Deformed symmetries in noncommutative and multifractional spacetimes

Abstract

We clarify the relation between noncommutative spacetimes and multifractional geometries, two quantum-gravity-related approaches where the fundamental description of spacetime is not given by a classical smooth geometry. Despite their different conceptual premises and mathematical formalisms, both research programs allow for the spacetime dimension to vary with the probed scale. This feature and other similarities led to ask whether there is a duality between these two independent proposals. In the absence of curvature and comparing the symmetries of both position and momentum space, we show that $\kappa$-Minkowski spacetime and the commutative multifractional theory with $q$-derivatives are physically inequivalent but they admit several contact points that allow one to describe certain aspects of $\kappa$-Minkowski noncommutative geometry as a multifractional theory and vice versa. Contrary to previous literature, this result holds without assuming any specific measure for $\kappa$-Minkowski. More generally, no well-defined $\star$-product can be constructed from the $q$-theory, although the latter does admit a natural noncommutative extension with a given deformed Poincaré algebra. A similar no-go theorem may be valid for all multiscale theories with factorizable measures. Turning gravity on, we write the algebras of gravitational first-class constraints in the multifractional theories with $q$- and weighted derivatives and discuss their differences with respect to the deformed algebras of $\kappa$-Minkowski spacetime and of loop quantum gravity.

12 pages. v2: physical motivations and consequences clarified, minor typos corrected

References in corpus (8)

• Group field theory with non-commutative metric variables
• Fractal properties of quantum spacetime
• Anomaly-free cosmological perturbations in effective canonical quantum gravity
• Cosmological perturbations in Hybrid Loop Quantum Cosmology: Mukhanov-Sasaki variables
• Diffusion on $\kappa$-Minkowski space
• Dimensional flow in discrete quantum geometries
• Symmetries and propagator in multifractional scalar field theory
• Spectral Dimension of kappa-deformed space-time

Cited by in corpus (6)

• Multifractional theories: an unconventional review
• Hypersurface-deformation algebroids and effective space-time models
• Measuring the propagation speed of gravitational waves with LISA
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Deformed general relativity
• Black-hole models in loop quantum gravity