Quantization of Length in Spaces with Position-Dependent Noncommutativity

TL;DR

This paper introduces a new method for quantizing length in noncommutative spaces with position-dependent noncommutativity, extending length quantization from a plane to three dimensions using ladder operators.

Contribution

It develops a novel operator algebra and ladder operators that enable length quantization in all three spatial directions in noncommutative spaces with positional dependence.

Findings

Length quantization occurs in all three directions, not just a plane.

Operator algebra allows raising and lowering of length eigenvalues.

Eigenvalues of the length operator are explicitly derived.

Abstract

We present a novel approach to quantizing the length in noncommutative spaces with positional-dependent noncommutativity. The method involves constructing ladder operators that change the length not only along a plane but also along the third direction due to a noncommutative parameter that is a combination of canonical/Weyl-Moyal type and Lie algebraic type. The primary quantization of length in canonical-type noncommutative space takes place only on a plane, while in the present case, it happens in all three directions. We establish an operator algebra that allows for the raising or lowering of eigenvalues of the operator corresponding to the square of the length. We also attempt to determine how the obtained ladder operators act on different states and work out the eigenvalues of the square of the length operator in terms of eigenvalues corresponding to the ladder operators. We conclude by discussing the results obtained.