pith. sign in

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
0
0 comments X

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.

pith-pipeline@v0.9.1-grok · 5711 in / 1295 out tokens · 26371 ms · 2026-06-25T21:52:09.587560+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 36 canonical work pages · 6 internal anchors

  1. [1]

    Bergmann, Observables in General Relativity, Rev

    P.G. Bergmann, Observables in General Relativity, Rev. Modern Phys. 33 (4) (1961) 510, http://dx.doi.org/10.1103/RevModPhys.33.510

    work page doi:10.1103/revmodphys.33.510 1961

  2. [2]

    Rovelli, What is observable in classical and quantum gravity? Classical Quantum Gravity 8 (2) (1991) 297, http://dx.doi.org/10.1088/0264-9381/8/2/ 011, Publisher: IOP Publishing

    C. Rovelli, What is observable in classical and quantum gravity? Classical Quantum Gravity 8 (2) (1991) 297, http://dx.doi.org/10.1088/0264-9381/8/2/ 011, Publisher: IOP Publishing

    work page doi:10.1088/0264-9381/8/2/ 1991

  3. [3]

    Comment on ‘Time in quantum gravity: An hypothesis’

    C. Rovelli, Quantum evolving constants. Reply to "Comment on ‘Time in quantum gravity: An hypothesis’", Phys. Rev. D 44 (4) (1991) 1339, http://dx.doi.org/10.1103/PhysRevD.44.1339

    work page doi:10.1103/physrevd.44.1339 1991

  4. [4]

    Rovelli, Time in quantum gravity: An hypothesis, Phys

    C. Rovelli, Time in quantum gravity: An hypothesis, Phys. Rev. D 43 (2) (1991) 442, http://dx.doi.org/10.1103/PhysRevD.43.442

    work page doi:10.1103/physrevd.43.442 1991

  5. [5]

    Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, 2004, http://dx.doi.org/10.1017/CBO9780511755804

    C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, 2004, http://dx.doi.org/10.1017/CBO9780511755804

    work page doi:10.1017/cbo9780511755804 2004

  6. [6]

    Forget time

    C. Rovelli, "Forget time" Essay written for the FQXi contest on the Nature of Time, Found. Phys. 41 (9) (2011) 1475–1490, http://dx.doi.org/10.1007/ s10701-011-9561-4, Publisher: Springer

    2011

  7. [7]

    Rovelli, F

    C. Rovelli, F. Vidotto, Philosophical Foundations of Loop Quantum Gravity, 2022, arXiv:2211.06718

    arXiv 2022

  8. [8]

    Earman, Thoroughly Modern Mctaggart: Or, What Mctaggart Would Have Said If He Had Read the General Theory of Relativity, Philos.’ Impr

    J. Earman, Thoroughly Modern Mctaggart: Or, What Mctaggart Would Have Said If He Had Read the General Theory of Relativity, Philos.’ Impr. 2 (3) (2002) 1–28

    2002

  9. [9]

    Earman, The Implications of General Covariance for the Ontology and Ideology of Spacetime, Philos

    J. Earman, The Implications of General Covariance for the Ontology and Ideology of Spacetime, Philos. Found. Phys. 1 (C) (2006) 3–23, http: //dx.doi.org/10.1016/S1871-1774(06)01001-1

    work page doi:10.1016/s1871-1774(06)01001-1 2006

  10. [10]

    Pons, D.C

    J.M. Pons, D.C. Salisbury, K.A. Sundermeyer, Observables in classical canonical gravity: Folklore demystified, J. Phys.: Conf. Ser. 222 (1) (2010) 012018, http://dx.doi.org/10.1088/1742-6596/222/1/012018

    work page doi:10.1088/1742-6596/222/1/012018 2010

  11. [11]

    Pitts, Change in Hamiltonian General Relativity From the Lack of a Time-Like Killing Vector Field, Stud

    J.B. Pitts, Change in Hamiltonian General Relativity From the Lack of a Time-Like Killing Vector Field, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 47 (2014) 68–89, http://dx.doi.org/10.1016/j.shpsb.2014.05.007

    work page doi:10.1016/j.shpsb.2014.05.007 2014

  12. [12]

    J.B. Pitts, Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity, Classical Quantum Gravity 34 (5) (2017) 1–23, http://dx.doi.org/10.1088/1361-6382/aa5ce8

    work page doi:10.1088/1361-6382/aa5ce8 2017

  13. [13]

