Composite system in deformed space with minimal length

TL;DR

This paper investigates the effects of deformed Heisenberg algebra on composite systems, deriving effective parameters, and setting upper bounds on minimal length scales for electrons and quarks based on experimental data.

Contribution

It introduces a method to define total momentum and center-of-mass position in deformed space for composite systems with different deformation parameters, and applies it to estimate minimal length bounds.

Findings

Upper bound of minimal length for electron: 3.3×10⁻¹⁸ m

Upper bound of minimal length for quarks: 2.4×10⁻¹⁷ m

Effective deformation parameter is reduced for macroscopic bodies

Abstract

For composite systems made of $N$ different particles living in a space characterized by the same deformed Heisenberg algebra, but with different deformation parameters, we define the total momentum and the center-of-mass position to first order in the deformation parameters. Such operators satisfy the deformed algebra with new effective deformation parameters. As a consequence, a two-particle system can be reduced to a one-particle problem for the internal motion. As an example, the correction to the hydrogen atom $n$S energy levels is re-evaluated. Comparison with high-precision experimental data leads to an upper bound of the minimal length for the electron equal to $3.3\times 10^{-18} {\rm m}$. The effective Hamiltonian describing the center-of-mass motion of a macroscopic body in an external potential is also found. For such a motion, the effective deformation parameter is substantially reduced due to a factor $1/N^2$. This explains the strangely small result previously obtained for the minimal length from a comparison with the observed precession of the perihelion of Mercury. From our study, an upper bound of the minimal length for quarks equal to $2.4\times 10^{-17}{\rm m}$ is deduced, which appears close to that obtained for electrons.