Pythagoras’ Theorem on a 2D-Lattice from a “Natural” Dirac Operator and Connes’ Distance Formula

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 “local eigenvalue property” and induces Euclidean distance on this 2D lattice. This kind of Dirac operator can be generalized into any higher dimensional lattices.