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

Abstract

Starting with a formalism of operators in extended Hilbert space we reformulate a two dimensional quantum harmonic oscillator in such a way the ground state is explicitly maximally entangled. Being a vector in the Hilbert space, it has also a non-trivial expansion in a bigger extended space, which is not free of negative norm states by itself, but allows 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 worldsheets of interacting closed strings. a

Quantum origin of the Minkowski space

“…The latter is a subspace in the maximally extended space free of the fiducial vector. As discussed in [1], the extended space plays a special role constructing the quantum harmonic oscillator.…”

“…Following motivations presented above and keeping in mind ER=EPR hypothesis, we continue the analysis outlined in [1], examining mathematical formalism based on the idea of operators in the extended Hilbert space. By definition, the extended Hilbert space is a generalization of the standard quantum mechanical concept incorporating negative norm states [18,19].…”

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We show that D = 4 Minkowski space can be found to be associated with a certain type of operators in extended Hilbert space. We start with the concept of external operators introduced in [1] to make the connection between quantum entanglement and geometry predicted by ER=EPR conjecture more transparent. We discuss eigenequations of the simplest operators and identify D = 4 Minkowski space as spanned by normalized eigenvectors corresponding to the zero eigenvalue.

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“…By definition, the extended Hilbert space is a generalization of the standard quantum mechanical concept incorporating negative norm states [18,19]. As it was shown in [1] one can construct a special type of operators in that space, the so-called external operators. They play a role of the standard position and momentum and can be used to reformulate the quantum harmonic oscillator in such a way that there is a link with spacetime.…”

“…This paper is organized as follows. In section II discuss eigenequations of the simplest operators, the external position e and the external momentum, −i∂ [1]. We show that there is an infinite set of finite-dimensional separate spaces spanned by normalized eigenvalues.…”