VII${}_a^{\hslash }$, III${}_{a=1}^{\hslash }$, VI${}_{a\ne 1}^{\hslash }$

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 $l_{min}=2^{5/6}$.

arXiv commentLaTeX2e, 9pp, 10 Refs. v10: the original version restored and improved, Refs updated. Text proceeds arXiv:0901.4064, arXiv:0807.0428, arXiv:0806.1349

References in corpus (5)

• Note on operadic harmonic oscillator
• 2D binary operadic Lax representation for harmonic oscillator
• Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator
• Quantum counterparts of 3d Lie algebras over harmonic oscillator
• Jacobi operators of quantum counterparts of three-dimensional real Lie algebras over the harmonic oscillator