Minimal Scales from an Extended Hilbert Space

Abstract

We consider an extension of the conventional quantum Heisenberg algebra, assuming that coordinates as well as momenta fulfil nontrivial commutation relations. As a consequence, a minimal length and a minimal mass scale are implemented. Our commutators do not depend on positions and momenta and we provide an extension of the coordinate coherent state approach to Noncommutative Geometry. We explore, as toy model, the corresponding quantum field theory in a (2+1)-dimensional spacetime. Then we investigate the more realistic case of a (3+1)-dimensional spacetime, foliated into noncommutative planes. As a result, we obtain propagators, which are finite in the ultraviolet as well as the infrared regime.

16 pages, version which matches that published on CQG

References in corpus (66)

• String Theory and Noncommutative Geometry
• Noncommutative Field Theory
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Quantum Field Theory on Noncommutative Spaces
• Noncommutative Perturbative Dynamics
• A Generalized Uncertainty Principle in Quantum Gravity
• Accelerated Universe from Gravity Leaking to Extra Dimensions
• Noncommutative geometry inspired Schwarzschild black hole
• Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review
• The algebraic structure of the generalized uncertainty principle
• Quantum Groups, Gravity, and the Generalized Uncertainty Principle
• Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry
• Feynman Path Integral on the Noncommutative Plane
• Noncommutative geometry inspired charged black holes
• The Standard Model on Non-Commutative Space-Time
• Noncommutative Black Hole Thermodynamics
• Degravitation of the Cosmological Constant and Graviton Width
• Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta
• Non-commutative geometry inspired higher-dimensional charged, black holes
• UV divergence-free QFT on noncommutative plane
• Thermodynamics of interacting holographic dark energy with apparent horizon as an IR cutoff
• Noncommutative geometry inspired dirty black holes
• Loop quantum gravity: an outside view
• Signals for Non-Commutative Interactions at Linear Colliders
• On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
• Review of the Phenomenology of Noncommutative Geometry
• Comments on Non-Commutative Phenomenology
• A model of radiating black hole in noncommutative geometry
• Noncommutative Inspired Black Holes in Extra Dimensions
• "Kerrr" black hole: the Lord of the String
• Spectral dimension of a quantum universe
• Quantum Gravity: General Introduction and Recent Developments
• Voros product, Noncommutative Schwarzschild Black Hole and Corrected Area Law
• Observational Limits on Quantum Geometry Effects
• Entropic force, noncommutative gravity and ungravity
• Trace Anomaly in Quantum Spacetime Manifold
• Topics in Noncommutative Geometry Inspired Physics
• Generalized holographic dark energy and the IR cutoff problem
• Vacuum Fluctuations of Energy Density can lead to the observed Cosmological Constant
• Self-sustained traversable wormholes in noncommutative geometry
• Experimental verification of the Heisenberg uncertainty principle for hot fullerene molecules
• The decay-time of non-commutative micro-black holes
• The Minimal Length and Large Extra Dimensions
• String Theory, Supersymmetry, Unification, and All That
• Noncommutative geometry inspired black holes in higher dimensions at the LHC
• On Noncommutative Geometric Regularisation
• Signals for Non-Commutative QED in $e\gamma$ and $\gamma \gamma$ Collisions
• Signals for Non-Commutative QED in $e\gamma$ and $\gamma \gamma$ Collisions
• Vacuum polarization in the Schwarzschild spacetime and dimensional reduction
• The Dimensional-Reduction Anomaly
• Compton scattering in Noncommutative Space-Time at the NLC
• Minimal Uncertainty in Momentum: The Effects of IR Gravity on Quantum Mechanics
• Signatures of Non-commutative QED at Photon Colliders
• Effective Field Theories on Non-Commutative Space-Time
• Fuzziness at the horizon
• Boundaries and the Casimir effect in non-commutative space-time
• Finite Field Theory on Noncommutative Geometries
• Some Phenomenological Consequences of the Time-Ordered Perturbation Theory of QED on Noncommutative Spacetime
• Signals for Noncommutative QED at Future $e^{+}{e}^{-}$ Colliders
• Thermodynamics on Noncommutative Geometry in Coherent State Formalism
• The IR-Completion of Gravity: What happens at Hubble Scales?
• Vacuum polarization in two-dimensional static spacetimes and dimensional reduction
• Quantum Electrodynamics on Noncommutative Spacetime
• Moving mirrors and black hole evaporation in non-commutative space-times
• Particle production and transplanckian problem on the non-commutative plane
• One more step toward the noncommutative brane inflation

Cited by in corpus (19)

• Physics on Smallest Scales - An Introduction to Minimal Length Phenomenology
• Self-Completeness and the Generalized Uncertainty Principle
• Neutrino oscillations as a novel probe for a minimal length
• Super-accelerating bouncing cosmology in asymptotically-free non-local gravity
• A noncommutative approach to the cosmological constant problem
• Hausdorff dimension of a particle path in a quantum manifold
• Spacetime singularity resolution in Snyder noncommutative space
• Natural Cutoffs via Compact Symplectic Manifolds
• Minimum length effects in black hole physics
• The final stage of gravitationally collapsed thick matter layers
• Minimum-length deformed QM/QFT, issues and problems
• Finite electrodynamics from T-duality
• Influence of the dispersion relation on the Unruh effect according to the relativistic Doppler shift method
• Quaternionic Quantization Principle in General Relativity and Supergravity
• Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity
• Regular black holes in isothermal cavity
• Equilibrium properties of blackbody radiation with an ultraviolet energy cut-off
• How strings can explain regular black holes
• Uncertainty relations for a non-canonical phase-space noncommutative algebra