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arxiv: 2606.25072 · v1 · pith:UAU73VDXnew · submitted 2026-06-23 · 🌀 gr-qc · physics.hist-ph

Diffeomorphism-Invariant Quantities in Phase Space: More than Correlations

\'Alvaro Mozota Frauca This is my paper

Pith reviewed 2026-06-25 21:52 UTC · model grok-4.3

classification 🌀 gr-qc physics.hist-ph
keywords diffeomorphism invariancephase spaceobservablescorrelationsspatiotemporal structuresgeneral relativityquantum gravityinvariants
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The pith

Spatiotemporal structures are indispensable for defining the invariant content of diffeomorphism-invariant models.

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

The paper shows that the physical content of diffeomorphism-invariant theories cannot be captured by correlations alone. It demonstrates that observables defined via vanishing Poisson brackets fail to apply to every phase space trajectory and cannot always be expressed as smooth functions. Spatiotemporal relations are proven to be invariant quantities, which makes reference to spatiotemporal structures necessary for a complete account of what remains unchanged under diffeomorphisms. This challenges the standard relational view in the foundations of these theories.

Core claim

In models with temporal diffeomorphism invariance, the standard definition of observables as phase space functions with vanishing Poisson brackets with the constraints does not cover all trajectories. Correlations can be shown to be invariant only in a generalized sense that does not yield smooth phase space functions. Spatiotemporal relations are also invariant. Spatiotemporal structures are therefore indispensable for defining the invariant content. These formal results are expected to extend to models invariant under d-dimensional diffeomorphisms.

What carries the argument

The proof that spatiotemporal structures are required in addition to correlations to define the full set of diffeomorphism invariants.

If this is right

  • The correlation-based definition of observables does not apply to all phase space trajectories in every diffeomorphism-invariant theory.
  • Correlations yield invariance only in a generalized manner that does not produce smooth phase space functions.
  • Spatiotemporal relations qualify as invariants under diffeomorphisms.
  • Any complete definition of the invariant content must incorporate spatiotemporal structures.
  • The findings challenge certain relational approaches in the foundations of general relativity and quantum gravity.

Where Pith is reading between the lines

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

  • Canonical quantization procedures that rely exclusively on relational observables may need to be extended to include explicit spatiotemporal structures.
  • The result connects the problem of defining physical quantities to the status of background structures in background-independent theories.
  • Specific models such as the parametrized particle or reduced gravity could be examined to check whether their invariants require spatiotemporal input.

Load-bearing premise

The formal results derived for models with temporal diffeomorphism invariance extend to the general case of d-dimensional diffeomorphism invariance.

What would settle it

A concrete diffeomorphism-invariant model in which the complete set of invariants can be defined using only smooth correlation functions with no reference to spatiotemporal structures.

read the original abstract

A popular view in the foundations of diffeomorphism-invariant theories is that their physical content is encoded in correlations or `observables': a set of phase space functions that have vanishing Poisson brackets with the constraints related to diffeomorphisms. In this article I study the phase space structure of models with a temporal diffeomorphism invariance and prove a series of formal results that challenge this view in a few ways. First, I show how this view is not applicable to all the phase space trajectories of every diffeomorphism-invariant theory. Second, I show how correlations can be proved to be invariant only in a way that generalizes the standard definition of invariance and in a way that does not provide smooth phase space functions. Third, I prove that spatiotemporal relations are also invariant. Fourth, I prove that spatiotemporal structures are indispensable for defining the invariant content of diffeomorphism-invariant models. Finally, I comment that these results are expected to be generalizable for models invariant under $d$-dimensional diffeomorphisms, which represents a challenge for some views in the foundations of general relativity and quantum gravity.

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 / 1 minor

Summary. The paper challenges the view that the physical content of diffeomorphism-invariant theories is encoded solely in correlations (phase-space functions with vanishing Poisson brackets with the diffeomorphism constraints). For models with temporal diffeomorphism invariance, it proves four formal results: (1) the correlation view does not apply to all phase-space trajectories, (2) correlations can be shown invariant only via a generalized definition that fails to yield smooth phase-space functions, (3) spatiotemporal relations are likewise invariant, and (4) spatiotemporal structures are indispensable for defining invariant content. The abstract comments that these results are expected to generalize to d-dimensional diffeomorphisms, posing a challenge to certain foundational views in general relativity and quantum gravity.

