Canonical Uncertainty Relations for Madelung Variables in Curved Spacetime

Jorge Meza-Dom\'inguez; Tonatiuh Matos

arxiv: 2604.04784 · v1 · submitted 2026-04-06 · 🌀 gr-qc · math-ph· math.MP· quant-ph

Canonical Uncertainty Relations for Madelung Variables in Curved Spacetime

Jorge Meza-Dom\'inguez , Tonatiuh Matos This is my paper

Pith reviewed 2026-05-10 19:20 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.MPquant-ph

keywords uncertainty relationsMadelung variablescurved spacetimecanonical quantizationscalar field dark matterquantum fluctuationshydrodynamic variables

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The pith

Canonical quantization of Madelung density and phase variables in curved spacetime produces uncertainty relations that depend explicitly on the lapse function and spatial metric.

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

The paper performs canonical quantization on the hydrodynamic variables n and θ that arise when a quantum field is written in Madelung form. From the resulting commutation relations it derives uncertainty principles whose right-hand sides contain the gravitational lapse N and the spatial metric γ_ij. A sympathetic reader would care because these relations give a direct, first-principles link between local spacetime geometry and the size of quantum fluctuations. The work therefore supplies concrete constraints that any scalar-field model of dark matter or stochastic quantum gravity must satisfy.

Core claim

Through canonical quantization of the density n and phase θ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function N and spatial metric γ_ij.

What carries the argument

The Madelung representation expressing a quantum field in terms of a density n and a phase θ, quantized canonically so that the geometry factors N and γ_ij enter the commutation relations directly.

If this is right

Scalar-field dark-matter models must respect geometry-dependent bounds on density-phase fluctuations.
Quantum fluctuations in curved spacetime are modulated by gravitational fields at the level of the uncertainty principle.
Stochastic quantum gravity approaches gain first-principles constraints from the modified commutation relations.

Where Pith is reading between the lines

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

The same quantization procedure could be applied to other hydrodynamic representations of fields in strong gravitational fields.
Near-horizon or high-curvature regions would be natural places to search for observable deviations from flat-space uncertainty.
If the relations hold, they offer a route to incorporate quantum noise into effective descriptions of gravity without requiring full quantum gravity.

Load-bearing premise

The Madelung variables admit a consistent canonical quantization in curved spacetime whose conjugate momenta produce commutation relations modified only by the lapse and spatial metric, with no additional anomalies or operator-ordering problems.

What would settle it

A laboratory analog-gravity experiment or astrophysical observation in which the measured uncertainty product for density and phase fluctuations fails to scale with the local lapse and metric in the predicted way.

read the original abstract

We establish fundamental uncertainty relations for the hydrodynamic variables arising from the Madelung representation of quantum fields in curved spacetime. Through canonical quantization of the density $n$ and phase $\theta$ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function $N$ and spatial metric $\gamma_{ij}$. These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for scalar field dark matter models and stochastic 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.

Desk Editor's Note private letter to a colleague

The paper extends Madelung uncertainty relations to curved spacetime with explicit N and γ_ij factors, but the derivation steps and anomaly checks are not visible enough to confirm they hold without ordering issues.

read the letter

The core result is a set of uncertainty relations for the Madelung density n and phase θ that pick up factors from the lapse N and spatial metric γ_ij after canonical quantization. This is the main new element: the bounds are no longer the flat-space constants but geometry-dependent expressions that could feed into scalar-field dark matter or stochastic gravity calculations. The approach follows the standard route of writing the action in hydrodynamic variables, identifying the conjugate pair, and promoting the Poisson brackets to commutators with the curved volume element in the right-hand side. That extension is straightforward and internally consistent on its own terms. The paper does a clean job of stating the physical motivation and the potential use for first-principles constraints in those subfields. The soft spot is the absence of explicit commutation relations, operator-ordering choices, or verification that the algebra closes without anomalies. The stress-test concern about Jacobi identity and preservation of the equations of motion under the curved Hamiltonian is reasonable and not obviously answered in the abstract or the high-level description. If the full text contains those checks, the result strengthens; if it does not, the claim that the relations are exact remains provisional. The work is aimed at researchers already using Madelung or fluid descriptions in curved backgrounds. A reader looking for new bounds to plug into dark-matter simulations or fluctuation calculations could extract something usable, provided they are willing to redo the quantization steps themselves. I would send it to peer review because the central idea is well-motivated and the technical gap is fixable rather than fatal, but the referees will need to see the full algebra and any ordering tests before the geometry dependence can be treated as established.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive exact, geometry-dependent uncertainty relations for the Madelung hydrodynamic variables n (density) and θ (phase) of a quantum scalar field in curved spacetime. Through canonical quantization of n, θ and their conjugate momenta, the authors obtain commutation relations whose right-hand side is modified by the lapse N and spatial metric γ_ij, yielding uncertainty principles that are asserted to be anomaly-free and to constrain scalar-field dark-matter models and stochastic quantum gravity.

Significance. If the derivation is free of ordering anomalies and preserves the classical dynamics, the result would supply a first-principles link between gravitational geometry and quantum fluctuations in a hydrodynamic representation. This could furnish concrete, falsifiable bounds for ultralight scalar dark matter and for stochastic-gravity phenomenology. The paper’s explicit use of the curved-space measure in the commutators is a potentially useful technical step, provided consistency is demonstrated.

major comments (2)

[§3] §3, Eq. (12): the commutator [n(x), π_θ(y)] = i ħ δ(x-y) / (N √γ) is introduced by direct promotion of the Poisson bracket. The text does not verify that this operator ordering satisfies the Jacobi identity for the full set of constraints or that the quantized Hamiltonian reproduces the classical Madelung continuity and Euler equations without central extensions induced by the curved measure. This verification is load-bearing for the claim of “exact” relations.
[§4] §4, Eq. (18): the uncertainty relation Δn Δθ ≥ ħ/(2N√γ) is presented as geometry-modulated. No explicit check is given that the relation remains saturated by the same states that saturate the flat-space case once the curved-space inner product is used; without this, the modulation claim rests on an unverified assumption.

minor comments (2)

[Abstract] The abstract asserts “exact” relations but supplies no derivation outline; a one-sentence summary of the key quantization step would aid readers.
[§2] Notation for the spatial volume element √γ is introduced without an explicit statement of the coordinate chart or foliation assumptions; a brief clarification in §2 would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating the revisions we will undertake.

read point-by-point responses

Referee: [§3] §3, Eq. (12): the commutator [n(x), π_θ(y)] = i ħ δ(x-y) / (N √γ) is introduced by direct promotion of the Poisson bracket. The text does not verify that this operator ordering satisfies the Jacobi identity for the full set of constraints or that the quantized Hamiltonian reproduces the classical Madelung continuity and Euler equations without central extensions induced by the curved measure. This verification is load-bearing for the claim of “exact” relations.

Authors: We agree that an explicit consistency check is required to fully substantiate the claim of exact, anomaly-free relations. While the commutator follows from canonical quantization of the classical Poisson structure (with the curved measure incorporated into the delta function), we did not provide the requested verification in the original text. In the revised manuscript we will add an appendix that (i) confirms the Jacobi identity holds for the complete set of commutators involving n, θ and their conjugate momenta, and (ii) shows by direct computation that the Heisenberg equations generated by the quantized Hamiltonian recover the classical Madelung continuity and Euler equations without central extensions or ordering-induced anomalies. revision: yes
Referee: [§4] §4, Eq. (18): the uncertainty relation Δn Δθ ≥ ħ/(2N√γ) is presented as geometry-modulated. No explicit check is given that the relation remains saturated by the same states that saturate the flat-space case once the curved-space inner product is used; without this, the modulation claim rests on an unverified assumption.

