  We propose a notion of functor of points for noncommutative spaces, valued in
categories of bimodules, and endowed with an action functional determined by a
notion of hermitian structures and height functions, modeled on an
interpretation of the classical functor of points as a physical sigma model. We
discuss different choices of such height functions, based on different notions
of volumes and traces, including one based on the Hattori-Stallings rank. We
show that the height function determines a dynamical time evolution on an
algebra of observables associated to our functor of points. We focus in
particular the case of noncommutative arithmetic curves, where the relevant
algebras are sums of matrix algebras over division algebras over number fields,
and we discuss a more general notion of noncommutative arithmetic spaces in
higher dimensions, where our approach suggests an interpretation of the Jones
index as a height function.
