VII$^{\hbar}_a$, III$_{a=1}^{\hbar}$, VI$_{a\neq1}^{\hbar}$

TL;DR

This paper constructs quantum versions of certain 3D Lie algebras using operadic Lax representations for the harmonic oscillator, studies their Jacobians, and suggests a quantized volume element and minimal length in three-dimensional space.

Contribution

It introduces a method to quantize specific 3D Lie algebras and explores their geometric properties, including volume quantization and minimal length.

Findings

Quantum algebra Jacobians are analyzed.

Volume element in R^3 is quantized as |(x,y,z)|=4√2(2n+1).

Elementary length in the model is approximately 2^{5/6}.

Abstract

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some 3d real Lie algebras in Bianchi classification. The Jacobians of these quantum algebras are studied. It is conjectured that the tangent algebras of these quantum algebras are the Heisenberg algebra. From this it follows that the volume element in $\mathbb{R}^{3}$ is quantized by $|(x,y,z)|=4\sqrt{2}(2n+1)$, ($n=0,1,2,\dots$). Thus, the elementary (minimal) length in this model is $l_{min}=2^{5/6}$.