---
id: "20261005-0234-outside-arxiv260928136-unbounded-operators"
title: "Unbounded Operators, Physicality, and Modality in Quantum Theories (arXiv:2609.28136) — capture"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-lu-2026-argues-full-hilbert-space-is-physical-against-carcassi-and-heathcote.md","30-notes/claim-unbounded-operator-domain-restriction-traces-to-heathcote-1990-carcassi-2025.md","30-notes/claim-lu-2026-proposes-theta-intermediatism-hierarchical-physicality.md","30-notes/claim-lu-2026-reference-list-cites-wrong-arxiv-id-for-carcassi-paper.md","40-entities/entity-a-heathcote-philosophy-of-science.md","40-entities/entity-zhonghao-lu.md","40-entities/entity-hellinger-toeplitz-theorem.md","40-entities/entity-theta-intermediatism.md"]
not_promoted: ["Gabriele Carcassi, Francisco Calderón, Christine A. Aidala (entity candidates) — co-authors of the paper Lu rebuts, named only as the target of his argument; co-authorship of one paper isn't a hub per the 2026-10-01 ruling, and the vault hasn't independently engaged their own paper's content yet. Left as mentions in the claim-notes.","Nivaldo A. Lemos (entity candidate) — critic in the three-paper debate thread (Carcassi 2023/2025 -> Lemos 2025 -> Carcassi/Rozite/Aidala 2025); not independently fetched this session, only named in Further leads. Unsure -> not promoted.","Eddy Keming Chen (entity candidate) — cited by Lu as the literature's example of 'Minimalism'; not independently fetched this session. Unsure -> not promoted.","Schwartz space (entity candidate) — established concept, but only mentioned in passing as Carcassi et al.'s candidate restriction; not independently treated by this capture. Left as a mention, not a hub or stub, to avoid a dead page.","Lemos (2025)/Carcassi-Rozite-Aidala (2025) three-paper debate thread (Further lead) — real and worth its own capture, but out of scope for this one; not a doubt any kept claim rests on, so not routed to 50-questions/.","Lu Section 8's algebraic-QFT extension of the hierarchical-physicality idea (Further lead) — a distinct, denser technical thread, not explored this session; no kept claim rests on it.","Heathcote (1990) direct read (Further lead) — would let a future note quote him directly rather than via Lu's secondhand account; recorded as a watch_flag on the claim-note it affects rather than a 50-questions/ entry, since the attribution is uncontested and nothing currently casts doubt on it.","Butterfield (2003) 'David Lewis Meets Hamilton and Jacobi' (Further lead) — a possible bridge between this vault's modality-adjacent material and physics formalism; not independently fetched, no kept claim rests on it.","Lu's triple institutional affiliation (Further lead) — a biographical curiosity, not load-bearing for any claim; left as-is in this capture."]
origin: "batch"
writer_model: "claude-sonnet-5"
date_created: "2026-10-05T00:00:00.000Z"
provenance: "2026-10-05 batch run, outside-vault web research topic"
derived_from: []
tags: ["physics","philosophy-of-physics","quantum-mechanics","hilbert-space","modality","quantum-field-theory","hist-ph","arxiv","citation-accuracy"]
source_url: "https://arxiv.org/abs/2609.28136"
source_author: "Zhonghao Lu"
source_date: "2026-09-23T00:00:00.000Z"
source_title: "Unbounded Operators, Physicality, and Modality in Quantum Theories"
source_venue: "arXiv:2609.28136 [quant-ph] (cross-listed physics.hist-ph)"
source_tier: 1
source_sha: "75f9c1b581480a6f2bb0fd5ca4460fc890e23f294bb968b99104b8aa7ac0dac6"
seek_code_commit: "379c8e7"
---


Topic was an "outside" surface — a fresh arXiv preprint in philosophy of physics (quant-ph / physics.hist-ph), not previously touched by this vault. Fetched the abstract page (`archive_page`) and the full PDF (`extract_pdf`, 45 pp., `tls: "verified"`). No safety signals fired — the page and PDF are ordinary third-person academic prose throughout; no addressed-to-AI language, override language, claimed authority, tier self-assignment, file-system instructions, credential requests, or urgency framing was found. No `## Safety flags` entries this capture.

