Spectral dimension explained

The spectral dimension is a real-valued quantity that characterizes a spacetime geometry and topology. It characterizes a spread into space over time, e.g. an ink drop diffusing in a water glass or the evolution of a pandemic in a population. Its definition is as follow: if a phenomenon spreads as

t ^{n} , with t the time, then the spectral dimension is 2 n . The spectral dimension depends on the topology of the space, e.g., the distribution of neighbors in a population, and the diffusion rate.

In physics, the concept of spectral dimension is used, among other things, in quantum gravity,^{[1] [2] [3] [4] [5]} percolation theory, superstring theory,^[6] or quantum field theory.^[7]

Examples

The diffusion of ink in an isotropic homogeneous medium like still water evolves as

t ^{3/2} , giving a spectral dimension of 3.

Ink in a 2D Sierpiński triangle diffuses following a more complicated path and thus more slowly, as

t ^{0.6826} , giving a spectral dimension of 1.3652.^[8]

See also

• Dimension
• Fractal dimension
• Hausdorff dimension

Notes and References

1. Ambjørn . J. . Jurkiewicz . J. . Loll . R. . The Spectral Dimension of the Universe is Scale Dependent . Physical Review Letters . 95 . 17 . 2005-10-20 . 0031-9007 . 10.1103/physrevlett.95.171301 . 171301. 16383815 . hep-th/0505113 . 2005PhRvL..95q1301A . 15496735 .
2. Modesto . Leonardo . Fractal spacetime from the area spectrum . Classical and Quantum Gravity . 26 . 24 . 2009-11-24 . 0264-9381 . 10.1088/0264-9381/26/24/242002 . 242002. 0812.2214. 118826379 .
3. Hořava . Petr . Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point . Physical Review Letters . 102 . 16 . 2009-04-20 . 0031-9007 . 10.1103/physrevlett.102.161301 . 161301. 19518693 . 0902.3657. 2009PhRvL.102p1301H . 8799552 .
4. Lauscher. Oliver. Reuter, Martin. Ultraviolet fixed point and generalized flow equation of quantum gravity. Physical Review D. 2001. 65. 2. 025013. 10.1103/PhysRevD.65.025013. hep-th/0108040. 2001PhRvD..65b5013L . 1926982.
5. Lauscher. Oliver. Reuter, Martin. Fractal spacetime structure in asymptotically safe gravity. Journal of High Energy Physics. 2005. 2005. 10. 050. 10.1088/1126-6708/2005/10/050. hep-th/0508202. 2005JHEP...10..050L . 14396108.
6. Atick . Joseph J. . Witten . Edward . The Hagedorn transition and the number of degrees of freedom of string theory . Nuclear Physics B . Elsevier BV . 310 . 2 . 1988 . 0550-3213 . 10.1016/0550-3213(88)90151-4 . 291–334. 1988NuPhB.310..291A .
7. Lauscher . Oliver . Reuter . Martin . Fractal spacetime structure in asymptotically safe gravity . Journal of High Energy Physics . 2005 . 10 . 2005-10-18 . 1029-8479 . 10.1088/1126-6708/2005/10/050 . 050. hep-th/0508202 . 2005JHEP...10..050L . 14396108 .
8. https://www2.icp.uni-stuttgart.de/~hilfer/publications/renormalisation-on-sierpinski-type-fractals-Hilfer.pdf R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals”