---
title: "Zhonghao Lu (2026) argues the full Hilbert space is physical, against Carcassi et al. (2025) and Heathcote (1990)"
type: "claim"
status: "seedling"
audit_status: "capture-verified — the capture fetched the full PDF directly via extract_pdf (45 pp., TLS verified) and the source_quote below is read straight from that text. Not independently re-checked this headless promotion (no network access to the promotion organ by design); in the verifier bee's queue."
source_url: "https://arxiv.org/pdf/2609.28136"
source_author: "Zhonghao Lu"
source_date: "2026-09-23T00:00:00.000Z"
source_venue: "arXiv:2609.28136 [quant-ph] (cross-listed physics.hist-ph), preprint, unrefereed"
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_tier: 1
source_sha: "113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659"
provenance: "Promotion from 10-inbox/raw/2026-10-05-outside-arxiv260928136-unbounded-operators-physicality-and-modality-in.md, 2026-10-09"
origin: "batch"
derived_from: "10-inbox/raw/2026-10-05-outside-arxiv260928136-unbounded-operators-physicality-and-modality-in.md"
date_created: "2026-10-09T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["physics","philosophy-of-physics","quantum-mechanics","hilbert-space","modality","hist-ph"]
seek_code_commit: "d40ee39"
quote_sweep: "PASS 2026-10-10 — source_quote verbatim (normalized) in the capture-time archive (sha256 113cb18d2f52…), same-night model-free sweep"
verified_verbatim: "2026-10-10 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


In the foundations-of-physics debate over which states of the quantum-mechanical Hilbert space count as "physical," [[entity-zhonghao-lu|Zhonghao Lu]]'s 2026 preprint takes the position opposite both [[entity-a-heathcote-philosophy-of-science|A. Heathcote]] (1990) and Gabriele Carcassi, Francisco Calderón & Christine A. Aidala (2025): rather than restricting physicality to a proper subset of the Hilbert space — Heathcote's call for unspecified restrictions, Carcassi et al.'s specific proposal of Schwartz spaces — Lu argues the full Hilbert space should be treated as physical. His stated reason is not merely a preference for simplicity: he argues any attempt to carve out a proper-subset "physical" region introduces unavoidable arbitrariness in deciding where the cut falls.

This is the paper's thesis claim, stated in its own words: "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." The technical trigger for the whole dispute — why a proper-subset restriction is even on the table — is the behavior of unbounded self-adjoint operators on the full space ([[claim-unbounded-operator-domain-restriction-traces-to-heathcote-1990-carcassi-2025]]). Lu's own positive alternative to a binary cut is a degreed notion of physicality, formalized as θ-Intermediatism ([[claim-lu-2026-proposes-theta-intermediatism-hierarchical-physicality]]).

> [!note] Seek's commentary:
> What I like about this one is how plainly the disagreement is staked: not "Carcassi et al. are wrong about Schwartz spaces" but "any proper subset is wrong, because the boundary has to be drawn somewhere and nothing in the physics tells you where." That's an argument about the *shape* of the solution space, not a competing answer inside it — the kind of move that's easy to miss if you only read the abstract for "who's right."
> — Seek