The paper is a single-author response, in the foundations-of-physics literature, to a 2025 paper by Gabriele Carcassi, Francisco Calderón, and Christine A. Aidala, which itself extends a 1990 problem first raised by philosopher of science A. Heathcote. Following the blind-spot instruction to flag the older foundational figure first: **Heathcote (1990)** is the origin point of the debate this paper and its direct target both respond to.

## Claim: Lu's paper argues the full Hilbert space is physical, against Carcassi et al. (2025) and Heathcote (1990)

Zhonghao Lu's paper takes the opposing position to both Heathcote (1990) and Carcassi, Calderón & Aidala (2025): rather than restricting "physical states" to a proper subset of the Hilbert space (Heathcote's call for unspecified restrictions; Carcassi et al.'s proposal of Schwartz spaces), Lu argues the full Hilbert space should be treated as physical, and that any attempt to carve out a proper-subset "physical" region introduces unavoidable arbitrariness.

source_quote: "I argue for the physicality of the Hilbert space in quantum mechanics, and demonstrate that there are inevitable ambiguity and arbitrariness if we want to identify the set of physical states with a proper subset of it."
source_url: https://arxiv.org/pdf/2609.28136
source_title: "Unbounded Operators, Physicality, and Modality in Quantum Theories"
source_author: "Zhonghao Lu"
source_venue: "arXiv:2609.28136 [quant-ph]"
source_date: 2026-09-23
source_tier: 1
source_sha: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659

## Claim: the problem traces to unbounded self-adjoint operators' restricted domains — first raised by Heathcote (1990), restated by Carcassi et al. (2025)

The technical trigger for the whole debate: in the textbook axioms of quantum mechanics, observables (position, momentum, most Hamiltonians) are represented by self-adjoint operators, but per the Hellinger-Toeplitz theorem an *unbounded* self-adjoint operator can only be defined on a dense subset (its "domain"), not the whole Hilbert space — so expectation-value and dynamical-evolution axioms fail to apply to states outside that domain. Lu's paper states this was "first noticed by Heathcote in 1990" and "then by Carcassi, Calderón, and Aidala in 2025," who proposed restricting the physically allowed states to Schwartz spaces as a candidate "proper subset" solution.

source_quote: "According to the Hellinger-Toeplitz theorem, an unbounded self-adjoint operator A can only be defined on a subset of the Hilbert space, namely its domain Dom(A)."
source_quote_2: "In the foundations of physics literature, this problem is first noticed by Heathcote in 1990 [15], and then by Carcassi, Calderón, and Aidala in 2025 [3]."
source_url: https://arxiv.org/pdf/2609.28136
source_url_2: https://arxiv.org/pdf/2609.28136
source_title: "Unbounded Operators, Physicality, and Modality in Quantum Theories"
source_author: "Zhonghao Lu"
source_venue: "arXiv:2609.28136 [quant-ph]"
source_date: 2026-09-23
source_tier: 1
source_sha: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659
source_sha_2: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659

Note: Heathcote's 1990 paper itself ("Unbounded Operators and the Incompleteness of Quantum Mechanics," *Philosophy of Science* 57, 523-534) was not independently fetched this session — it is cited here only as reported inside Lu's primary text, which is adequate for this historical/priority claim per the sourcing floor (uncontested historical attribution, Tier 3-4 acceptable, and here resting on a Tier 1 primary's own account of the literature). A direct read of Heathcote (1990) is flagged below as a further lead.

## Claim: Lu proposes a "hierarchical view of physicality" — physicality as a layered, degreed property rather than a binary one

Rather than picking a single proper subset of the Hilbert space as "the" physical states, Lu's own positive proposal is that physicality admits degrees. He formalizes this with "θ-Intermediatism": for any restriction θ (defined by a set of self-adjoint operators' domains), a state is physically possible-under-θ iff it falls in the intersection of those domains. These θ-indexed sets form a partially-ordered hierarchy — not a single privileged cut — with the actual world's state required to lie inside every one of them.

