On the tensor product structure of general covariant systems

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Defining a generic quantum system requires, together with a Hilbert space and a Hamiltonian, the introduction of an algebra of observables, or equivalently a tensor product structure. Assuming a background time variable, Cotler, Penington and Ranard showed that the Hamiltonian selects an almost-unique tensor product structure. This result has been advocated by Carrol and collaborators as supporting the Everettian interpretation of quantum mechanics and providing a pivotal tool for quantum gravity. In this talk I argue against this, on the basis of the fact that the Cotler-Penington-Ranard result does not hold in the generic background-independent case where the Hamiltonian is replaced by a Hamiltonian constrain. This reinforces the understanding that entropy and entanglement, that in the quantum theory depend on the tensor product structure, are quantities that are observable dependent. To conclude, I would like to pose the question of whether clocks can be thought as a resource, and how thinking of time in terms of physical clocks can inform our interpretation of quantum mechanics