Quantum geodesics on $\lambda$-Minkowski spacetime

Abstract

We apply a recent formalism of quantum geodesics to the well-known bicrossproduct model $\lambda$-Minkowski quantum spacetime $[x^i,t]=ı{\lambda }_p{x}^i$ with its flat quantum metric as a model of quantum gravity effects, with ${\lambda }_p$ the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order ${\lambda }_p$ frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity $v$ of a Gaussian with width $\sqrt{2\beta }$ initially centred at the origin changes its shape but its centre of mass moves with $\frac{<x>}{<t>}=v(1+\frac{{\lambda }_p^2}{2\beta }+O({\lambda }_p^3))$, an order ${\lambda }_p^2$ correction. This implies, at least within perturbation theory, that a `point particle' cannot be modelled as an infinitely sharp Gaussian due to quantum gravity corrections. For contrast, we also look at quantum geodesics on the noncommutative torus with a 2D curved weak quantum Levi-Civita connection.

arXiv comment28 pages amslatex, 3 pdf figures; some small corrections and clarified the notation in a couple of places

References in corpus (12)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness
• Connections on central bimodules
• Linear Connections in Non-Commutative Geometry
• Waves on Noncommutative Spacetime and Gamma-Ray Bursts
• Noncommutative Differential Calculus on the Kappa-Minkowski Space
• Gravity induced from quantum spacetime
• Quantum gravity on a square graph
• Quantum Riemannian geometry and particle creation on the integer line
• Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model
• Fuzzy and discrete black hole models

Cited by in corpus (3)

• Quantum geodesic flows and curvature
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model