Thermal and spectral dimension of (generalized) Snyder noncommutative spacetimes

Giovanni Amelino-Camelia,Flaminia Giacomini,Giulia Gubitosi

doi:10.1016/j.physletb.2018.07.030

Abstract

We report an investigation of the Snyder noncommutative spacetime and of some of its most natural generalizations, also looking at them as a powerful tool for comparing different notions of dimensionality of a quantum spacetime. It is known that (generalized-)Snyder noncommutativity, while having rich off-shell implications (kinematical Hilbert space), does not affect noninteracting on-shell particles (physical Hilbert space), and we argue that physically meaningful notions of dimensionality should describe such spacetimes as trivially four-dimensional, without any running with scales. By studying the thermodynamics of a gas of massless particles living on these spacetimes, we find that indeed the Snyder model and its generalizations have constant thermal dimension of four. We also compute the spectral dimension of the Snyder model and its generalizations, finding that, as a result of its sensitivity to off-shell properties, it runs from the standard value of four in the infrared towards lower values in the ultraviolet limit.

Highlights

The Snyder noncommutative spacetime [1] is the earliest and one of the most studied proposals for describing quantum properties of spacetime at the Planck scale
It is known thatSnyder noncommutativity, while having rich off-shell implications, does not affect on-shell particles, and we argue that physically meaningful notions of dimensionality should describe such spacetimes as trivially four-dimensional, without any running with scales
By studying the thermodynamics of a gas of massless particles living on these spacetimes, we find that the Snyder model and its generalizations have constant thermal dimension of four

Summary

INTRODUCTION

The Snyder noncommutative spacetime [1] is the earliest and one of the most studied proposals for describing quantum properties of spacetime at the Planck scale It is generally understood as an interesting case of spacetime whose coordinates have discrete spectrum[2, 3], as one can most rigorously establish within a manifestlycovariant analysis [4,5,6,7] on the kinematical Hilbert space. The thermal dimension, first proposed in [13], is computed by exploiting the fact that some thermodynamical properties of a gas of massless particles living on standard Minkowski spacetime scale with temperature in a way that depends on the dimensionality of the spacetime itself. In this case the spectral dimension it not always well defined, because of divergences introduced by the Euclideanized d’Alembertian

THE SNYDER MODEL AND ITS GENERALIZATIONS

CLOSING REMARKS