SpaceTime from Hilbert Space: Decompositions of Hilbert Space as Instances of Time

TL;DR

This paper explores how different decompositions of Hilbert space into tensor factors can give rise to various spatial geometries, proposing that time acts as a parameter for these decompositions, demonstrated through toy models based on Kitaev's toric code.

Contribution

It introduces a novel perspective that time can parameterize Hilbert space decompositions, leading to emergent spatial geometries, supported by toy model demonstrations.

Findings

Hilbert space decompositions influence emergent geometry

Time can parameterize changes in Hilbert space structure

Toy models show dynamical topology and dimension changes

Abstract

There has been recent interest in identifying entanglement as the fundamental concept from which space may emerge. We note that the particular way that a Hilbert space is decomposed into tensor factors is important in what the resulting geometry looks like. We then propose that time may be regarded as a variable that parameterizes a family of such decompositions, thus giving rise to a family of spatial geometries. As a proof of concept, this idea is demonstrated in two toy models based on Kitaev's toric code, which feature a dynamical change of dimension and topology.