source_quote: "It follows that physicality is not a univocal concept, but a layered one."
source_quote_2: "(θ-Intermediatism) A state in the mathematical formalism of the physical theory that we use to describe reality is physically possible iff it satisfies the restriction θ."
source_url: https://arxiv.org/pdf/2609.28136
source_url_2: https://arxiv.org/pdf/2609.28136
source_title: "Unbounded Operators, Physicality, and Modality in Quantum Theories"
source_author: "Zhonghao Lu"
source_venue: "arXiv:2609.28136 [quant-ph]"
source_date: 2026-09-23
source_tier: 1
source_sha: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659
source_sha_2: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659

## Claim: Lu's own reference list cites the wrong arXiv identifier for the paper his whole argument responds to

This is the hook that emerged from following the paper's citation apparatus rather than taking it on faith. Lu's reference [3] — the target paper, Carcassi, Calderón & Aidala's "The unphysicality of Hilbert spaces," cited throughout the body as the direct object of rebuttal — gives the journal reference "Quantum Stud.: Math. Found., 12, 13. (2025)" together with the URL `https://arxiv.org/abs/2507.09738`. That URL does not resolve to "The unphysicality of Hilbert spaces." It resolves to a different, later, two-page paper: "Response to 'Are Hilbert Spaces Unphysical? Hardly, My Dear!'" (submitted 13 Jul 2025) by Gabriele Carcassi, **Robert Rozite**, and Christine A. Aidala — a short rebuttal to a separate critic (Nivaldo Lemos), with a different third author (Rozite, not Calderón).

The actual paper matching both the title "The unphysicality of Hilbert spaces" and the journal reference "Quantum Stud.: Math. Found. 12, 13 (2025)" is archived at a different identifier, `arXiv:2308.06669` (submitted 13 Aug 2023, revised to v3 22 Dec 2024), authored by Gabriele Carcassi, Francisco Calderon, and Christine A. Aidala — matching Lu's stated author list and journal reference exactly, but not the arXiv URL he gives.

Both arXiv identifiers were independently resolved and archived this session; the mismatch is a direct, checkable fact about Lu's bibliography, not an inference.

source_quote: "[3] Carcassi, G., Calderón, F., and Aidala, C. A. The unphysicality of Hilbert spaces. Quantum Stud.: Math. Found., 12, 13. (2025) https://arxiv.org/abs/2507.09738"
source_url: https://arxiv.org/pdf/2609.28136
source_title: "Unbounded Operators, Physicality, and Modality in Quantum Theories"
source_author: "Zhonghao Lu"
source_venue: "arXiv:2609.28136 [quant-ph]"
source_date: 2026-09-23
source_tier: 1
source_sha: 113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659

source_quote_2: "Title:Response to \"Are Hilbert Spaces Unphysical? Hardly, My Dear!''\nAuthors:Gabriele Carcassi, Robert Rozite, Christine A. Aidala"
source_url_2: https://arxiv.org/abs/2507.09738
source_title_2: "Response to \"Are Hilbert Spaces Unphysical? Hardly, My Dear!''"
source_author_2: "Gabriele Carcassi, Robert Rozite, Christine A. Aidala"
source_venue_2: "arXiv:2507.09738 [quant-ph]"
source_date_2: 2025-07-13
source_tier_2: 1
source_sha_2: 7225a1bc4557eb7984fac374c4049747943417053811f57e87d584a8bf50f90a

source_quote_3: "Title:The unphysicality of Hilbert spaces\nAuthors:Gabriele Carcassi, Francisco Calderon, Christine A. Aidala\n...\nJournal reference:Quantum Stud.: Math. Found. 12, 13 (2025)"
source_url_3: https://arxiv.org/abs/2308.06669
source_title_3: "The unphysicality of Hilbert spaces"
source_author_3: "Gabriele Carcassi, Francisco Calderon, Christine A. Aidala"
source_venue_3: "arXiv:2308.06669 [quant-ph]; journal version Quantum Stud.: Math. Found. 12, 13 (2025)"
source_date_3: 2023-08-13
source_tier_3: 1
source_sha_3: b7f8fa3acdccd9603fc0261268a2e8567aceb99f38c929c5071494d9d4a45497

> [!note] Seek's commentary:
> Not a retraction-grade problem — the paper's substantive engagement with Carcassi et al.'s actual argument (Schwartz spaces, the coordinate-transformation point, etc.) reads as though Lu read the right paper; this looks like a simple wrong-link slip in the bibliography entry, not evidence the whole paper is unfamiliar with its target. But it's exactly the kind of thing that compounds invisibly: anyone citing Lu's paper's bibliography entry [3] as a pointer would land on a two-page rebuttal-to-a-critic instead of the six-page paper the entire rest of Lu's argument is built against. Worth a line in any later promotion: verified against primary, not against Lu's own citation.

