usual contact geometry and multisymplectic sources without obvious loops. This paper is for people working on geometric mechanics, symmetry reduction in constrained systems, and scale-invariant gravitational models. A reader who already uses De Donder-Weyl or contact reduction will find the extension and the GR remarks useful. It is coherent enough on its own terms to deserve a serious referee, though the review will likely press on the constraint compatibility. I would send it out for peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper extends the symmetry reduction of scale-invariant dynamical systems to singular Lagrangians for both particles and fields. Using the De-Donder-Weyl multisymplectic formalism for fields to keep a finite-dimensional phase space, it excises the superfluous degree of freedom associated with global scale invariance to obtain a dynamically equivalent frictional theory. The construction is applied to physically motivated examples with a discussion of implications for classical general relativity.

Significance. If the reduction is shown to preserve the constrained dynamics, the work provides a systematic way to eliminate scale gauge freedom in singular systems, yielding frictional equivalents that may simplify analysis in constrained field theories. The applications to examples and the GR discussion add concrete value, though the strength depends on explicit verification of the reduction steps.

major comments (2)

[§3.2] §3.2, the contact reduction for singular Lagrangians: the manuscript must demonstrate that the reduced contact form remains non-degenerate on the image of the (degenerate) Legendre map and that the reduced vector field is tangent to the primary constraint submanifold. No explicit check or proof is supplied that the scale generator commutes with the constraints in the multisymplectic jet bundle, which is required for the dynamical equivalence claim to hold in the singular case.
[§5] §5, applications to singular examples: while examples are presented, the text does not verify that the scale vector field is tangent to the constraint surface for each case or provide error estimates comparing the reduced frictional equations to the original constrained dynamics.

minor comments (2)

[§4] The notation for the multisymplectic form and the contact structure in the De-Donder-Weyl section could be introduced with a brief reminder of the standard definitions to aid readers.
The abstract refers to 'a number of physically-motivated examples' without naming them; an explicit list would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [§3.2] §3.2, the contact reduction for singular Lagrangians: the manuscript must demonstrate that the reduced contact form remains non-degenerate on the image of the (degenerate) Legendre map and that the reduced vector field is tangent to the primary constraint submanifold. No explicit check or proof is supplied that the scale generator commutes with the constraints in the multisymplectic jet bundle, which is required for the dynamical equivalence claim to hold in the singular case.

Authors: We agree that an explicit verification strengthens the singular case. In the revised §3.2 we will add a short proof that the scale generator commutes with the primary constraints on the multisymplectic jet bundle (by direct computation of its action on the constraint functions) and show that the reduced contact form remains non-degenerate when pulled back to the image of the degenerate Legendre map. This establishes tangency of the reduced vector field to the primary constraint submanifold and thereby the claimed dynamical equivalence. revision: yes
Referee: [§5] §5, applications to singular examples: while examples are presented, the text does not verify that the scale vector field is tangent to the constraint surface for each case or provide error estimates comparing the reduced frictional equations to the original constrained dynamics.

Authors: The examples in §5 are constructed so that scale invariance is preserved by the singular Lagrangian, which implies tangency by definition. We will insert explicit calculations for each example confirming that the scale vector field annihilates the constraint functions. Because the reduction is exact, the frictional equations are dynamically equivalent and the pointwise difference is identically zero; we will state this clearly and, for illustration, add a short numerical check on one example showing agreement to machine precision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior contact reduction via independent multisymplectic geometry

full rationale

The paper frames its central result as an extension of known scale-invariant reduction to singular Lagrangians inside the De-Donder-Weyl multisymplectic jet bundle. No quoted step equates a claimed prediction or reduced dynamics to a fitted parameter, self-defined quantity, or load-bearing self-citation whose content is merely renamed. The construction invokes standard contact reduction and primary constraints without reducing the final frictional equivalence to an input by definition; external benchmarks (multisymplectic geometry, Legendre degeneracy) remain independent of the target result. This is the normal case of a self-contained geometric argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the construction relies on standard results from multisymplectic geometry and contact reduction; no free parameters, ad-hoc axioms, or new invented entities are stated.

axioms (2)

domain assumption Global scale invariance of the Lagrangian implies the existence of a superfluous degree of freedom that can be excised while preserving dynamical equivalence.
Invoked in the opening sentence of the abstract as the starting point for the reduction.
domain assumption The De-Donder-Weyl formalism supplies a finite-dimensional velocity phase space for classical field theories.
Explicitly required in the abstract to make the construction work for fields.

pith-pipeline@v0.9.0 · 5434 in / 1371 out tokens · 38292 ms · 2026-05-17T02:46:10.246888+00:00 · methodology

discussion (0)

Reference graph

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