Zhonghao Lu (2026) argues the full Hilbert space is physical, against Carcassi et al. (2025) and Heathcote (1990)

physicsphilosophy-of-physicsquantum-mechanicshilbert-spacemodalityhist-ph

In the foundations-of-physics debate over which states of the quantum-mechanical Hilbert space count as "physical," Zhonghao Lu's 2026 preprint takes the position opposite both 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).