Quantum Spacetime – Dimensionality |

**In General ** > s.a. dimension
of a space / models of spacetime;
quantum spacetime proposals.

* __Idea__: In many approaches
to quantum gravity (but not in canonical-quantization based approaches such as
Wheeler's geometrodynamics or loop quantum gravity, for example), there is no
spacetime manifold of fixed dimensonality as part of the background structure,
and the definition of spacetime dimensionality must be of a different type than
that of a manifold.

* __Fractal dimension__: From
a quantum field theory point of view, the fractal (Hausdorff) dimension of spacetime
is determined by the exponential falloff of the 2-point function with geodesic distance,
from which one can extract critical behavior; Based on the anomalous dimension
of Newton's constant and the spectral dimension, at sub-Planckian distances
spacetime is a fractal with effective dimension 2 (dimensional reduction).

* __Complex dimension__:
A non-zero imaginary dimension can correspond to a discrete scale invariance
at short distances.

@ __General references__: Horiguchi et al PLB(95)ht/94 [small-scale 3D structure];
Khorrami et al gq/95;
Antoniadis et al PLB(98)ht [Hausdorff, from quantum field theory];
Mansouri & Nasseri PRD(99)gq [variable];
Castro CSF(00)ht [infinite-dimensional];
Sakellariadou ht/07-conf [Kaluza-Klein theory and large extra dimensions];
Maziashvili IJMPA(08)-a0709 [operational definition];
Maziashvili IJMPD(09)-a0905-GRF [running];
Nicolini & Spallucci PLB(11)-a1005 [un-spectral dimension];
Carlip et al PRL(11)-a1103 [vacuum fluctuations and small-scale structure];
Stojković a1406-MPLA [rev];
González-Ayala & Angulo-Brown a1502 [3+1 dimensions and the second law of thermodynamics];
Amelino-Camelia et al a1805 [notions of dimensionality, Snyder spacetime].

@ __Spectral dimension__:
Hořava PRL(09)-a0902 [at a Lifshitz Point];
Modesto & Nicolini PRD(10)-a0912;
> s.a. quantum geometry.

@ __In string theory__: Rama PLB(07)ht/06.

@ __In other approaches__: Alencar et al PLB(15)-a1505 [Hořava-Lifshitz gravity].

@ __Related topics__: Calcagni PRD(17)-a1705 [complex dimensionality with discrete scaling symmetry];
news na(17)oct [early-universe knotted flux tube networks].

> __In other approaches__:
see brane world; causal sets;
discrete spacetime models; fractals
in physics [multiscale spacetimes]; kaluza-klein theory.

> __Related topics__:
see models in canonical quantum gravity [fractional dimensions].

> __And phenomenology__:
see inflationary scenarios; particle
phenomenology in quantum gravity; quantum gravity and geometry.

**Scale-Dependent Dimension / Dimensional Reduction**

* __Idea__: The fact that in several approaches
to quantum gravity the effective dimensionality of spacetime is length-scale dependent, and
in particular decreases to 2 at the lowest length scales.

* __Rem__: There is a concern that the research
work on dimensional reduction has been mostly based on analyses of the spectral dimension, which
involves an unphysical Euclideanization of spacetime and is highly sensitive to the off-shell
properties of a theory; As a consequence, different formulations of the same physical theory
may lead to different spectral dimensions.

@ __General references__: Carlip a0909-proc,
a1009-in;
Reuter & Saueressig JHEP(11)-a1110 [detailed renormalization-group study];
Giasemidis et al PRD(12)-a1202,
JPA(12)-a1202;
Stoica AP(14)-a1205 [and singularities];
Stojković MPLA(13)-a1304;
Amelino-Camelia et al PLB(14)-a1311 [running spectral dimension without a preferred frame];
Stojković a1406-MPLA [rev];
Musser Nautilus(15);
Amelino-Camelia et al PLB(17)-a1602 [in terms of thermal dimension];
Carlip IJMPD(16)-a1605-GRF,
CQG(17)-a1705;
Ronco AHEP(16)-a1605 [in lqg];
Hossenfelder Forbes(16)jul [I]; Arzano & Calcagni IJMPD(17)-a1710 [and entanglement entropy]; > s.a. quantum gravity and geometry [metric fluctuations]; variation of constants [*c*].

@ __In specific approaches__: Afshordi & Stojković PLB(14)-a1405 [string theory and evolving dimensions];
Padmanabhan et al GRG(16)-a1507 [from renormalized metric tensor];
Amelino-Camelia et al PLB(17)-a1705 [multifractional theories];
Lizzi & Pinzul a1711
[dimensional deception from non-commutative tori];
Steinhaus & Thüringen a1803 [spectral dimension in a simplified spin foam model];
> s.a. spin-foam models.

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