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arxiv: 2605.22260 · v1 · pith:BITHCRPJnew · submitted 2026-05-21 · ⚛️ physics.hist-ph · gr-qc

The Limitations of the Notion of `Observable' in Diffeomorphism-Invariant Models

\'Alvaro Mozota Frauca This is my paper

Pith reviewed 2026-05-22 01:58 UTC · model grok-4.3

classification ⚛️ physics.hist-ph gr-qc
keywords diffeomorphism invariancegauge symmetryobservablesgeneral relativitycorrelationsgauge transformationsquantum gravity
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The pith

Diffeomorphism invariance is a gauge symmetry only globally, restricting the observable concept in theories like general relativity.

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

The paper separates gauge transformations into local and global senses. Observables appear naturally only in the local sense. Diffeomorphism invariance fits the global sense of gauge symmetry. Therefore the observable notion applies only restrictively, leaving claims that correlations encode all physical content without support.

Core claim

Diffeomorphism invariance can be understood as a gauge symmetry only from a global point of view, and hence that the concept of observable applies only in a restricted manner. This has the consequence that some popular claims in the literature, such as the claim that the physical content of diffeomorphism-invariant models is encoded in correlations, are unfounded.

What carries the argument

The distinction between local and global senses of gauge transformation, which determines where the observable concept applies naturally.

If this is right

  • The standard notion of observable has only restricted applicability in diffeomorphism-invariant models.
  • Claims that physical content is encoded in correlations between quantities are unfounded.
  • Interpretations of classical and quantum versions of these theories must respect this limitation on observables.
  • Approaches to quantizing general relativity that rely on unrestricted observables require adjustment.

Where Pith is reading between the lines

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

  • The local-global distinction might prompt similar scrutiny of observables in other background-independent physical theories.
  • It suggests adapting gauge-theory tools when symmetries are global rather than local.
  • Specific models such as certain cosmological spacetimes could be checked to see whether local observables emerge under this view.
  • This framing may shift how the problem of defining physical quantities is approached in quantum gravity.

Load-bearing premise

The local versus global distinction for gauge transformations is the decisive factor for when the observable concept applies, and diffeomorphisms count exclusively as the global kind.

What would settle it

A consistent treatment of diffeomorphisms as local gauge transformations that permits definition of standard observables capturing the full physical content.

Figures

Figures reproduced from arXiv: 2605.22260 by \'Alvaro Mozota Frauca.

Figure 1
Figure 1. Figure 1: Configuration space trajectories of two harmonic oscillators [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
read the original abstract

The application of the notion of `observable' from gauge theory to diffeomorphism-invariant theories -- most relevantly to general relativity -- has led to numerous conceptual and technical issues when interpreting classical theories with this symmetry and building quantum versions of them. In this article I distinguish between two senses of gauge transformation: local and global, and I argue that the notion of observable appears more naturally in the local sense of gauge transformation. Then, I argue that diffeomorphism invariance can be understood as a gauge symmetry only from a global point of view, and hence, that the concept of observable applies only in a restricted manner. This has the consequence that some popular claims in the literature, such as the claim that the physical content of diffeomorphism-invariant models is encoded in correlations, are unfounded.

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.

Referee Report

2 major / 2 minor

Summary. The paper distinguishes two senses of gauge transformation—local and global—and argues that the notion of observable applies naturally only in the local sense. It claims that diffeomorphism invariance can be understood as a gauge symmetry only from a global point of view, with the consequence that the concept of observable applies only in a restricted manner and that popular claims (such as the physical content of diffeomorphism-invariant models being encoded in correlations) are unfounded.

Significance. If the local/global distinction and the classification of diffeomorphisms hold, the paper would provide a novel philosophical framework for interpreting gauge symmetries in general relativity and related models. This could prompt reevaluation of standard treatments of Dirac observables and relationalism in the foundations of quantum gravity, while highlighting limitations in applying gauge-theoretic notions of observability to diffeomorphism-invariant theories.