    Gryb, Jacobi’s principle and the disappearance of time, Phys

    S. Gryb, Jacobi’s principle and the disappearance of time, Phys. Rev. D 81 (4) (2010) 044035, http://dx.doi.org/10.1103/PhysRevD.81.044035, Publisher: American Physical Society

    work page doi:10.1103/physrevd.81.044035 2010

  14. [14]

    Gryb, K.P

    S. Gryb, K.P. Thébault, Time remains, Br. J. Phil. Sci. 67 (3) (2016) 663–705, http://dx.doi.org/10.1093/bjps/axv009

    work page doi:10.1093/bjps/axv009 2016

  15. [15]

    Pooley, Background Independence, Diffeomorphism Invariance and the Meaning of Coordinates, in: D

    O. Pooley, Background Independence, Diffeomorphism Invariance and the Meaning of Coordinates, in: D. Lehmkuhl, G. Schiemann, E. Scholz (Eds.), Towards a Theory of Spacetime Theories, in: Einstein Studies, Springer, New York, NY, 2017, pp. 105–143, http://dx.doi.org/10.1007/978-1-4939-3210-8_4

    work page doi:10.1007/978-1-4939-3210-8_4 2017

  16. [16]

    Garc´ ıa-Parrado and E

    Á. Mozota Frauca, Reassessing the problem of time of quantum gravity, Gen. Relativity Gravitation 55 (1) (2023) 21, http://dx.doi.org/10.1007/s10714- 023-03067-x, arXiv:2301.07973

    work page doi:10.1007/s10714- 2023

  17. [17]

    Mozota Frauca, The Problem of Time for Non-Deparametrizable Models and Quantum Gravity, in: F

    Á. Mozota Frauca, The Problem of Time for Non-Deparametrizable Models and Quantum Gravity, in: F. Bianchini, V. Fano, P. Graziani (Eds.), Current Topics in Logic and the Philosophy of Science. Papers from SILFS 2022 Postgraduate Conference, in: The SILFS series, vol. 4, College Publications, 2024

    2022

  18. [18]

    Mozota Frauca, Time is Order, in: S

    Á. Mozota Frauca, Time is Order, in: S. De Bianchi, M. Forgione, L. Marongiu (Eds.), Time and Timelessness in Fundamental Physics and Cosmology: Historical, Philosophical, and Mathematical Perspectives, Springer Nature Switzerland, Cham, 2024, pp. 49–67, http://dx.doi.org/10.1007/978-3-031-61860- 4_4

    work page doi:10.1007/978-3-031-61860- 2024

  19. [19]

    Review of Ansatz Designing Techniques for Variational Quantum Algorithms,

    Á. Mozota Frauca, Quantum Cosmology and the Age of the Universe, J. Phys.: Conf. Ser. 2948 (1) (2025) 012008, http://dx.doi.org/10.1088/1742- 6596/2948/1/012008

    work page doi:10.1088/1742- 2025

  20. [20]

    Mozota Frauca, Against Radical Relationalism: in Defense of the Ordinal Structure of Time, Found

    Á. Mozota Frauca, Against Radical Relationalism: in Defense of the Ordinal Structure of Time, Found. Phys. 55 (3) (2025) 37, http://dx.doi.org/10.1007/ s10701-025-00850-5

    2025

  21. [21]

    Mozota Frauca, GPS observables in Newtonian spacetime or why we do not need ‘physical’ coordinate systems, Eur

    Á. Mozota Frauca, GPS observables in Newtonian spacetime or why we do not need ‘physical’ coordinate systems, Eur. J. Philos. Sci. 14 (4) (2024) 51, http://dx.doi.org/10.1007/s13194-024-00611-7

    work page doi:10.1007/s13194-024-00611-7 2024

  22. [22]

    Mozota Frauca, Does Quantum Cosmology Predict the Age of the Universe? J

    Á. Mozota Frauca, Does Quantum Cosmology Predict the Age of the Universe? J. Gen. Philos. Sci. (2026) http://dx.doi.org/10.1007/s10838-025-09754-4

    work page doi:10.1007/s10838-025-09754-4 2026

  23. [23]

