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Open Problems in Geometric Nonlinear Quantum Dynamics

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
A Hilbert manifold is a differentiable manifold modeled on a Hilbert space. A densely defined vector field is defined on a dense domain and a flow solves its evolution equation. Stone's theorem identifies strongly continuous unitary groups with self-adjoint generators in the linear case. For a composite quantum system, no signaling means that local operations on one subsystem do not alter another subsystem's observable statistics. Dissipative hydrodynamics permits irreversible evolution, rather than only Hamiltonian transport.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Develop a theory of densely defined vector fields on Hilbert manifolds that generate flows, generalizing Stone’s theorem. What are the best special-relativistic and general-relativistic analogues of the diffeomorphism group’s role in quantum mechanics? Develop nonlinear gauge transformations that may leave Hilbert space, extend nonlinear quantum time evolutions to composite systems with or without the no-signaling property, and apply differential geometry to generalized, possibly dissipative, quantum hydrodynamics.
Source scope and qualification.
The target consists of all five linked construction questions in the supplied statement. It is a programme, not one uniquely specified conjecture; no particular nonlinear evolution or signaling axiom is imposed beyond what the question asks to investigate. The primary numbered list also begins with a request for a general theory of quasi-invariant measures on spaces of embeddings and immersions, omitted from the abbreviated catalogue paragraph; this is identified as an additional source research direction rather than silently discarded. A quasi-invariant measure and its transforms have the same null sets under the relevant diffeomorphism action.

Short English statement

Build geometric foundations for nonlinear quantum evolution, its relativistic symmetries, gauge changes, composites and dissipative hydrodynamics.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-279-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-279

Open Problems in Geometric Nonlinear Quantum Dynamics

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
A Hilbert manifold is a differentiable manifold modeled on a Hilbert space. A densely defined vector field is defined on a dense domain and a flow solves its evolution equation. Stone's theorem identifies strongly continuous unitary groups with self-adjoint generators in the linear case. For a composite quantum system, no signaling means that local operations on one subsystem do not alter another subsystem's observable statistics. Dissipative hydrodynamics permits irreversible evolution, rather than only Hamiltonian transport.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Develop a theory of densely defined vector fields on Hilbert manifolds that generate flows, generalizing Stone’s theorem. What are the best special-relativistic and general-relativistic analogues of the diffeomorphism group’s role in quantum mechanics? Develop nonlinear gauge transformations that may leave Hilbert space, extend nonlinear quantum time evolutions to composite systems with or without the no-signaling property, and apply differential geometry to generalized, possibly dissipative, quantum hydrodynamics.
Source scope and qualification.
The target consists of all five linked construction questions in the supplied statement. It is a programme, not one uniquely specified conjecture; no particular nonlinear evolution or signaling axiom is imposed beyond what the question asks to investigate. The primary numbered list also begins with a request for a general theory of quasi-invariant measures on spaces of embeddings and immersions, omitted from the abbreviated catalogue paragraph; this is identified as an additional source research direction rather than silently discarded. A quasi-invariant measure and its transforms have the same null sets under the relevant diffeomorphism action.

Short English statement

Build geometric foundations for nonlinear quantum evolution, its relativistic symmetries, gauge changes, composites and dissipative hydrodynamics.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Computes decomposition-dependent evolution of the maximally mixed qubit and its conditional no-signaling failure for a specified nonlinear extension rule.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
An explicit ensemble-dependence obstruction.
On a qubit pure state $|\psi\rangle$, consider nonlinear evolution $i\dot\psi=\langle\psi,Z\psi\rangle Z\psi$. In Bloch-vector coordinates $r=(x,y,z)$ with $|r|=1$, the effective Hamiltonian is $zZ$, so $\dot r=2z(e_z\times r)$. The maximally mixed density matrix has an ensemble of opposite $z$-axis states, each stationary, so its ensemble-averaged derivative is zero.

Choose instead $r_\pm=\pm(\sin\theta,0,\cos\theta)$ with $0<\theta<\pi/2$. Their equal mixture is the same maximally mixed state, but both Bloch derivatives equal $(0,2\sin\theta\cos\theta,0)$. Thus evolving the pure members and averaging gives a nonzero mixed-state derivative. The evolution is not determined by the density matrix alone.

A composite-system consequence under an explicit rule.
A maximally entangled pair allows the remote subsystem to be decomposed into either of those equal antipodal ensembles by measuring the other subsystem in the corresponding basis. If the nonlinear law is applied separately to those conditional pure states and outcomes are ignored, the local predictions depend on the remote basis. This violates no signaling under that ensemble-evolution rule. Another extension rule would need to be defined and checked; the example is not a theorem against every nonlinear formulation.
All source directions retained.
The first gap is a well-defined composite-state evolution with a consistent operational interpretation. A nonlinear Stone-type flow theory on dense domains, relativistic analogues of diffeomorphism actions, nonlinear gauges leaving Hilbert space, dissipative hydrodynamics, and the additional quasi-invariant measures on embeddings/immersions are not constructed here. The finite-dimensional calculation only tests one key obstruction within that broad program.
Verification and adversarial audit.
Checked both antipodal ensembles, Bloch derivative signs and conditional remote-preparation rule; retained every source direction including quasi-invariant measures. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked both antipodal ensembles, Bloch derivative signs and conditional remote-preparation rule; retained every source direction including quasi-invariant measures. Retained unresolved status and zero original-target progress; A consistent composite-state rule and the source general flow/geometric/measure constructions remain missing.

Coordinator checked the opposite-state Bloch derivatives and ensemble-dependent reduced evolution under the explicitly specified conditional-state rule. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. A consistent composite-state rule and the source general flow/geometric/measure constructions remain missing.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. A consistent composite-state rule and the source general flow/geometric/measure constructions remain missing.

Sources

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