Does the Hamiltonian alone determine which subsystems behave classically?
In plain words
Decoherence arguments start by splitting the world into an object and its environment, but quantum theory does not say where that split lies. The question is whether the energy function of the whole system (its Hamiltonian) by itself picks out one natural split in which classical behavior appears.
Precise statement
Given a Hamiltonian H on a finite-dimensional Hilbert space with no prior tensor-product structure (TPS, a factorization into subsystems), determine whether a criterion such as quasi-classicality of the induced subsystem dynamics, stability of pointer states, or minimal entanglement growth (Carroll and Singh, PRA 2021) selects a TPS unique up to local unitaries, and whether it coincides with the TPS in which H is local. Cotler, Penington and Ranard (CMP 2019) showed that the spectrum alone almost always determines a unique local TPS when one exists; the open part is whether quasi-classicality rather than locality fixes the split, and for which H the two agree.
What would settle it
A theorem that a stated quasi-classicality criterion selects a unique TPS for a defined class of H, or explicit H admitting inequivalent TPSs that are equally quasi-classical.
Status in the literature
Unverified note
Carroll and Singh (2021) proposed a quasi-classicality criterion and Loizeau and Sels (Foundations of Physics 2025, doi:10.1007/s10701-024-00813-2) studied subsystems determined by the spectrum; no criterion is accepted.