Significance. If the four formal results are correct and the generalization to full spacetime diffeomorphism invariance is rigorously established, the work would be significant for foundations of GR and quantum gravity: it would demonstrate that the 'observables' or 'correlations' program is incomplete and that spatiotemporal structures play an indispensable role beyond correlations. The paper supplies explicit formal results on phase-space structure for the temporal case, which is a strength, but the current limitation to temporal invariance restricts immediate applicability to the target theories.

major comments (2)
  1. [Abstract] Abstract (final sentence): the claim that the four results 'are expected to be generalizable' to d-dimensional diffeomorphism invariance is stated only as a comment with no supporting derivation, lemma, or argument showing how the temporal proofs extend when spatial diffeomorphism constraints are added. This is load-bearing for the paper's challenge to views in GR and QG, which require full spacetime invariance.
  2. [Abstract] The fourth result (that spatiotemporal structures are indispensable for defining the invariant content of diffeomorphism-invariant models) is proven only for temporal diffeomorphism invariance. Without an explicit extension, the indispensability conclusion does not follow for the general d-dimensional case that is the target of the abstract's final claim.
minor comments (1)
  1. The abstract asserts a series of formal proofs; the manuscript should ensure that each of the four results includes explicit statements of assumptions, scope, and any error-handling or limiting cases to aid verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and precise comments on the scope of our results. We agree that the abstract's phrasing regarding generalization to d-dimensional diffeomorphism invariance lacks supporting derivation and that the fourth result is established only for temporal invariance. We will revise the abstract to remove any implication that the indispensability conclusion or challenge to foundational views in GR/QG has been shown beyond the temporal case, and to present the generalization explicitly as a conjecture for future work.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that the four results 'are expected to be generalizable' to d-dimensional diffeomorphism invariance is stated only as a comment with no supporting derivation, lemma, or argument showing how the temporal proofs extend when spatial diffeomorphism constraints are added. This is load-bearing for the paper's challenge to views in GR and QG, which require full spacetime invariance.

    Authors: We accept this assessment. The manuscript establishes the four formal results exclusively for models with temporal diffeomorphism invariance and offers no derivation or lemma for the extension to spatial constraints. The final sentence of the abstract will be revised to state that generalization to d-dimensional diffeomorphisms remains an open conjecture without proof in the present work, rather than an expectation that underpins the paper's conclusions. revision: yes

  2. Referee: [Abstract] The fourth result (that spatiotemporal structures are indispensable for defining the invariant content of diffeomorphism-invariant models) is proven only for temporal diffeomorphism invariance. Without an explicit extension, the indispensability conclusion does not follow for the general d-dimensional case that is the target of the abstract's final claim.

    Authors: The fourth result is proven only within the temporal setting, as the manuscript's analysis is restricted to that case. The abstract will be updated so that statements about the indispensability of spatiotemporal structures are confined to temporal diffeomorphism-invariant models, with no claim that this conclusion extends to the general d-dimensional case without further work. revision: yes

Circularity Check

0 steps flagged

No circularity; formal phase-space proofs stand independently

full rationale

The manuscript derives four explicit formal results (non-applicability of correlation view to all trajectories, limited invariance of correlations, invariance of spatiotemporal relations, and indispensability of spatiotemporal structures) strictly for models possessing temporal diffeomorphism invariance, using standard phase-space definitions of constraints and Poisson brackets. These steps rely on direct mathematical argument rather than parameter fitting, self-definition, or load-bearing self-citation. The final comment that the results are 'expected to be generalizable' to d-dimensional diffeomorphisms is presented as an unproven expectation, not as a premise required for the temporal-case theorems. No quoted equation or step reduces to its own input by construction, satisfying the criteria for a self-contained formal derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard elements of constrained Hamiltonian mechanics (Poisson brackets with diffeomorphism constraints) that are drawn from prior literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Physical content of diffeomorphism-invariant theories is encoded in phase space functions with vanishing Poisson brackets with the constraints
    This is the popular view being challenged; invoked as the starting point for the analysis.
  • standard math Poisson bracket formalism applies to models with temporal diffeomorphism invariance
    Standard background assumption in the field of canonical gravity.

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discussion (0)

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Reference graph

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