Spectral dimension

This article is about the spacetime concept. Not to be confused with the independent variable in spectral analysis.

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 ${\displaystyle t^{n}}$, with ${\displaystyle t}$ the time, then the spectral dimension is ${\displaystyle 2n}$. 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 ${\displaystyle t^{3/2}}$, giving a spectral dimension of 3.

Ink in a 2D Sierpiński triangle diffuses following a more complicated path and thus more slowly, as ${\displaystyle t^{0.6826}}$, giving a spectral dimension of 1.3652.[8]

See also

• Dimension
• Fractal dimension
• Hausdorff dimension

References

1. Ambjørn, J.; Jurkiewicz, J.; Loll, R. (2005-10-20). "The Spectral Dimension of the Universe is Scale Dependent". Physical Review Letters. 95 (17): 171301. arXiv:hep-th/0505113. Bibcode:2005PhRvL..95q1301A. doi:10.1103/physrevlett.95.171301. ISSN 0031-9007. PMID 16383815. S2CID 15496735.
2. Modesto, Leonardo (2009-11-24). "Fractal spacetime from the area spectrum". Classical and Quantum Gravity. 26 (24): 242002. arXiv:0812.2214. doi:10.1088/0264-9381/26/24/242002. ISSN 0264-9381. S2CID 118826379.
3. Hořava, Petr (2009-04-20). "Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point". Physical Review Letters. 102 (16): 161301. arXiv:0902.3657. Bibcode:2009PhRvL.102p1301H. doi:10.1103/physrevlett.102.161301. ISSN 0031-9007. PMID 19518693. S2CID 8799552.
4. Lauscher, Oliver; Reuter, Martin (2001). "Ultraviolet fixed point and generalized flow equation of quantum gravity". Physical Review D. 65 (2): 025013. arXiv:hep-th/0108040. Bibcode:2001PhRvD..65b5013L. doi:10.1103/PhysRevD.65.025013. S2CID 1926982.
5. Lauscher, Oliver; Reuter, Martin (2005). "Fractal spacetime structure in asymptotically safe gravity". Journal of High Energy Physics. 2005 (10): 050. arXiv:hep-th/0508202. Bibcode:2005JHEP...10..050L. doi:10.1088/1126-6708/2005/10/050. S2CID 14396108.
6. Atick, Joseph J.; Witten, Edward (1988). "The Hagedorn transition and the number of degrees of freedom of string theory". Nuclear Physics B. 310 (2). Elsevier BV: 291–334. Bibcode:1988NuPhB.310..291A. doi:10.1016/0550-3213(88)90151-4. ISSN 0550-3213.
7. Lauscher, Oliver; Reuter, Martin (2005-10-18). "Fractal spacetime structure in asymptotically safe gravity". Journal of High Energy Physics. 2005 (10): 050. arXiv:hep-th/0508202. Bibcode:2005JHEP...10..050L. doi:10.1088/1126-6708/2005/10/050. ISSN 1029-8479. S2CID 14396108.
8. R. Hilfer and A. Blumen (1984) “Renormalisation on Sierpinski-type fractals” J. Phys. A: Math. Gen. 17