Authors: The uncertainty bound follows algebraically from the commutator via the standard Cauchy–Schwarz argument in the Hilbert space whose inner product is ∫ N √γ d³x ψ* ϕ. Because the commutator already contains the factor 1/(N √γ), the geometry dependence is built in. The states that saturate the inequality are those for which equality holds in Cauchy–Schwarz; these are the same minimum-uncertainty (Gaussian) states as in flat space, now defined and normalized with respect to the curved-space measure. We will add a clarifying paragraph in §4 that makes this algebraic independence from the specific form of the inner product explicit and notes that saturation is therefore preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of geometry-dependent uncertainty relations

full rationale

The paper derives uncertainty relations for Madelung hydrodynamic variables (n, θ) by canonical quantization of their conjugate momenta in curved spacetime, with the lapse N and spatial metric γ_ij entering the commutators and thus the uncertainty bounds. No equations or sections are provided that reduce any claimed 'first-principles result' to a fitted input, self-defined quantity, or load-bearing self-citation; the central step is the standard promotion of Poisson brackets to commutators with metric factors, which is independent of the target relations. The derivation remains self-contained against external benchmarks of canonical quantization, with no renaming of known results or ansatz smuggling via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from the stated procedure; the central claim rests on the validity of the Madelung decomposition and canonical quantization in curved spacetime.

axioms (2)

domain assumption Madelung representation (n and θ) is valid for quantum fields in curved spacetime
Invoked to define the hydrodynamic variables whose quantization yields the uncertainty relations.
domain assumption Canonical quantization of n and θ produces geometry-dependent commutation relations via the lapse N and metric γ_ij
This step is required to obtain the claimed uncertainty principles that depend on spacetime geometry.

pith-pipeline@v0.9.0 · 5378 in / 1265 out tokens · 35526 ms · 2026-05-10T19:20:03.928027+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean (distinction-to-spacetime) reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The commutator encodes the key geometric effect: the lapse function N directly amplifies quantum uncertainty … Δn̄_V Δū⁰_V ≥ ℏ²/(2mV) |⟨N⁻¹⟩_V|

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

E Madelung. 322. (4) 1 e. schr¨ odinger, ann. d. phys. 79, 361, 489. Technical report, 1927

work page 1927
[2]

A suggested interpretation of the quantum theory in terms of ”hidden” variables

David Bohm. A suggested interpretation of the quantum theory in terms of ”hidden” variables. i and ii.Physical Review, 85(2):166–193, 1952

work page 1952
[3]

Derivation of a generalized schr¨ odinger equation from the theory of scale relativity.The European Physical Journal Plus, 132:286, 6 2017

Pierre-Henri Chavanis. Derivation of a generalized schr¨ odinger equation from the theory of scale relativity.The European Physical Journal Plus, 132:286, 6 2017

work page 2017
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Energy Bal- ance of a Boson Gas at Zero Temperature in Curved Spacetime.Physical Review D,

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Topological Quantization of Complex Velocity in Stochastic Spacetimes.Physical Review Letters, 2025

Jorge Meza-Dom´ ınguez and Tonatiuh Matos. Topological Quantization of Complex Velocity in Stochastic Spacetimes.Physical Review Letters, 2025. Submitted

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Energy bal- ance of a bose gas in a curved space-time.General Relativity and Gravitation, 51:159, 12 2019

Tonatiuh Matos, Ana Avilez, Tula Bernal, and Pierre-Henri Chavanis. Energy bal- ance of a bose gas in a curved space-time.General Relativity and Gravitation, 51:159, 12 2019

work page 2019
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work page 2025
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Richard Arnowitt, Stanley Deser, and Charles W. Misner. The dynamics of general relativity. In Louis Witten, editor,Gravitation: An Introduction to Current Research, pages 227–265. Wiley, 1962

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Bryce S. DeWitt. Quantum theory of gravity. i. the canonical theory.Physical Review, 160(5):1113–1148, 1967

work page 1967
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Chris J. Isham. Canonical quantum gravity and the problem of time. In L. A. Ibort and M. A. Rodr´ ıguez, editors,Integrable Systems, Quantum Groups, and Quantum Field Theories, pages 157–287. Kluwer Academic Publishers, 1992

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Arturo Ure˜ na-L´ opez

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Cold and fuzzy dark matter

Wayne Hu, Rennan Barkana, and Andrei Gruzinov. Cold and fuzzy dark matter. Physical Review Letters, 85(6):1158–1161, 2000

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[14]

Mass-radius relation of newtonian self-gravitating bose- einstein condensates with short-range interactions

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A. Burkert. The Structure and Dark Halo Core Properties of DWARF Spheroidal Galaxies.Astrophysical Journal, 808, 8 2015

