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arXiv:2103.15104

Refutation of Hilbert Space Fundamentalism

Ovidiu Cristinel Stoica

Published Mar 28, 2021• 0 citations
arXivHTMLPDF

Abstract

According to the "Hilbert Space Fundamentalism" Thesis, all features of a physical system, including the 3D-space, a preferred basis, and factorization into subsystems, uniquely emerge from the state vector and the Hamiltonian alone. I give a simplified account of the proof from arXiv:2102.08620 showing that such emerging structures cannot be both unique and physically relevant.

DOI
10.48550/arxiv.2103.15104

Related papers

3D-space and the preferred basis cannot uniquely emerge from the quantum structure

Ovidiu Cristinel Stoica

Hilbert-Space Fundamentalism (HSF) states that the only fundamental structures are the quantum state vector and the Hamiltonian, and from them everything else emerge uniquely, including the 3D-space, a preferred basis, and a preferred factorization of the Hilbert space. In this article it is shown that whenever such a structure emerges from the Hamiltonian and the state vector alone, if it is physically relevant, it is not unique. Moreover, HSF leads to strange effects like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space. The results from this article affect all theories that adhere to HSF, whether they assume branching or state vector reduction (in particular the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism"), various proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity.

arXiv:2102.08620•Advances in Theoretical and Mathematical Physics•Jan 1, 2022•6 citations

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No change in Hilbert space fundamentalism

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Hilbert space fundamentalism (HSF) states that everything about the physical world is encoded in the Hamiltonian operator and the state vector (as a unit vector, not a wavefunction, which requires additional specification of a configuration space, a position basis, or the position observables). That all structures needed to describe reality, including subsystems, space, fields, emerge from these. I show that HSF can't account for our observations that the physical world changes in time.

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Comment on "On the emergence of preferred structures in quantum theory" by Soulas, Franzmann, and Di Biagio

Ovidiu Cristinel Stoica

This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian $\mathsf{H}$ and the unit vector $|ψ\rangle$ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from $\mathsf{H}$ and $|ψ\rangle$ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between $|ψ(t)\rangle$ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.

quant-pharXiv:2603.07674•Mar 10, 2026•0 citations
According to the "Hilbert Space Fundamentalism" Thesis, all features of a physical system, including the 3D-space, a preferred basis, and factorization into subsystems, uniquely emerge from the state vector and the Hamiltonian alone. I give a simplified account of the proof from arXiv:2102.08620 showing that such emerging structures cannot be both unique and physically relevant.
Hilbert-Space Fundamentalism (HSF) states that the only fundamental structures are the quantum state vector and the Hamiltonian, and from them everything else emerge uniquely, including the 3D-space, a preferred basis, and a preferred factorization of the Hilbert space. In this article it is shown that whenever such a structure emerges from the Hamiltonian and the state vector alone, if it is physically relevant, it is not unique. Moreover, HSF leads to strange effects like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space. The results from this article affect all theories that adhere to HSF, whether they assume branching or state vector reduction (in particular the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism"), various proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity.
Abstract It is often claimed that, from the abstract quantum state vector and Hamiltonian, everything emerges uniquely (for example the 3D-space, a preferred factorization of the Hilbert space, and a preferred basis). This thesis, called Hilbert-Space Fundamentalism (HSF), is assumed for example in Carroll and Singh’s “Mad-dog Everettianism”, some proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity. I show that whenever the 3D-space, a preferred factorization, or any other structure emerges like this, there are infinitely many solutions. This refutes Hilbert-Space Fundamentalism. Implications for various research programs aiming to show that certain structures emerge uniquely, and for various interpretations of Quantum Mechanics, are discussed.
We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.
The quantum world is described by a unit vector in the Hilbert space and the Hamiltonian. Do these abstract basis-independent objects give a complete description of the physical world, or should we include observables like positions and momenta and the decomposition into subsystems? According to "Hilbert-space fundamentalism" they give a complete description, and all other features of the physical world emerge from them (Carroll, arXiv:2103.09780). Here I will give a concrete refutation of this thesis based on the symmetries of the theory of quantum measurements. These results show that even if a tensor product structure is assumed along with the unit vector and the Hamiltonian, concrete physically distinct worlds can be described by the same structures.
This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian H\mathsf{H}H and the unit vector ∣ψ⟩|ψ\rangle∣ψ⟩ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from H\mathsf{H}H and ∣ψ⟩|ψ\rangle∣ψ⟩ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.