Abstract
In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length ℓ in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of un-spectral dimensionDU of a d-dimensional, euclidean (quantum) spacetime, as the spectral dimension measured by an “un-particle” probe. We find a general expression for the un-spectral dimension DU labelling different spacetime phases: a semi-classical phase, where ordinary spectral dimension gets contribution from the scaling dimension dU of the un-particle probe; a critical “Planckian phase”, where four-dimensional spacetime can be effectively considered two-dimensional when dU=1; a “Trans-Planckian phase”, which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning.
Highlights
In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length l in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side
We find a general expression for the un-spectral dimension DU labelling different spacetime phases: a semi-classical phase, where ordinary spectral dimension gets contribution from the scaling dimension dU of the un-particle probe ; a critical ”Planckian phase”, where four-dimensional spacetime can be effectively considered two-dimensional when dU = 1; a ”Trans-Planckian phase”, which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning
If we look at a fractal, for instance the Cantor set in Fig. 1a), we can grasp the meaning of what could be the spacetime in the presence of strong quantum gravity fluctuations
Summary
In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length l in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. This special feature is accompanied by the fact that in two dimensions the spacetime is conformally flat and field theories more naturally enjoy properties like conformal invariance. An effective way to measure the actual dimension of a quantum manifold consists in studying the diffusion of a test particle.
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