Spectral dimension

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

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

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 , giving a spectral dimension of 3.

Ink in a 2D Sierpiński triangle diffuses following a more complicated path and thus more slowly, as , giving a spectral dimension of 1.3652.[8]

See also

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. Elsevier BV. 310 (2): 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
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