Fractional and noncommutative spacetimes

Abstract

We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of κ-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncommutative spacetime defined by a certain nonlinear algebra of the coordinates, which interpolates between κ-Minkowski and canonical spacetime. These results are based upon a braiding formula valid for any nonlinear algebra which can be mapped onto the Heisenberg algebra.

15 pages. v2: typos corrected

References in corpus (18)

• The principle of relative locality
• Horava-Lifshitz Cosmology: A Review
• Horava-Lifshitz gravity: a status report
• Fractal properties of quantum spacetime
• Diffeomorphisms in group field theories
• Statistics and UV-IR Mixing with Twisted Poincare Invariance
• Quantum Gravity on the Lattice
• Fock space, quantum fields and kappa-Poincaré symmetries
• Gravity on a multifractal
• 3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
• The Emergence of Spacetime, or, Quantum Gravity on Your Desktop
• Anatomy of a deformed symmetry: field quantization on curved momentum space
• Twisted Poincaré Invariant Quantum Field Theories
• Discrete to continuum transition in multifractal spacetimes
• Rainbow statistics
• Braided quantum field theories and their symmetries
• Quantum fields, non-locality and quantum group symmetries
• Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime

Cited by in corpus (36)

• Quantum Einstein Gravity
• Geometry and field theory in multi-fractional spacetime
• Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
• The R^2 phase-diagram of QEG and its spectral dimension
• Multifractional theories: an unconventional review
• Multi-scale gravity and cosmology
• Diffusion on $\kappa$-Minkowski space
• Diffusion in multiscale spacetimes
• Asymptotic Safety, Fractals, and Cosmology
• Quantum gravity and gravitational-wave astronomy
• Quantum mechanics in fractional and other anomalous spacetimes
• Diffusion in quantum geometry
• Multifractional theories: an updated review
• Cosmic microwave background and inflation in multi-fractional spacetimes
• Primordial Black Hole Evaporation and Spontaneous Dimensional Reduction
• Lorentz violations in multifractal spacetimes
• ABC of multi-fractal spacetimes and fractional sea turtles
• Spectral dimension and diffusion in multi-scale spacetimes
• Multifractional spacetimes, asymptotic safety and Hořava-Lifshitz gravity
• Metric dimensional reduction at singularities with implications to Quantum Gravity
• Varying electric charge in multiscale spacetimes
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Symmetries and propagator in multifractional scalar field theory
• Quantum field theory with varying couplings
• Deformed symmetries in noncommutative and multifractional spacetimes
• Standard Model in multiscale theories and observational constraints
• Introduction to multifractional spacetimes
• Particle-physics constraints on multifractal spacetimes
• Unitary Standard Model from Spontaneous Dimensional Reduction and Weak Boson Scattering at the LHC
• Singular General Relativity
• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Relativistic particle in multiscale spacetimes
• Geometry of fractional spaces
• Entangling Fractals
• Gravitational Foundations and Exact Solutions in $n$-Dimensional Fractional Cosmology
• Three-dimensional gravity and deformations of relativistic symmetries