Entangled Harmonic Oscillators and Space-time Entanglement

TL;DR

This paper explores the mathematical relationship between Gaussian entanglement and harmonic oscillators within a relativistic framework, revealing implications for space-time structure, entanglement entropy, and symmetries in bound states.

Contribution

It demonstrates the transfer of entanglement concepts to Lorentz-covariant bound states and clarifies the mathematical role of symmetries like Sp(4) and shear transformations.

Findings

Gaussian entanglement shares formulas with harmonic oscillators in relativistic contexts

Entanglement of space and time variables affects entropy and temperature of bound states

Symmetries like Sp(4) and shear transformations are relevant for future entanglement studies

Abstract

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of O(3,2) de Sitter group containing two O(3,1) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the Sp(4) group which serves as the basic language for two-mode squeezed states. Since the Sp(4) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.