Noncommutative Geometry of Lattices and Staggered Fermions

Differential structure of a d-dimensional lattice, which essentially is a noncommutative exterior algebra, is defined using both reductions in 1-order plus 2-order of universal differential calculus in the context of NCG developed by Dimakis et al , and the formalism adopting a Dirac-Connes operator proposed by us recently within Connes’ NCG. Metric structure on this lattice and on differential forms follows in the conventional literature of NCG. Within specific matrix representations, our Dirac-Connes operator coincides with staggered Dirac operator , in the case that dimension of the lattice equals to 1, 2 and 4.