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arXiv:2102.08620 · Advances in Theoretical and Mathematical Physics

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

Ovidiu Cristinel Stoica

Published Jan 1, 2022• 6 citations
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Abstract

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.

DOI
10.4310/atmp.2022.v26.n10.a12

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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.
Abstract. In the many worlds community seems to exist a belief that the physics of a quantum theory is completely defined by it’s Hamilton operator given in an abstract Hilbert space, especially that the position basis may be derived from it as preferred using decoherence techniques. We show, by an explicit example of non-uniqueness, taken from the theory of the KdV equation, that the Hamilton operator alone is not sufficient to fix the physics. We need the canonical operators ˆp, ˆq as well. As a consequence, it is not possible to derive a “preferred basis ” from the Hamilton operator alone, without postulating some additional structure like a “decomposition into systems”. We argue that this makes such a derivation useless for fundamental physics. 1.