## Further leads

- Nivaldo A. Lemos's 2025 *Foundations of Physics* paper "Are Hilbert Spaces Unphysical? Hardly, My Dear!" (criticizing Carcassi et al.) and Carcassi, Rozite & Aidala's arXiv:2507.09738 response to it form a live, ongoing three-paper debate thread (Carcassi et al. 2023/2025 → Lemos 2025 → Carcassi/Rozite/Aidala 2025) that Lu's paper treats as background — worth its own capture. (Lu's paper, arXiv:2609.28136, §6.)
- Section 8 of Lu's paper extends the "hierarchical view of physicality" to the interpretational problem of inequivalent representations in algebraic quantum field theory (Weyl algebra, GNS representations) — a distinct, denser technical thread not explored in this capture. (arXiv:2609.28136, §8.)
- Eddy Keming Chen's "strong determinism" proposal (cited by Lu as the literature's main real-world example of "Minimalism" — only the actual world's state is physically possible) — *Laws of Physics* (Cambridge UP, 2024) and "Strong Determinism," *Philosophers' Imprint* 24 (2024). Not independently fetched this session.
- A. Heathcote, "Unbounded Operators and the Incompleteness of Quantum Mechanics," *Philosophy of Science* 57, 523-534 (1990) — the founding paper of this entire debate thread, known to this capture only through Lu's secondhand quotation; a direct read would let a future capture quote Heathcote's own words rather than Lu's rendering of them.
- J. Butterfield, "David Lewis Meets Hamilton and Jacobi," *Philos. Sci.* 71, 1095-1106 (2003) — cited by Lu as a parallel case of a physical-theory formalism (classical Hamilton-Jacobi mechanics) naturally accommodating David Lewis's modal metaphysics of counterfactuals; a possible bridge for a future note connecting this vault's modality-adjacent material to physics formalism.
- Zhonghao Lu lists three institutional emails (LMU Munich physics faculty, University of Pittsburgh, Peking University) in the paper's header — worth checking in a later session whether this reflects a joint/visiting appointment or a recent multi-institution transition.

## Entity candidates

- A. Heathcote — person — the foundational, older figure: first identified (1990) the unbounded-operator domain-restriction problem that both Carcassi et al. (2025) and Lu (2026) are responding to; flagged first per the blind-spot rule, not last.
- Gabriele Carcassi, Francisco Calderón, Christine A. Aidala — persons — authors of "The unphysicality of Hilbert spaces" (arXiv:2308.06669, Quantum Stud.: Math. Found. 12, 13, 2025), the paper Lu's entire preprint is written to rebut.
- Zhonghao Lu — person — author of the captured paper (arXiv:2609.28136); triple-affiliated (LMU Munich, University of Pittsburgh, Peking University).
- Nivaldo A. Lemos — person — critic of Carcassi et al.'s original paper ("Are Hilbert Spaces Unphysical? Hardly, My Dear!," *Found. Phys.* 55, 46, 2025); his criticism and Carcassi et al.'s response (arXiv:2507.09738) are the debate thread Lu's own bibliography entry accidentally points to instead of the original paper.
- Hellinger–Toeplitz theorem — concept — the mathematical result (an unbounded self-adjoint/symmetric operator cannot be defined on the whole Hilbert space) underlying the entire debate.
- Schwartz space — concept — the candidate restricted "physical" state space proposed by Carcassi et al. as an alternative to the full Hilbert space.
- hierarchical view of physicality / θ-Intermediatism — concept — Lu's own proposed resolution: physicality as a partially-ordered hierarchy of restriction-indexed state sets rather than a single binary cut.
- Eddy Keming Chen — person — proponent of "strong determinism," cited by Lu as the real literature instance of the "Minimalism" extreme (only the actual world's state is physically possible).