Lu (2026) proposes θ-Intermediatism — a hierarchical, layered view of physicality rather than a single binary cut

physicsphilosophy-of-physicsquantum-mechanicshilbert-spacemodalitymodal-metaphysicshist-ph

Rather than picking one proper subset of the Hilbert space as "the" physical states — the move Zhonghao Lu's paper argues against (claim-lu-2026-argues-full-hilbert-space-is-physical-against-carcassi-and-heathcote) — Lu's own positive proposal treats physicality as a matter of degree. He states the governing intuition plainly: "It follows that physicality is not a univocal concept, but a layered one." He formalizes this as θ-Intermediatism: for any restriction θ, defined by a set of self-adjoint operators' domains (the structure behind the domain-restriction problem), a state is physically possible under θ if and only if it falls inside the intersection of those domains — "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."

Because different choices of θ pick out different restriction sets, each θ generates its own notion of "physically possible," and these θ-indexed sets form a partially ordered hierarchy rather than a single privileged cut. The actual world's state is required to lie inside every one of them — so no single θ is "the" correct restriction; physicality becomes a nested structure of how-restricted-can-you-be questions, each with its own answer, rather than a yes/no property of a state. Lu extends the same hierarchical move to the interpretational problem of inequivalent representations in algebraic quantum field theory (Weyl algebra, GNS representations) — a further technical thread this capture did not explore.