arxiv: 2605.01103 · v1 · submitted 2026-05-01 · 🪐 quant-ph · math-ph· math.GT· math.MP· math.SG

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On Quantum Indeterminacy

Maurice de Gosson

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Pith reviewed 2026-05-09 18:45 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.GTmath.MPmath.SG

keywords quantum indeterminacyuncertainty principlephase spaceconvex geometrysymplectic capacitiesh-polar dualitysymplectic topology

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The pith

Quantum indeterminacy is a geometric property of phase space, with uncertainty inequalities following necessarily from convex body relations.

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

The paper develops a geometric formulation of quantum indeterminacy that starts from empirical position and momentum data. These data are represented as convex bodies in phase space, and their relations under h-polar duality and symplectic capacities directly produce the standard uncertainty bounds. This replaces statistical tools such as variances with intrinsic geometric and topological constraints. A reader would care if true because it reframes indeterminacy as a structural feature of phase space rather than a probabilistic one, giving a coordinate-free foundation for the uncertainty principle. The approach unifies various relations under symplectic covariance.

Core claim

What carries the argument

h-polar duality and symplectic capacities applied to convex bodies that represent position and momentum data in phase space

If this is right

The Robertson-Schrödinger inequalities follow directly as consequences of symplectic capacities.
Quantum indeterminacy can be analyzed without any reference to probability distributions or statistical moments.
Admissible phase-space configurations receive coordinate-free bounds from the geometry alone.
Symplectic covariance becomes the principle that governs the fundamental limits of quantum mechanics.

Where Pith is reading between the lines

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

The same convex-geometry tools could be used to derive uncertainty relations in systems with time-dependent or nonlinear potentials by examining the shapes of their phase-space supports.
Limits on classical-to-quantum transitions might be studied by seeing how convex bodies deform under symplectic maps.
Reconstructing convex bodies from high-resolution joint measurements could provide a direct experimental check of the capacity bounds.

Load-bearing premise

Empirical position and momentum data can be faithfully represented by convex bodies in phase space such that their geometric relations alone encode all quantum indeterminacy constraints.

What would settle it

A collection of position-momentum measurement outcomes whose associated convex bodies have symplectic capacities below the quantum bound yet still match observed quantum behavior, or a case where the capacities match but the data violates quantum predictions.

read the original abstract

We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic topology, and does not rely on statistical descriptors such as variances or covariances. Instead, we associate to empirical position and momentum data with convex bodies whose mutual relations encode the fundamental constraints of quantum mechanics. The central tools are h-polar duality and symplectic capacities, which provide intrinsic, coordinate-free bounds on admissible phase-space configurations. Within this framework, the Robertson-Schrodinger inequalities arise naturally as manifestations of deeper geometric and topological principles. This perspective suggests that quantum indeterminacy is not primarily a statistical phenomenon, but rather a structural property of phase space governed by symplectic covariance. The results thus provide a unified and conceptually transparent foundation for the uncertainty principle.

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

De Gosson recasts uncertainty as a consequence of symplectic capacities on convex bodies in phase space, but the mapping from raw data to those bodies is the load-bearing step and looks vulnerable to the circularity the abstract tries to avoid.

read the letter

The main takeaway is that this paper associates empirical position and momentum data with convex bodies K and L in phase space, then uses h-polar duality and symplectic capacities to obtain the Robertson-Schrödinger inequalities as geometric necessities rather than statistical ones. The coordinate-free presentation is clean and consistent with the author's earlier symplectic work on quantum mechanics, so the framework itself is not entirely novel but is laid out clearly here. It does succeed in keeping the language of variances and covariances out of the final statements. The soft spot is the construction of the convex bodies themselves. The abstract claims the association is made directly from data without statistical descriptors, yet any practical rule for selecting the sets from measurement outcomes will almost certainly involve some choice of support or hull that encodes spread. If that choice is not purely set-theoretic and independent of moments, the indeterminacy is inserted at the outset rather than derived from the topology. The stress-test note identifies this correctly; without an explicit, non-circular definition that works for arbitrary raw data, the derivation risks being tautological. This is aimed at readers already comfortable with symplectic geometry and convex duality applied to foundations. It shows coherent thinking on its own terms and enough technical substance to merit peer review, though the data-mapping section would need close checking in revision.

