Question

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∈H
is labeled as |n1,n2,⋯>, ni∈Z+. The labels 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 { nij } (or G) one could define certain equivalences ∼, for example; |n1,n2,⋯>∼|n11,n12,⋯>+|n21,n22,⋯>+…
Based on these equivalences, one may try to define a "geometric" structure (H,<>,∼)
.

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.

Answer 1

The discussion of spacetime's emergence from quantum entanglement involves theoretical physics and quantum gravity research, wherein geometry may arise from equivalences in a Hilbert space without pre-defined spacetime structure. No definitive model has been accepted, but numerous theories such as string theory and loop quantum gravity explore these concepts.

Step-by-step explanation:

The exploration of spacetime and the geometry of Hilbert Space through equivalences of quantum states is a complex area of modern theoretical physics, likely involving facets of quantum gravity research. The program of understanding emergence of spacetime from entanglement is a sophisticated attempt to reconcile general relativity and quantum mechanics.

This research is often tied to attempts to formulate a theory of quantum gravity, where the fabric of spacetime itself arises from more fundamental quantum mechanical principles.

The idea that geometry might emerge from identifying equivalence classes in a Hilbert space, without pre-defined spacetime structure, is indeed a part of some approaches in theoretical physics. In such models, geometry and notions of space might emerge from more primitive quantum mechanical entities, breaking away from classical assumptions that space and time are fundamental and immutable frameworks.

While several theoretical frameworks have been suggested - including various forms of string theory and loop quantum gravity - the exact implementation of Hilbert space geometries without presupposing spacetime is a subject of ongoing research and discussion.

As of now, no single approach has been universally accepted, but numerous proposals exist that explore the deep relationship between quantum entanglement, Hilbert space structures, and the emergence of classical space-like geometry.