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Spectral dimension

From Wikipedia, the free encyclopedia

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 , 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.

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

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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

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References

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  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