I was just watching the public talk: Sean Caroll (2020). A Brief History of Quantum Mechanics. The Royal Institution.
Around the end of the talk there was a brief description of the idea of how one would go about understanding the emergence of spacetime from entanglement. I am interested to know if any implementations of this program go along the lines of the following description.
Consider a Hilbert Space $(H, < >)$, where a "state" $h \in H$ is labeled as $|n_1, n_2, \dots >$, ${n_i} \in \mathbb{Z}^+ $. The labels $\mathbb{Z}^+$ could be generalized later, for example, being replaced by a group $G$ such that the entries are related to each other in some manner. Then by putting in some more structure on $H$ or on { $n_{ij}$ } (or $G$) one could define certain equivalences $\sim$, for example; $|n_1, n_2, \dots > \sim |n_{11}, n_{12}, \dots > + |n_{21}, n_{22}, \dots > + \dots$.
Based on these equivalences, one may try to define a "geometric" structure $(H, < >, \sim)$.
I was curious as to what projects have tried to define a geometric (space-like) structure on a Hilbert Space using equivalences of certain "states" without previously putting in a structure of vector bundle of states over a space to begin with. Or does it turn out that the some more structure required has to be necessarily space-like (like in the usual case one defines the vector bundle over a space).
My motivation comes from a certain affinity towards the belief that geometry is a construct that comes about when one chooses to make identifications of an equivalence class; like the idea that the neighborhood of a point is isotropic/homogeneous in some sense, or likewise when one chooses to/has to ignore certain degrees of freedom and treat some of them as the same.
