On the Quantization of Length in Noncommutative Spaces

TL;DR

This paper demonstrates that in certain noncommutative spaces, lengths and areas are quantized, extending the analysis to three dimensions and exploring the limitations when including time noncommutativity.

Contribution

It introduces a ladder operator approach to quantize length in 3-D noncommutative spaces and clarifies conditions under which length quantization is feasible.

Findings

Length and area are quantized in 2-D noncommutative spaces.

The ladder operator method extends to 3-D length quantization.

Quantization of spacetime length is not possible with time-space noncommutativity.

Abstract

We consider canonical/Weyl-Moyal type noncommutative (NC) spaces with rectilinear coordinates. Motivated by the analogy of the formalism of the quantum mechanical harmonic oscillator problem in quantum phase-space with that of the canonical-type NC 2-D space, and noting that the square of length in the latter case is analogous to the Hamiltonian in the former case, we arrive at the conclusion that the length and area are quantized in such an NC space, if the area is expressed entirely in terms of length. We extend our analysis to 3-D case and formulate a ladder operator approach to the quantization of length in 3-D space. However, our method does not lend itself to the quantization of spacetime length in 1+1 and 2+1 Minkowski spacetimes if the noncommutativity between time and space is considered. If time is taken to commute with spatial coordinates and the noncommutativity is maintained only among the spatial coordinates in 2+1 and 3+1 dimensional spacetime, then the quantization of spatial length is possible in our approach.