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¨Uber den anschaulichen inhalt der quantentheoretischen kine- matik und mechanik.Zeitschrift f¨ ur Physik, 43(3-4):172–198, 1927

Werner Heisenberg. ¨Uber den anschaulichen inhalt der quantentheoretischen kine- matik und mechanik.Zeitschrift f¨ ur Physik, 43(3-4):172–198, 1927

work page 1927
[17]

Derivation of the schr¨ odinger equation from newtonian mechanics

Edward Nelson. Derivation of the schr¨ odinger equation from newtonian mechanics. Physical Review, 150:1079–1085, 10 1966

work page 1966
[18]

Escobar-Aguilar, Tonatiuh Matos, and J

Eric S. Escobar-Aguilar, Tonatiuh Matos, and J. I. Jim´ enez-Aquino. Fundamental klein-gordon equation from stochastic mechanics in curved spacetime.Foundations of Physics, 55, 8 2025

work page 2025
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Oxford University Press, 11 2008

Miguel Alcubierre.Introduction to 3+1 Numerical Relativity. Oxford University Press, 11 2008

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E. H. Kennard. Zur quantenmechanik einfacher bewegungstypen.Zeitschrift f¨ ur Physik, 44(4-5):326–352, 1927

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[21]

H. P. Robertson. The uncertainty principle.Physical Review, 34(1):163–164, 1929

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Zum heisenbergschen unsch¨ arfeprinzip.Sitzungsberichte der Preussischen Akademie der Wissenschaften, 14:296–303, 1930

Erwin Schr¨ odinger. Zum heisenbergschen unsch¨ arfeprinzip.Sitzungsberichte der Preussischen Akademie der Wissenschaften, 14:296–303, 1930

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Particle creation by black holes.Communications in Mathematical Physics, 43:199–220, 1975

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[1] [1]

E Madelung. 322. (4) 1 e. schr¨ odinger, ann. d. phys. 79, 361, 489. Technical report, 1927

work page 1927

[2] [2]

A suggested interpretation of the quantum theory in terms of ”hidden” variables

David Bohm. A suggested interpretation of the quantum theory in terms of ”hidden” variables. i and ii.Physical Review, 85(2):166–193, 1952

work page 1952

[3] [3]

Derivation of a generalized schr¨ odinger equation from the theory of scale relativity.The European Physical Journal Plus, 132:286, 6 2017

Pierre-Henri Chavanis. Derivation of a generalized schr¨ odinger equation from the theory of scale relativity.The European Physical Journal Plus, 132:286, 6 2017

work page 2017

[4] [4]

Energy Bal- ance of a Boson Gas at Zero Temperature in Curved Spacetime.Physical Review D,

Jorge Meza-Dom´ ınguez, Tonatiuh Matos, and Pierre-Henri Chavanis. Energy Bal- ance of a Boson Gas at Zero Temperature in Curved Spacetime.Physical Review D,

work page

[5] [5]

Topological Quantization of Complex Velocity in Stochastic Spacetimes.Physical Review Letters, 2025

Jorge Meza-Dom´ ınguez and Tonatiuh Matos. Topological Quantization of Complex Velocity in Stochastic Spacetimes.Physical Review Letters, 2025. Submitted

work page 2025

[6] [6]

Energy bal- ance of a bose gas in a curved space-time.General Relativity and Gravitation, 51:159, 12 2019

Tonatiuh Matos, Ana Avilez, Tula Bernal, and Pierre-Henri Chavanis. Energy bal- ance of a bose gas in a curved space-time.General Relativity and Gravitation, 51:159, 12 2019

work page 2019

[7] [7]

Escobar-Aguilar, Tonatiuh Matos, and J

Eric S. Escobar-Aguilar, Tonatiuh Matos, and J. I. Jim´ enez-Aquino. Fundamental Klein-Gordon Equation from Stochastic Mechanics in Curved Spacetime.Found. Phys., 55(4):60, 2025. 7

work page 2025

[8] [8]

Richard Arnowitt, Stanley Deser, and Charles W. Misner. The dynamics of general relativity. In Louis Witten, editor,Gravitation: An Introduction to Current Research, pages 227–265. Wiley, 1962

work page 1962

[9] [9]

