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

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.