On the ultraviolet behavior of quantum fields over noncommutative manifolds

By exploiting the relation between Fredholm modules and the Segal-Shale-Stinespring version of canonical quantization, and taking as starting paint the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behavior of second-quantized fields over noncommutative three-tori, and discuss what behavior should be expected on other noncommutative spin manifolds.