On the Quantization of Length in Noncommutative Spaces
arXiv:2206.07972 · doi:10.1155/2022/8009789
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.
arXiv comment9 pages, minor changes, version accepted for publication in Advances in High Energy Physics
References in corpus (21)
- String Theory and Noncommutative Geometry
- Hilbert Space Representation of the Minimal Length Uncertainty Relation
- Quantum Field Theory on Noncommutative Spaces
- The quantum structure of spacetime at the Planck scale and quantum fields
- Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review
- A Gravity Theory on Noncommutative Spaces
- Quantum mechanics on the noncommutative plane and sphere
- A length operator for canonical quantum gravity
- Two phases of the noncommutative quantum mechanics
- Non-commutative Oscillators and the commutative limit
- Noncommutative Gravity
- Quanta of Geometry: Noncommutative Aspects
- Noncommutative Two Dimensional Gravities
- On uncertainty relations in noncommutative quantum mechanics
- Noncommutative Geometry for Pedestrians
- Supersymmetric Quantum Mechanics on Non-Commutative Plane
- Emergence of Geometric phase shift in Planar Non-commutative Quantum Mechanics
- U(1) gauge invariant noncommutative Schrödinger theory and gravity
- Remarks on the Formulation of Quantum Mechanics on Noncommutative Phase Spaces
- On the Quantization of Length in Noncommutative Spaces
- Novel Ambiguities in the Seiberg-Witten Map and the Emergent Gravity