Dirac operators and spectral triples for some fractal sets built on curves

By Erik Christensen, Cristina Ivan, Michel L. Lapidus. 2006

79 citations recorded by the source corpus.

Abstract We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log 3 / log 2 .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba9badee65c8649253b3964455050abcffc20a68

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In Applied Mathematics

1. Matilde Marcolli
2. J. Bellissard
3. Joseph Keir
4. Kamran Reihani

In Mathematical Physics

1. Matilde Marcolli
2. J. Bellissard
3. Joseph Keir
4. Kamran Reihani
5. Yuri I. Manin

Citations in the corpus

• Dynamical Systems on Spectral Metric Spaces (2010, background): “Several important classes of noncommutative spectral triples have been constructed by Rieffel [63] and by Christensen, Antonescu-Ivan and various collaborators [13, 14, 15, 16].”
• Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, methodology): “The shift π/2l(e)I to the Dirac operator Dl e ( ) is introduced in [12] to avoid a kernel, so that the zeta function   z = s Tr D s D D ( ) (∣ ∣ ) is well defined.”