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.

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

References in corpus (1)

• Fuzzy and discrete black hole models

Cited by in corpus (3)

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