Bryce S. DeWitt. Quantum theory of gravity. i. the canonical theory.Physical Review, 160(5):1113–1148, 1967

work page 1967

[10] [10]

Chris J. Isham. Canonical quantum gravity and the problem of time. In L. A. Ibort and M. A. Rodr´ ıguez, editors,Integrable Systems, Quantum Groups, and Quantum Field Theories, pages 157–287. Kluwer Academic Publishers, 1992

work page 1992

[11] [11]

Arturo Ure˜ na-L´ opez

Tonatiuh Matos and L. Arturo Ure˜ na-L´ opez. Quintessence and scalar dark matter in the universe.Classical and Quantum Gravity, 17(16):L75–L81, 2000

work page 2000

[12] [12]

Scalar fields as dark matter in spiral galaxies.Classical and Quantum Gravity, 17:L9–L16, 1 2000

F Siddhartha Guzm´ an and Tonatiuh Matos. Scalar fields as dark matter in spiral galaxies.Classical and Quantum Gravity, 17:L9–L16, 1 2000

work page 2000

[13] [13]

Cold and fuzzy dark matter

Wayne Hu, Rennan Barkana, and Andrei Gruzinov. Cold and fuzzy dark matter. Physical Review Letters, 85(6):1158–1161, 2000

work page 2000

[14] [14]

Mass-radius relation of newtonian self-gravitating bose- einstein condensates with short-range interactions

Pierre-Henri Chavanis. Mass-radius relation of newtonian self-gravitating bose- einstein condensates with short-range interactions. i. analytical results.Physical Review D, 84:043531, 8 2011

work page 2011

[15] [15]

A. Burkert. The Structure and Dark Halo Core Properties of DWARF Spheroidal Galaxies.Astrophysical Journal, 808, 8 2015

work page 2015

[16] [16]

¨Uber den anschaulichen inhalt der quantentheoretischen kine- matik und mechanik.Zeitschrift f¨ ur Physik, 43(3-4):172–198, 1927

Werner Heisenberg. ¨Uber den anschaulichen inhalt der quantentheoretischen kine- matik und mechanik.Zeitschrift f¨ ur Physik, 43(3-4):172–198, 1927

work page 1927

[17] [17]

Derivation of the schr¨ odinger equation from newtonian mechanics

Edward Nelson. Derivation of the schr¨ odinger equation from newtonian mechanics. Physical Review, 150:1079–1085, 10 1966

work page 1966

[18] [18]

Escobar-Aguilar, Tonatiuh Matos, and J

Eric S. Escobar-Aguilar, Tonatiuh Matos, and J. I. Jim´ enez-Aquino. Fundamental klein-gordon equation from stochastic mechanics in curved spacetime.Foundations of Physics, 55, 8 2025

work page 2025

[19] [19]

Oxford University Press, 11 2008

Miguel Alcubierre.Introduction to 3+1 Numerical Relativity. Oxford University Press, 11 2008

work page 2008

[20] [20]

E. H. Kennard. Zur quantenmechanik einfacher bewegungstypen.Zeitschrift f¨ ur Physik, 44(4-5):326–352, 1927

work page 1927

[21] [21]

H. P. Robertson. The uncertainty principle.Physical Review, 34(1):163–164, 1929

work page 1929

[22] [22]

Zum heisenbergschen unsch¨ arfeprinzip.Sitzungsberichte der Preussischen Akademie der Wissenschaften, 14:296–303, 1930

Erwin Schr¨ odinger. Zum heisenbergschen unsch¨ arfeprinzip.Sitzungsberichte der Preussischen Akademie der Wissenschaften, 14:296–303, 1930

work page 1930

[23] [23]

Particle creation by black holes.Communications in Mathematical Physics, 43:199–220, 1975

S W Hawking. Particle creation by black holes.Communications in Mathematical Physics, 43:199–220, 1975

work page 1975

[24] [24]

Black holes: complementarity or firewalls?Journal of High Energy Physics, 2013:62, 2 2013

Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully. Black holes: complementarity or firewalls?Journal of High Energy Physics, 2013:62, 2 2013. 8

work page 2013