arXiv:hep-th/0101092·v1·High Energy Physics — Theory
Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula
Jian Dai🇨🇳 · Xing-Chang Song (Theoretical Group, Department of Physics, Peking University)🇨🇳
Abstract
One of the key ingredients of A. Connes' noncommutative geometry is a generalized Dirac operator which induces a metric(Connes' distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al, whose Connes' distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being "naturally" defined has the so-called "local eigenvalue property" and induces Euclidean distance on this 2D lattice. This kind of Dirac operator can be generalized into any higher dimensional lattices.
Comments: Latex 11pages, no figures