Connections between Weyl geometry, quantum potential and quantum entanglement

TL;DR

This paper explores the deep connections between Weyl geometry, quantum potential, and quantum entanglement, revealing a dimension-dependent relationship that offers insights into quantum gravity and a geometric method to detect entanglement.

Contribution

It introduces a formulation linking Weyl scalar curvature, quantum potential, and entanglement, highlighting the special role of 3D space in quantum gravity.

Findings

Weyl scalar curvature correlates with quantum potential only in 3D space.

Weyl scalar curvature exhibits a negative peak for separable states and positive for entangled states.

Numerical analysis of oscillators demonstrates geometric signals of quantum entanglement.

Abstract

The Weyl geometry promises potential applications in gravity and quantum mechanics. We study the relationships between the Weyl geometry, quantum entropy and quantum entanglement based on the Weyl geometry endowing the Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl scalar curvature in the $n$-dimensional system. The Weyl scalar field plays a bridge role to connect the Weyl scalar curvature, quantum potential and quantum entanglement. We also give the Einstein-Weyl tensor and the generalized field equation in 3D vacuum case, which reveals the relationship between Weyl geometry and quantum potential. Particularly, we find that the correspondence between the Weyl scalar curvature and quantum potential is dimension-dependent and works only for the 3D space, which reveals a clue to quantize gravity and a understanding why our space must be 3D if quantum gravity is compatible with quantum mechanics. We analyze numerically a typical example of two orthogonal oscillators to reveal the relationships between the Weyl scalar curvature, quantum potential and quantum entanglement based on this formulation. We find that the Weyl scalar curvature shows a negative dip peak for separate state but becomes a positive peak for the entangled state near original point region, which can be regarded as a geometric signal to detect quantum entanglement.