A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
arXiv:1105.1898 · doi:10.1103/PhysRevD.84.027502
Abstract
Quantum groups and non-commutative spaces have been repeatedly utilized in approaches to quantum gravity. They provide a mathematically elegant cut-off, often interpreted as related to the Planck-scale quantum uncertainty in position. We consider here a different geometrical interpretation of this cut-off, where the relevant non-commutative space is the space of directions around any spacetime point. The limitations in angular resolution expresses the finiteness of the angular size of a Planck-scale minimal surface at a maximum distance 1/\sqrt{Lambda} related the cosmological constant Lambda. This yields a simple geometrical interpretation for the relation between the quantum deformation parameter q=exp[i Lambda l_{Planck}^2 ] and the cosmological constant, and resolves a difficulty of more conventional interpretations of the physical geometry described by quantum groups or fuzzy spaces.
2 pages, 1 figure
References in corpus (28)
- Discreteness of area and volume in quantum gravity
- Quantum Groups, Gravity, and the Generalized Uncertainty Principle
- Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
- The principle of relative locality
- 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory
- Gravity in Non-Commutative Geometry
- Symmetry, Gravity and Noncommutativity
- Noncommutative Quantum Gravity
- Quantum gravity with a positive cosmological constant
- Quantum deformation of quantum gravity
- 4-dimensional Spin-foam Model with Quantum Lorentz Group
- Quantum deformation of two four-dimensional spin foam models
- Cosmological Deformation of Lorentzian Spin Foam Models
- Operators for quantized directions
- Noncommutative deformation of four dimensional Einstein gravity
- Harmonic Analysis on the quantum Lorentz group
- Cosmological Constant in LQG Vertex Amplitude
- Noncommutative Gravity in two Dimensions
- Gauge and Einstein Gravity from Non-Abelian Gauge Models on Noncommutative Spaces
- Gravity and the Structure of Noncommutative Algebras
- Quantum noncommutative gravity in two dimensions
- Noncommutative gravity: fuzzy sphere and others
- The Fuzzy Sphere Star-Product and Spin Networks
- Drinfeld Twist and General Relativity with Fuzzy Spaces
- Gravity on a fuzzy sphere
- Noncommutative spheres and the AdS/CFT correspondence
- Quantizing speeds with the cosmological constant
- Complex Ashtekar variables, the Kodama state and spinfoam gravity
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- Observables in Loop Quantum Gravity with a cosmological constant
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- Quantum hyperbolic geometry in loop quantum gravity with cosmological constant
- Calculating statistical distributions from operator relations: the statistical distributions of various intermediate statistics
- Quantum deformation of quantum cosmology: A framework to discuss the cosmological constant problem
- Phenomenology of curvature-induced quantum-gravity effects
- Typicality in spin network states of quantum geometry
- Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
- Quantum Hubble horizon
- On the Hydrogen Atom in the Holographic Universe
- Relational dynamics and Page-Wootters formalism in group field theory
- Spin foam propagator: A new perspective to include a cosmological constant
- Quantum Groups and Polymer Quantum Mechanics
- Doubly Quantum Mechanics
- Infinities as a measure of our ignorance
- Quantum Euler angles and agency-dependent spacetime
- Magnetic moments of octet baryons with deformed Uq(su(3)) flavor symmetry
- Remarks on SU_q(2) fermions
- Deformations of spacetime and internal symmetries