What determines the subsystem decomposition of a general covariant quantum system: canonical tensor product structure, selection criteria for observable subalgebras, and their rigorous extension beyond finite dimensions

Carral, Garay, and Vidotto (Found. Phys. 56, 33 (2026)) argued that in general covariant quantum systems, whose dynamics is a Hamiltonian constraint $C$ on an extended Hilbert space $\mathcal{K}$, the dynamical operator carries no information about the decomposition into subsystems: the constraint acts trivially on the physical state space, and even in ordinary quantum mechanics a Hamiltonian can be k-local with respect to inequivalent tensor product structures. They concluded that a quantum system is minimally specified by the triple $(\{\mathcal{A}_i\}, C, \mathcal{K})$ of partial-observable subalgebras, constraint, and extended space — but they explicitly declined to answer what the tensor product structure (TPS) means physically, whether a canonical factorization of the world into subsystems exists, and whether that decomposition is primary, perspectival, or dynamics-determined. This problem asks: does any dynamical or structural data canonically determine the subsystem decomposition (in particular in infinite-dimensional and background-independent theories), and if not, what physical criteria select the subalgebras $\{\mathcal{A}_i\}$? The core remains open: the finite-dimensional question of whether a Hamiltonian (plus state) selects a unique TPS is under active, unresolved dispute (Cotler-Penington-Ranard vs. Stoica vs. Soulas-Franzmann-Di Biagio, 2019-2026), the rigorous extension to infinite-dimensional Hilbert spaces and quantum field theory is explicitly missing, and no canonical selection principle for observable subalgebras exists in background-independent quantum gravity, where the quantum-reference-frame program corroborates a frame-relative (non-canonical) reading without settling it.

Background

In quantum theory, a composite system is described by a Hilbert space that factorizes as a tensor product, $\mathcal{H}=\mathcal{H}_1\otimes\cdots\otimes\mathcal{H}_m$; the choice of this tensor product structure (TPS) is what identifies subsystems, and with it locality and entanglement. The factorization is not intrinsic to the Hilbert space itself, and two lines of thought address where it comes from. On one view, associated with Zanardi, Lidar, and Lloyd, the TPS is observable-induced and operational: the partition into subsystems is dictated by the set of interactions and measurements that are accessible, so entanglement and subsystem identity are relative to those capabilities. On another view, the dynamics could determine the decomposition: a Hamiltonian is called k-local with respect to a TPS if it is a sum of terms each acting nontrivially on at most $k$ factors, and Cotler, Penington, and Ranard proved for finite-dimensional Hilbert spaces that a generic Hamiltonian admitting a k-local presentation admits essentially only one such TPS, suggesting that the pair (Hilbert space, Hamiltonian) might single out a preferred factorization — e.g., the one minimizing $k$. The obverse of this idea is the clock ambiguity of Albrecht and Iglesias: in a closed universe any variable can serve as the internal clock, and different choices lead to inequivalent descriptions.

Carral, Garay, and Vidotto challenged the dynamics-determination idea in two steps. First, for two coupled harmonic oscillators, the same Hamiltonian is 2-local in the natural TPS of the two oscillators and 1-local after the canonical transformation to uncoupled normal modes; choosing the TPS by minimizing $k$ discards the phenomenology (beats, slow energy exchange) tied to the original partition. Second, in general covariant (background-independent) systems — the setting of canonical quantum gravity — dynamics is not generated by a Hamiltonian but by a Hamiltonian constraint $C$ acting on an extended Hilbert space $\mathcal{K}$; physical states solve the Wheeler-DeWitt-type equation $C\psi=0$ (a partial observable, in Rovelli's terminology, is a quantity measurable but not necessarily predictable, such as a clock reading; the extended space carries the algebra of partial observables). Since $C$ annihilates all physical states, its restriction to the physical space is the zero operator, so the constraint alone cannot encode any subsystem structure; the physical content sits in the transition amplitudes of the partial observables. Their covariant two-oscillator model, with $C=(N_1+\tfrac{1}{2})\Omega_1+(N_2+\tfrac{1}{2})\Omega_2-E$ (with $N_j$ oscillator number operators), makes the point concrete. They conclude that the minimal description of a general covariant quantum system is the triple $(\{\mathcal{A}_i\},C,\mathcal{K})$: subalgebras of partial observables, the constraint, and the extended space. However, the operational perspective they inherit from Zanardi-Lidar-Lloyd is circular when taken as a selection principle — an 'interaction Hamiltonian' presupposes the very system split it is meant to select. They also note that time is a type of TPS (the clock-rest split) and that the observer/observed split underlying most interpretations of quantum mechanics is a TPS as well. Related obstructions are known beyond their setting: in gauge theories and gravity, constraints prevent a spacetime-local tensor product factorization altogether, which blocks standard quantum-information results of the LOCC type (LOCC, 'local operations and classical communication', is the standard restricted-interaction class of entanglement theory).

Problem Statement

For a general covariant quantum system — an extended Hilbert space $\mathcal{K}$ carrying the partial observables, with dynamics given by a Hamiltonian constraint $C$ whose kernel defines the physical states — the triple $(\{\mathcal{A}_i\},C,\mathcal{K})$ has been proposed as the minimal complete description, where $\{\mathcal{A}_i\}$ are subalgebras of partial observables that define the subsystems. The unresolved problem, raised by Carral-Garay-Vidotto and left unanswered by them, is:

Is there a canonical, physically preferred decomposition of a quantum system into subsystems — a distinguished tensor product structure, or equivalently a distinguished family of observable subalgebras — or is the decomposition always relative to a choice that must be supplied independently of the dynamics, or is it partially determined by dynamics together with additional structural inputs? The question comprises three linked components that jointly serve this single objective:

(a) The mathematical determination question. Can dynamical data alone ever single out the subsystem structure? In ordinary quantum mechanics this asks whether the pair (Hilbert space, Hamiltonian) — possibly with a state — determines a preferred TPS, where the finite-dimensional generic-uniqueness theorem of Cotler-Penington-Ranard is explicitly unextended to infinite-dimensional Hilbert spaces and quantum field theory (continuous spectra and breakdown of analyticity being the flagged obstacles). In the general covariant case it asks whether the constraint data $(C,\mathcal{K})$, together with transition amplitudes of partial observables, can distinguish among inequivalent subalgebra choices $\{\mathcal{A}_i\}$, given that $C$ acts as the zero operator on the physical states.

(b) The selection question. If no canonical decomposition exists, what operational, physical criteria select the partial-observable subalgebras $\{\mathcal{A}_i\}$ in a background-independent quantum theory, where no external time, spacetime region, observer, or interaction Hamiltonian is given a priori? (The known operational answer — subsystems are what a given agent or system can interact with — is circular as a selection principle, since an interaction Hamiltonian presupposes the split it should select.)

(c) The ontological question. Is the decomposition of the universe into subsystems ontologically primary, purely perspectival/operational, or partially determined by dynamics together with additional structural inputs?

A resolution takes one of three forms, each settling all three components in a coherent way: either an affirmative determination that dynamical data canonically single out the subsystem decomposition — the Hamiltonian, possibly with a state, or, in the general covariant case, the constraint data $(C,\mathcal{K})$ together with the transition amplitudes of partial observables, fixing a distinguished tensor product structure and hence a distinguished family of subalgebras — demonstrated in at least one nontrivial model, in particular an infinite-dimensional or background-independent one — or a principled selection rule that makes the triple $(\{\mathcal{A}_i\},C,\mathcal{K})$ determinate — demonstrated in nontrivial general covariant models (such as parametrized systems, i.e. ordinary systems reformulated with time as a dynamical variable subject to a constraint, or the constraint two-oscillator model) and reducing to the standard clock-rest or spatial factorization when an external time or background is restored — or a general no-go/underdetermination theorem establishing that no admissible dynamical or structural data canonically determine the subsystem decomposition, together with a characterization of the extra input any complete theory must supply. Here 'admissible data' comprise at least the data canvassed in component (a): in the general covariant case, the constraint data $(C,\mathcal{K})$ together with the transition amplitudes of partial observables; in ordinary quantum mechanics, the Hamiltonian, possibly with a state. Any of these outcomes directly constrains the ontological reading (c), which is settled only to the extent the mathematical results force one of the alternatives.

References

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