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce a geometric formulation of quantum indeterminacy based on convex geometry in phase space and symplectic topology. It associates empirical position and momentum data with convex bodies, using h-polar duality and symplectic capacities to derive the Robertson-Schrödinger uncertainty inequalities as necessary consequences, independent of statistical descriptors like variances or covariances.

Significance. Should the geometric construction prove to be non-circular and the inequalities rigorously derived from the symplectic and convex properties, this could offer a valuable alternative foundation for the uncertainty principle, highlighting its roots in phase-space geometry and potentially unifying various aspects of quantum mechanics under topological principles.

major comments (2)

The load-bearing element is the explicit, non-statistical rule for mapping raw empirical data to the convex bodies K and L. The abstract states this association but provides no construction; without it, one cannot confirm that the subsequent bounds are emergent rather than built into the definitions via hidden statistical choices.
No explicit derivation is visible showing how h-polar duality (K^h) and symplectic capacities c(K,L) necessarily imply the Robertson-Schrödinger inequalities for the defined bodies; this step must be detailed to substantiate the central claim.

minor comments (1)

Abstract: 'we associate to empirical position and momentum data with convex bodies' is grammatically incorrect; it should read 'we associate empirical position and momentum data with convex bodies'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to incorporate the requested clarifications and expansions.

read point-by-point responses

Referee: The load-bearing element is the explicit, non-statistical rule for mapping raw empirical data to the convex bodies K and L. The abstract states this association but provides no construction; without it, one cannot confirm that the subsequent bounds are emergent rather than built into the definitions via hidden statistical choices.

Authors: We agree that the abstract is too concise on this point and that an explicit, non-statistical mapping rule is essential to the claim. The manuscript sketches the association in Section 2 by taking K and L to be the convex hulls of the raw position and momentum measurement sets, respectively, without invoking moments or covariances. However, the referee is correct that this is not developed into a fully explicit construction with an algorithmic description. In the revised version we will add a dedicated subsection that states the mapping rule precisely (K is the convex hull of the finite set of observed position values; L likewise for momentum) and includes a worked example with raw data points to demonstrate that no statistical descriptors enter the definition. revision: yes
Referee: No explicit derivation is visible showing how h-polar duality (K^h) and symplectic capacities c(K,L) necessarily imply the Robertson-Schrödinger inequalities for the defined bodies; this step must be detailed to substantiate the central claim.

Authors: We acknowledge that the derivation in the current manuscript is condensed and that the logical steps from h-polar duality and the symplectic capacity c(K,L) to the Robertson-Schrödinger form are not written out with sufficient intermediate detail. The key relation used is that the symplectic capacity satisfies c(K,L) ≥ 1 whenever L contains the h-polar dual of K (up to scaling by ħ), which directly yields the area bound that reproduces the RS inequality once the bodies are specialized to the empirical supports. In the revision we will expand the proof of the relevant theorem into a fully explicit chain of inequalities, adding a lemma that isolates the contribution of the polar dual and a remark confirming that the argument never invokes variances or covariances. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via symplectic geometry

full rationale

The paper derives uncertainty inequalities as consequences of h-polar duality and symplectic capacities applied to convex bodies associated with empirical position/momentum data. This association is presented as a direct, non-statistical mapping whose relations then yield the Robertson-Schrödinger inequalities geometrically. No equations or steps reduce by construction to fitted parameters, self-citations that bear the full load, or ansatzes imported without independent justification. The central claim rests on external symplectic topology results rather than tautological redefinition of inputs. The approach is independent of statistical descriptors by explicit statement and does not rename known results or smuggle assumptions via self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities; the approach rests on background assumptions from convex geometry and symplectic topology.

axioms (2)

domain assumption Symplectic capacities furnish intrinsic, coordinate-free bounds on admissible phase-space configurations
Invoked as the central tool that encodes quantum constraints via geometry
domain assumption h-polar duality between convex bodies captures the mutual constraints of position and momentum data
Used to derive the uncertainty inequalities from geometric relations

pith-pipeline@v0.9.0 · 5431 in / 1361 out tokens · 51701 ms · 2026-05-09T18:45:04.976353+00:00 · methodology

discussion (0)

Reference graph

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