FIAS Scientific Report 2011 - Frankfurt Institute for Advanced Studies ...

Fractal dimensions and micro-structure of a quantum universe
Collaborators: P. Nicolini 1 , B. Niedner 2 , E. Spallucci 3
1 Frankfurt Institute for Advanced Studies, 2 University of Oxford, UK, 3 University of Trieste, Italy.
Spacetime is expected to change its nature when probed at high energies/short scales. Roughly speaking spacetime
is like a surface which from afar (low energies) appears smooth, while it turns out to be rough at close
distance (high energies). As a result, instead of the classical description in terms of a smooth differential manifold,
spacetime in such extreme regime can be accurately modeled by a fractal surface. There exist two major
indicators of the fractal character of a manifold: the Hausdorff dimension and the spectral dimension.
To discolse further features of quantum spacetimes, in 2011 we have introduced another indicator namely the
un-spectral dimension. The latter consist in the spectral dimension due to the propagation of an un-particle
probe. Un-particles are a conjectured new sector of the standard model of particle physics which is supposed to
show up at energies higher than 1 TeV. The main feature of un-particles is the possibility of generalizing to the
massive case the concept of neutrino: un-particles are supposed to be weakly interacting with other particles
and have a scale-invariant mass spectrum controlled by a non-integer scaling dimension.
When considering the simplest realization of the un-particle diffusion equation, we found that the problem is
equivalent to that of a one-dimensional heat conduction along a bar subject to a time dependent heat source.
On the gravity side we considered the diffusion on a black hole event horizon: by taking into account the
curvature effects we found a complete fractalization of the horizon, corresponding to the geometry built up
by un-gravitons (i.e. the un-particle analogue of bosons that mediate the gravitational force) trapped at the
Schwarzschild radius.
We also showed that the spacetime manifold undergoes a new “trans-Planckian” regime characterized by an
un-spectral dimension lower than two. This new phase we can dub “spacetime vapor” to be consistent with the
thermal interpretation of the diffusion process cannot be probed by any sort of ordinary matter.
a)
b)
In (a) we have a Cantor set, i.e. a fractal surface which resembles the quantum spacetime.
In (b), there is an artistic picture of the spacetime, whose holed structure shows a fractal self-similarity.
Related publications in 2011:
1) P. Nicolini and E. Spallucci, Un-spectral dimension and quantum spacetime phases, Physics Letters B 695
(2011) 290.
2) P. Nicolini and B. Niedner, Hausdorff dimension of a particle path in a quantum manifold, Physical Review
D 83 (2011) 024017.
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