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].
@ 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 a1705 [complex dimensionality with discrete scaling symmetry].
> In other approaches: see brane world; causal sets; discrete geometries; 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, a1705; Ronco AHEP(16)-a1605 [in lqg]; Hossenfelder Forbes(16)jul [I]; > 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 a1705 [multifractional theories]; > s.a. spin-foam models.

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