Thermal and spectral dimension of (generalized) Snyder noncommutative spacetimes

TL;DR

This paper investigates the dimensionality of Snyder noncommutative spacetimes, showing that their physical dimension remains constant at four, while spectral dimension varies with scale, revealing different aspects of quantum spacetime structure.

Contribution

It provides a detailed comparison of physical and spectral dimensions in Snyder noncommutative spacetimes, highlighting their scale-dependent and scale-independent behaviors.

Findings

Thermal dimension of Snyder spacetimes remains constant at four.

Spectral dimension runs from four in infrared to lower values in ultraviolet.

Physical dimensionality is trivially four, unaffected by scale.

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 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.