Quantum harmonic oscillator, entanglement in the vacuum and its geometric interpretation

TL;DR

This paper explores a mathematical framework linking quantum entanglement in the vacuum state of a harmonic oscillator to the geometry of spacetime, specifically $AdS_3$, inspired by the ER=EPR conjecture.

Contribution

It introduces a novel approach using extended Hilbert spaces with indefinite metrics to interpret quantum ground states geometrically in $AdS_3$.

Findings

Ground state of a two-dimensional harmonic oscillator is maximally entangled.

The geometric interpretation relates boundary world lines to massless particles.

Bulk structures resemble interacting closed strings.

Abstract

Inspired by ER=EPR conjecture we present a mathematical tool providing a link between quantum entanglement and the geometry of spacetime. We start with the idea of operators in extended Hilbert space which, by definition, has no positive definite scalar product. Adopting several simple postulates we show that a quantum harmonic oscillator can be constructed as a positive definite sector in that space. We discuss the two-dimensional oscillator constructed in such a way that the ground state is maximally entangled. Being a vector in the Hilbert space, it has also a non-trivial expansion in a bigger extended space. On one hand, the space is not free of negative norm states. On the other hand, it allows one to interpret the ground state geometrically in terms of $AdS_3$. The interpretation is based solely on the form of the expansion, revealing certain structures at the boundary and in the bulk of $AdS_3$. The former correspond to world lines of massless particles at the boundary. The latter resemble interacting closed strings.