    Kuchař, Time and interpretations of quantum gravity, in: G

    K.V. Kuchař, Time and interpretations of quantum gravity, in: G. Kunstatter, D. Vincent, J. Williams (Eds.), Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, World Scientific Publishing Company, Singapore, 1992, http://dx.doi.org/10.1142/S0218271811019347, ISSN: 02182718

    work page doi:10.1142/s0218271811019347 1992

  24. [24]

    Canonical Quantum Gravity and the Problem of Time

    C.J. Isham, Canonical Quantum Gravity and the Problem of Time, in: Integrable Systems, Quantum Groups, and Quantum Field Theories, Springer Netherlands, 1993, pp. 157–287, http://dx.doi.org/10.1007/978-94-011-1980-1_6, [arXiv:gr-qc/9210011]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-94-011-1980-1_6 1993

  25. [25]

    Chakraborty, P

    A. Chakraborty, P. Nandi, B. Chakraborty, Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase, Nuclear Phys. B 975 (2022) 115691, http://dx.doi.org/10.1016/j.nuclphysb.2022.115691

    work page doi:10.1016/j.nuclphysb.2022.115691 2022

  26. [26]

    Nandi, F.G

    P. Nandi, F.G. Scholtz, The hidden Lorentz covariance of quantum mechanics, Ann. Physics 464 (2024) 169643, http://dx.doi.org/10.1016/j.aop.2024. 169643

    work page doi:10.1016/j.aop.2024 2024

  27. [27]

    Partial observables

    C. Rovelli, Partial observables, Phys. Rev. D 65 (12) (2002) 124013, http://dx.doi.org/10.1103/PhysRevD.65.124013, arXiv:gr-qc/0110035. Publisher: American Physical Society

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.65.124013 2002

  28. [28]

    A simple background-independent hamiltonian quantum model

    D. Colosi, C. Rovelli, Simple background-independent Hamiltonian quantum model, Phys. Rev. D 68 (10) (2003) http://dx.doi.org/10.1103/PhysRevD.68. 104008, arXiv:gr-qc/0306059

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.68 2003

  29. [29]

    Mozota Frauca, The Limitations of the Notion of ‘Observable’ in Diffeomorphism-Invariant Models, British J

    Á. Mozota Frauca, The Limitations of the Notion of ‘Observable’ in Diffeomorphism-Invariant Models, British J. Philos. Sci. (2026) http://dx.doi.org/10. 1086/741910

    2026

  30. [30]

    Dittrich, Partial and complete observables for canonical general relativity, Classical Quantum Gravity 23 (22) (2006) 6155, http://dx.doi.org/10.1088/ 0264-9381/23/22/006

    B. Dittrich, Partial and complete observables for canonical general relativity, Classical Quantum Gravity 23 (22) (2006) 6155, http://dx.doi.org/10.1088/ 0264-9381/23/22/006

    2006

  31. [31]

    Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen

    B. Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen. Relativity Gravitation 39 (11) (2007) 1891–1927, http://dx.doi. org/10.1007/s10714-007-0495-2

    work page doi:10.1007/s10714-007-0495-2 2007

  32. [32]

    Dittrich, J

    B. Dittrich, J. Tambornino, A perturbative approach to Dirac observables and their spacetime algebra, Classical Quantum Gravity 24 (4) (2007) 757, http://dx.doi.org/10.1088/0264-9381/24/4/001

    work page doi:10.1088/0264-9381/24/4/001 2007

  33. [33]

    Chataignier, P.A

    L. Chataignier, P.A. Hoehn, M.P.E. Lock, F.M. Mele, Relational Dynamics with Periodic Clocks, 2025, http://dx.doi.org/10.48550/arXiv.2409.06479, [arXiv:2409.06479 [quant-ph]]. Comment: 32 + 16 pages, 2 figures. References updated, and minor corrections

    work page doi:10.48550/arxiv.2409.06479 2025

  34. [34]

    Dittrich, P.A

    B. Dittrich, P.A. Höhn, T.A. Koslowski, M.I. Nelson, Can chaos be observed in quantum gravity? Phys. Lett. B 769 (2017) 554–560, http://dx.doi.org/10. 1016/j.physletb.2017.02.038

    2017

  35. [35]

    Donnelly, S.B

    W. Donnelly, S.B. Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, Phys. Rev. D 93 (2) (2016) 024030, http://dx.doi.org/10. 1103/PhysRevD.93.024030. Annals of Physics 493 (2026) 170565 14 Á. Mozota Frauca

    2016

  36. [36]

