spectral dimension

{{Short description|Type of geometric quantity}}

{{about-distinguish-text|the spacetime concept|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 $t^n$, with $t$ the time, then the spectral dimension is $2n$. 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,{{cite journal | last1=Ambjørn | first1=J. | last2=Jurkiewicz | first2=J. | last3=Loll | first3=R. | title=The Spectral Dimension of the Universe is Scale Dependent | journal=Physical Review Letters | volume=95 | issue=17 | date=2005-10-20 | issn=0031-9007 | doi=10.1103/physrevlett.95.171301 | page=171301| pmid=16383815 | arxiv=hep-th/0505113 | bibcode=2005PhRvL..95q1301A | s2cid=15496735 }}{{cite journal | last=Modesto | first=Leonardo | title=Fractal spacetime from the area spectrum | journal=Classical and Quantum Gravity | volume=26 | issue=24 | date=2009-11-24 | issn=0264-9381 | doi=10.1088/0264-9381/26/24/242002 | page=242002|arxiv=0812.2214| s2cid=118826379 }}{{cite journal | last=Hořava | first=Petr | title=Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point | journal=Physical Review Letters | volume=102 | issue=16 | date=2009-04-20 | issn=0031-9007 | doi=10.1103/physrevlett.102.161301 | page=161301| pmid=19518693 |arxiv=0902.3657| bibcode=2009PhRvL.102p1301H | s2cid=8799552 }}{{cite journal|last=Lauscher|first=Oliver|author2=Reuter, Martin|title=Ultraviolet fixed point and generalized flow equation of quantum gravity|journal=Physical Review D|year=2001|volume=65|issue=2|pages=025013|doi=10.1103/PhysRevD.65.025013|arxiv=hep-th/0108040|bibcode = 2001PhRvD..65b5013L |s2cid=1926982}}{{cite journal|last=Lauscher|first=Oliver|author2=Reuter, Martin|title=Fractal spacetime structure in asymptotically safe gravity|journal=Journal of High Energy Physics|year=2005|volume=2005|issue=10|pages=050|doi=10.1088/1126-6708/2005/10/050|arxiv=hep-th/0508202|bibcode = 2005JHEP...10..050L |s2cid=14396108}}

percolation theory,

superstring theory,{{cite journal | last1=Atick | first1=Joseph J. | last2=Witten | first2=Edward | title=The Hagedorn transition and the number of degrees of freedom of string theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=310 | issue=2 | year=1988 | issn=0550-3213 | doi=10.1016/0550-3213(88)90151-4 | pages=291–334| bibcode=1988NuPhB.310..291A }} or

quantum field theory.{{cite journal | last1=Lauscher | first1=Oliver | last2=Reuter | first2=Martin | title=Fractal spacetime structure in asymptotically safe gravity | journal=Journal of High Energy Physics | volume=2005 | issue=10 | date=2005-10-18 | issn=1029-8479 | doi=10.1088/1126-6708/2005/10/050 | pages=050| arxiv=hep-th/0508202 | bibcode=2005JHEP...10..050L | s2cid=14396108 }}