major comments (2)
  1. [Section on local vs. global gauge transformations] The section introducing the local/global distinction for gauge transformations: the claim that observables appear more naturally in the local sense is presented as following from the distinction itself, but the manuscript does not derive this from the standard definition of observables as quantities invariant under the full gauge group action (as in constrained Hamiltonian systems). Without addressing why the split is the sole relevant criterion, the subsequent restriction on observables remains framework-dependent rather than derived.
  2. [Argument on diffeomorphism invariance as global gauge] The argument classifying diffeomorphism invariance as gauge only globally: the manuscript asserts that diffeomorphisms correspond exclusively to the global sense, yet standard treatments treat diffeomorphisms as local (position-dependent) symmetries whose action is not restricted to a global viewpoint. This classification appears stipulative rather than derived from the transformation properties, undermining the conclusion that the observable concept applies only restrictedly and that correlation-based claims are unfounded.
minor comments (2)
  1. Add explicit comparison to Dirac's definition of observables and to standard references on gauge symmetries in GR to clarify how the proposed distinction departs from or extends existing literature.
  2. Provide concrete examples (e.g., from electromagnetism or Yang-Mills) when defining 'local' and 'global' to improve accessibility and precision of the central distinction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify areas where the derivations of our claims could be made more explicit in relation to standard definitions in gauge theory and constrained systems. We address each major comment below and indicate the revisions that will be incorporated to strengthen the manuscript while preserving its central arguments.

read point-by-point responses

  1. Referee: [Section on local vs. global gauge transformations] The section introducing the local/global distinction for gauge transformations: the claim that observables appear more naturally in the local sense is presented as following from the distinction itself, but the manuscript does not derive this from the standard definition of observables as quantities invariant under the full gauge group action (as in constrained Hamiltonian systems). Without addressing why the split is the sole relevant criterion, the subsequent restriction on observables remains framework-dependent rather than derived.

    Authors: We acknowledge that an explicit link to the standard definition of observables as quantities invariant under the full gauge group would strengthen the argument. The manuscript motivates the naturalness of observables in the local sense by noting that local gauge transformations permit the identification of invariants that remain well-defined under position-dependent changes without requiring simultaneous invariance under non-local transformations. In the revised manuscript we will add a new paragraph in the relevant section that derives this directly from the requirement of invariance under the complete gauge group action, as in the Dirac procedure. We will show that the local/global split is the operative criterion because only in the local case can one construct quantities that are invariant under the full group while still allowing for a non-trivial notion of locality; this is not merely framework-dependent but follows from the structure of the gauge action itself. revision: yes

  2. Referee: [Argument on diffeomorphism invariance as global gauge] The argument classifying diffeomorphism invariance as gauge only globally: the manuscript asserts that diffeomorphisms correspond exclusively to the global sense, yet standard treatments treat diffeomorphisms as local (position-dependent) symmetries whose action is not restricted to a global viewpoint. This classification appears stipulative rather than derived from the transformation properties, undermining the conclusion that the observable concept applies only restrictedly and that correlation-based claims are unfounded.

    Authors: The classification is derived from the transformation properties rather than stipulated. Diffeomorphisms act as automorphisms of the entire manifold and, unlike internal gauge transformations with compact support, cannot be varied independently over arbitrary disjoint regions without altering the global geometry. Although standard treatments emphasize their spacetime dependence, our distinction isolates the global character by focusing on whether the transformation can be localized while preserving the background-independent structure. We will expand the relevant section with additional references to the literature on background independence to make this derivation clearer. This supports the restricted applicability of the observable concept and the assessment that correlation-based accounts presuppose a local gauge interpretation that does not hold here. revision: partial

Circularity Check

0 steps flagged

No significant circularity; conceptual framework is self-contained

full rationale

The paper presents a philosophical distinction between local and global senses of gauge transformations, then classifies diffeomorphism invariance as applying only in the global sense to restrict the observable concept. This is an analytical premise introduced for the argument rather than a result derived from or reducing to any equations, fitted parameters, or self-citations by construction. No load-bearing steps equate outputs to inputs via definition or renaming; the dismissal of correlation-based claims follows directly from the proposed framework without circular reduction. The derivation is self-contained as interpretive analysis against standard treatments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the local/global gauge distinction is the appropriate lens for assessing observables in diffeomorphism-invariant theories; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The distinction between local and global gauge transformations is valid and determines the applicability of the observable concept.
    This premise is invoked when the abstract separates the two senses of gauge transformation and concludes that diffeomorphism invariance is only global.

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