Unbounded Operators, Physicality, and Modality in Quantum Theories (arXiv:2609.28136) — capture

physicsphilosophy-of-physicsquantum-mechanicshilbert-spacemodalityquantum-field-theoryhist-pharxivcitation-accuracy

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

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).