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MP: Theoretische und Mathematische Grundlagen der Physik

MP 7: Strukturelle Fragen der Quantenfeldtheorie

MP 7.6: Fachvortrag

Donnerstag, 26. März 1998, 16:00–16:20, H45

A New Approach to Functional Analysis on Graphs, the Connes
Spectral Triple and its Distance Function — •Manfred Requardt — Institut für Theoretische Physik, Universität Göttingen, Bunsenstraße 9, 37073 Göttingen

We develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a’spectral triple’, i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled ’Dirac operator’, encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the ’Connes-distance function’ and show that it is in general bounded from above by the ordinary distance on graphs (being, typically, strictly smaller(!) than the latter). We exhibit, among other things, the underlying reason for this phenomenon.