Spacetime Dimensionality |

**In General** > s.a. dimension of a space.

@ __General references__:
Lämmerzahl & Macías JMP(93);
Tegmark CQG(97)gq [and strings];
Mankoč Borštnik & Nielsen JPA(02) [why 3+1];
Ghaboussi gq/03 [??];
Rama PLB(07) [reason, from strings];
Wesson ch(07)-a0712 [conceptual];
Maziashvili PLB(09)-a0809,
IJMPD(09)-a0905-GRF [running];
Bojowald in(07)-a0807 [and canonical gravity];
Lee NPB(10) [as a complex variable];
Altshuler PRD(12)-a1205 [argument for 4 dimensions from Mach's principle];
Kaviani & Atyabi a1401 [for different fields];
Deser PRS(20)-a1909 [why is *D* = 4?];
> s.a. quantum spacetime.

@ __Anthropic arguments__: Caruso a0806-fs [3+1];
Scargill PRR(20)-a1906 [can life exist in 2+1 dimensions?].

**In Quantum Gravity**
> s.a. 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 PLB(18)-a1805 [notions of dimensionality, Snyder spacetime].

@ __Spectral dimension__:
Hořava PRL(09)-a0902 [at a Lifshitz Point];
Modesto & Nicolini PRD(10)-a0912;
Eckstein & Trześniewski a2005 [and dimension spectra];
> 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**
> s.a. higher-dimensional gravity.

* __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__: Svozil JPA(86) [fractals and dimensional shadowing];
Manogue & Dray MPLA(99)ht/98 [without compactification];
Cognola & Zerbini NPB(01);
Pons JPCS(07)ht/06 [truncation, constraints and consistency];
Maraner & Pachos AP(08)-a0704 [from breaking of general covariance];
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 ["vanishing dimensions" and high energies];
Amelino-Camelia et al PLB(14)-a1311 [running spectral dimension without a preferred frame];
Stojković a1406-MPLA [rev];
Musser Nautilus(15);
Coumbe a1509-MG14 [and variable speed of light];
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];
Carlip Univ-a1904-proc [rev];
> 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 PRD(17)-a1711 [dimensional deception from non-commutative tori];
Steinhaus & Thüringen PRD(18)-a1803 [spectral dimension in a simplified spin foam model];
Cooperman & Dorghabekov PRD(19)-a1812 [causal dynamical triangulations, setting the scale];
Becker et al a1911 [in higher-derivative gravity];
> s.a. spin-foam models.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 3 may 2021