Discreteness of area in noncommutative space

Abstract

We introduce an area operator for the Moyal noncommutative plane. We find that the spectrum is discrete, but, contrary to the expectation formulated by other authors, not characterized by a "minimum-area principle". We show that an intuitive analysis of the uncertainty relations obtained from Moyal-plane noncommutativity is fully consistent with our results for the spectrum, and we argue that our area operator should be generalizable to several other noncommutative spaces. We also observe that the properties of distances and areas in the Moyal plane expose some weaknesses in the line of reasoning adopted in some of the heuristic analyses of the measurability of geometric spacetime observables in the quantum-gravity realm.

References in corpus (11)

• Discreteness of area and volume in quantum gravity
• The quantum structure of spacetime at the Planck scale and quantum fields
• Quantum gravity and minimum length
• Loop Quantum Gravity
• Gauge Theory on Noncommutative Spaces
• Lectures on Loop Quantum Gravity
• Quantum Measurement, Gravitation, and Locality
• Limits on the Measurability of Space-time Distances in (the Semi-classical Approximation of) Quantum Gravity
• Noncommutative Differential Calculus on the Kappa-Minkowski Space
• On Local Observations in Quantum Gravity
• A note about the quantum of area in a non-commutative space

Cited by in corpus (20)

• Entropy/Area spectra of the charged black hole from quasinormal modes
• Fock representation of gravitational boundary modes and the discreteness of the area spectrum
• The spectral distance on the Moyal plane
• Pauli-Jordan Function and Scalar Field Quantization in $\kappa$-Minkowski Noncommutative Spacetime
• The Momentum Spaces of $\kappa$-Minkowski noncommutative spacetime
• Quantum deformation of quantum cosmology: A framework to discuss the cosmological constant problem
• Non-singular Brans-Dicke collapse in deformed phase space
• Minimal length in quantum space and integrations of the line element in Noncommutative Geometry
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Modified Newton's Law of Gravitation Due to Minimal Length in Quantum Gravity
• Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
• Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries
• Length and distance on a quantum space
• Lifshitz field theories, Snyder noncomutative space-time and momentum dependent metric
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality
• The Weyl-Mellin quantization map for $\kappa$-Minkowski Noncommutative Spacetime
• Doubly Quantum Mechanics
• Fuzzy worldlines with $\kappa$-Poincaré symmetries
• A Twisted ${\mathcal{C}}^{\star }$-algebra formulation of Quantum Cosmology with application to the Bianchi I model
• Snyder and their representation with creation and annihilation operators