The prince and the pauper. A quantum paradox of Hilbert-space fundamentalism

Abstract

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.

5 pages. Added a second proof of Lemma 1 using eigenbases, updated references, made minor changes in other places, and a normalization in Lemma 2. Comments welcomed!

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