    Donnelly, S.B

    W. Donnelly, S.B. Giddings, Observables, gravitational dressing, and obstructions to locality and subsystems, Phys. Rev. D 94 (10) (2016) 104038, http://dx.doi.org/10.1103/PhysRevD.94.104038

    work page doi:10.1103/physrevd.94.104038 2016

  37. [37]

    Harlow, J.-q

    D. Harlow, J.-q. Wu, Covariant phase space with boundaries, J. High Energy Phys. 2020 (10) (2020) 146, http://dx.doi.org/10.1007/JHEP10(2020)146

    work page doi:10.1007/jhep10(2020)146 2020

  38. [38]

    Goeller, P.A

    C. Goeller, P.A. Hoehn, J. Kirklin, Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance, 2022, http://dx.doi.org/10.48550/arXiv.2206.01193, [arXiv:2206.01193 [hep-th]]. Comment: 90+13 pages. Comments welcome and appreciated

    work page doi:10.48550/arxiv.2206.01193 2022

  39. [39]

    Sundermeyer, Constrained Dynamics, vol

    K.A. Sundermeyer, Constrained Dynamics, vol. 169, Springer-Verlag, 1982, http://dx.doi.org/10.1007/BFb0036225, [Series Title: Lecture Notes in Physics Publication Title: Lecture Notes in Physics]

    work page doi:10.1007/bfb0036225 1982

  40. [40]

    Henneaux, C

    M. Henneaux, C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, Princeton, 1992, http://dx.doi.org/10.1515/9780691213866

    work page doi:10.1515/9780691213866 1992

  41. [41]

    Rothe, K.D

    H.J. Rothe, K.D. Rothe, Classical and Quantum Dynamics of Constrained Hamiltonian Systems, vol. 81, WORLD SCIENTIFIC, Singapore, 2010, http: //dx.doi.org/10.1142/7689, [Series Title: World Scientific Lecture Notes in Physics]

    work page doi:10.1142/7689 2010

  42. [42]

    Pitts, Does a second-class primary constraint generate a gauge transformation? Electromagnetisms and gravities, massless and massive, Ann

    J.B. Pitts, Does a second-class primary constraint generate a gauge transformation? Electromagnetisms and gravities, massless and massive, Ann. Physics 462 (2024) 169621, http://dx.doi.org/10.1016/j.aop.2024.169621

    work page doi:10.1016/j.aop.2024.169621 2024

  43. [43]

    Mozota Frauca, In Which Sense Can We Say That First-Class Constraints Generate Gauge Transformations? Philos

    Á. Mozota Frauca, In Which Sense Can We Say That First-Class Constraints Generate Gauge Transformations? Philos. Phys. (2024) http://dx.doi.org/10. 31389/pop.48

    2024

  44. [44]

    Undecided state dynamics with stubborn agents

    O. Pooley, D. Wallace, First-class constraints generate gauge transformations in electromagnetism (reply to Pitts), 2022, http://dx.doi.org/10.48550/arxiv. 2210.09063, [arXiv:2210.09063]

    work page internal anchor Pith review doi:10.48550/arxiv 2022

  45. [45]

    A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism

    J.B. Pitts, A first class constraint generates not a gauge transformation, but a bad physical change: The case of electromagnetism, Ann. Physics 351 (2014) 382–406, http://dx.doi.org/10.1016/j.aop.2014.08.014, [arXiv:1310.2756]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aop.2014.08.014 2014

  46. [46]

    Kuchař, Canonical Quantum Gravity, Gen

    K.V. Kuchař, Canonical Quantum Gravity, Gen. Relativity Gravitation 1992 (1993) 119

    1992

  47. [47]

    GPS observables in general relativity

    C. Rovelli, GPS observables in general relativity, Phys. Rev. D 65 (4) (2002) 044017, http://dx.doi.org/10.1103/PhysRevD.65.044017, arXiv:gr-qc/0110003. Publisher: American Physical Society

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.65.044017 2002

  48. [48]

    Pitts, Equivalent Theories and Changing Hamiltonian Observables in General Relativity, Found

    J.B. Pitts, Equivalent Theories and Changing Hamiltonian Observables in General Relativity, Found. Phys. 48 (5) (2018) 579–590, http://dx.doi.org/10. 1007/s10701-018-0148-1. Annals of Physics 493 (2026) 170565 15

    2018